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eigenmath/integral.cpp

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#include "stdafx.h"
#include "defs.h"
static void __integral(void);
static void integral_of_sum(void);
static void integral_of_product(void);
static void integral_of_form(void);
static int match(U *, U *, U *, int, int);
int flag;
void scan_integrals(int);
void print_integrals(void);
void
integral(void)
{
save();
__integral();
restore();
}
static void
__integral(void)
{
if (flag == 0) {
flag = 1;
scan_integrals(1);
}
p2 = pop();
p1 = pop();
if (isadd(p1)) {
integral_of_sum();
return;
}
if (car(p1) == symbol(MULTIPLY)) {
integral_of_product();
return;
}
integral_of_form();
}
static void
integral_of_form(void)
{
int h;
push(yya->u.sym.binding);
push(yyb->u.sym.binding);
push(yyx->u.sym.binding);
yyx->u.sym.binding = p2;
h = tos;
push(_one);
push(p1);
push(p2);
distill();
p3 = table;
while (iscons(p3)) {
p4 = car(p3);
if (match(p1, car(p4), cddr(p4), h, tos))
break;
p3 = cdr(p3);
}
if (iscons(p3)) {
push(cadr(p4));
eval();
} else {
push_symbol(INTEGRAL);
push(p1);
push(p2);
list(3);
}
p3 = pop();
tos = h;
yyx->u.sym.binding = pop();
yyb->u.sym.binding = pop();
yya->u.sym.binding = pop();
push(p3);
}
static void
integral_of_sum(void)
{
int h;
h = tos;
p1 = cdr(p1);
while (iscons(p1)) {
push(car(p1));
push(p2);
integral();
p1 = cdr(p1);
}
addk(tos - h);
}
static void
integral_of_product(void)
{
int h1, h2;
h1 = tos;
p3 = cdr(p1);
while (iscons(p3)) {
if (!find(car(p3), p2))
push(car(p3));
p3 = cdr(p3);
}
h2 = tos;
p3 = cdr(p1);
while (iscons(p3)) {
if (find(car(p3), p2))
push(car(p3));
p3 = cdr(p3);
}
if (tos - h2 == 0)
push(p2);
else {
multiply_all(tos - h2);
p1 = pop();
integral_of_form();
}
multiply_all(tos - h1);
}
static int
match(U *actual, U *formal, U *caveats, int h1, int h2)
{
int i, j;
save();
for (i = h1; i < h2; i++) {
for (j = h1; j < h2; j++) {
yya->u.sym.binding = stack[i];
yyb->u.sym.binding = stack[j];
// check caveats
p1 = caveats;
while (iscons(p1)) {
push(car(p1));
eval();
p2 = pop();
if (!isnum(p2) || iszero(p2))
break;
p1 = cdr(p1);
}
if (iscons(p1))
continue;
// actual == formal?
push(formal);
eval();
if (equal(actual, pop())) {
restore();
return 1;
}
}
}
restore();
return 0;
}
char *integrals[] = {
// 1
"a",
"a*x",
NULL,
// 9 (need a caveat for 7 so we can put 9 after 7)
"1/x",
"log(x)",
NULL,
// 7
"x^a",
"x^(a+1)/(a+1)",
NULL,
// 12
"exp(a*x)",
"1/a*exp(a*x)",
NULL,
"exp(a*x+b)",
"1/a*exp(a*x+b)",
NULL,
"x*exp(a*x^2)",
"exp(a*x^2)/(2*a)",
NULL,
"x*exp(a*x^2+b)",
"exp(a*x^2+b)/(2*a)",
NULL,
// 14
"log(a*x)",
"x*log(a*x)-x",
NULL,
// 15
"a^x",
"a^x/log(a)",
"a>0",
NULL,
// 16
"1/(a+x^2)",
"1/sqrt(a)*arctan(x/sqrt(a))",
"a>0",
NULL,
// 17
"1/(a-x^2)",
"1/sqrt(a)*arctanh(x/sqrt(a))",
NULL,
// 19
"1/sqrt(a-x^2)",
"arcsin(x/(sqrt(a)))",
NULL,
// 20
"1/sqrt(a+x^2)",
"log(x+sqrt(a+x^2))",
NULL,
// 27
"1/(a+b*x)",
"1/b*log(a+b*x)",
NULL,
// 28
"1/(a+b*x)^2",
"-1/(b*(a+b*x))",
NULL,
// "1/(2*a*b*x+a^2+b^2*x^2)", // can't math expanded poly
// "-1/b*1/(a+b*x)",
// NULL,
// 29
"1/(a+b*x)^3",
"-1/(2*b)*1/(a+b*x)^2",
NULL,
// 30
"x/(a+b*x)",
"x/b-a*log(a+b*x)/b/b",
NULL,
// 31
"x/(a+b*x)^2",
"1/b^2*(log(a+b*x)+a/(a+b*x))",
NULL,
// 33
"x^2/(a+b*x)",
"1/b^2*(1/2*(a+b*x)^2-2*a*(a+b*x)+a^2*log(a+b*x))",
NULL,
// 34
"x^2/(a+b*x)^2",
"1/b^3*(a+b*x-2*a*log(a+b*x)-a^2/(a+b*x))",
NULL,
// 35
"x^2/(a+b*x)^3",
"1/b^3*(log(a+b*x)+2*a/(a+b*x)-1/2*a^2/(a+b*x)^2)",
NULL,
// 37
"1/x*1/(a+b*x)",
"-1/a*log((a+b*x)/x)",
NULL,
// 38
"1/x*1/(a+b*x)^2",
"1/a*1/(a+b*x)-1/a^2*log((a+b*x)/x)",
NULL,
// 39
"1/x*1/(a+b*x)^3",
"1/a^3*(1/2*((2*a+b*x)/(a+b*x))^2+log(x/(a+b*x)))",
NULL,
// 40
"1/x^2*1/(a+b*x)",
"-1/(a*x)+b/a^2*log((a+b*x)/x)",
NULL,
// 41
"1/x^3*1/(a+b*x)",
"(2*b*x-a)/(2*a^2*x^2)+b^2/a^3*log(x/(a+b*x))",
NULL,
// 42
"1/x^2*1/(a+b*x)^2",
"-(a+2*b*x)/(a^2*x*(a+b*x))+2*b/a^3*log((a+b*x)/x)",
NULL,
// 60
"1/(a+b*x^2)",
"1/sqrt(a*b)*arctan(x*sqrt(a*b)/a)",
"a*b>0",
NULL,
// 61
"1/(a+b*x^2)",
"1/(2*sqrt(-a*b))*log((a+x*sqrt(-a*b))/(a-x*sqrt(-a*b)))",
"a*b<0",
NULL,
// 62 is the same as 60
// 63
"x/(a+b*x^2)",
"1/2*1/b*log(a+b*x^2)",
NULL,
//64
"x^2/(a+b*x^2)",
"x/b-a/b*integral(1/(a+b*x^2),x)",
NULL,
//65
"1/(a+b*x^2)^2",
"x/(2*a*(a+b*x^2))+1/2*1/a*integral(1/(a+b*x^2),x)",
NULL,
//66 is covered by 61
//70
"1/x*1/(a+b*x^2)",
"1/2*1/a*log(x^2/(a+b*x^2))",
NULL,
//71
"1/x^2*1/(a+b*x^2)",
"-1/(a*x)-b/a*integral(1/(a+b*x^2),x)",
NULL,
//74
"1/(a+b*x^3)",
"1/3*1/a*(a/b)^(1/3)*(1/2*log(((a/b)^(1/3)+x)^3/(a+b*x^3))+sqrt(3)*arctan((2*x-(a/b)^(1/3))*(a/b)^(-1/3)/sqrt(3)))",
NULL,
//76
"x^2/(a+b*x^3)",
"1/3*1/b*log(a+b*x^3)",
NULL,
//77
"1/(a+b*x^4)",
"1/2*1/a*(a/b/4)^(1/4)*(1/2*log((x^2+2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2))/(x^2-2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2)))+arctan(2*(a/b/4)^(1/4)*x/(2*(a/b/4)^(1/2)-x^2)))",
"a*b>0",
NULL,
//78
"1/(a+b*x^4)",
"1/2*(-a/b)^(1/4)/a*(1/2*log((x+(-a/b)^(1/4))/(x-(-a/b)^(1/4)))+arctan(x*(-a/b)^(-1/4)))",
"a*b<0",
NULL,
//79
"x/(a+b*x^4)",
"1/2*sqrt(b/a)/b*arctan(x^2*sqrt(b/a))",
"a*b>0",
NULL,
//80
"x/(a+b*x^4)",
"1/4*sqrt(-b/a)/b*log((x^2-sqrt(-a/b))/(x^2+sqrt(-a/b)))",
"a*b<0",
NULL,
//81
"x^2/(a+b*x^4)",
"1/4*1/b*(a/b/4)^(-1/4)*(1/2*log((x^2-2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4))/(x^2+2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4)))+arctan(2*(a/b/4)^(1/4)*x/(2*sqrt(a/b/4)-x^2)))",
"a*b>0",
NULL,
//82
"x^2/(a+b*x^4)",
"1/4*1/b*(-a/b)^(-1/4)*(log((x-(-a/b)^(1/4))/(x+(-a/b)^(1/4)))+2*arctan(x*(-a/b)^(-1/4)))",
"a*b<0",
NULL,
//83
"x^3/(a+b*x^4)",
"1/4*1/b*log(a+b*x^4)",
NULL,
//124
"sqrt(a+b*x)",
"2/3*1/b*sqrt((a+b*x)^3)",
NULL,
//125
"x*sqrt(a+b*x)",
"-2*(2*a-3*b*x)*sqrt((a+b*x)^3)/15/b^2",
NULL,
//126
"x^2*sqrt(a+b*x)",
"2*(8*a^2-12*a*b*x+15*b^2*x^2)*sqrt((a+b*x)^3)/105/b^3",
NULL,
//128
"sqrt(a+b*x)/x",
"2*sqrt(a+b*x)+a*integral(1/x*1/sqrt(a+b*x),x)",
NULL,
//129
"sqrt(a+b*x)/x^2",
"-sqrt(a+b*x)/x+b/2*integral(1/x*1/sqrt(a+b*x),x)",
NULL,
//131
"1/sqrt(a+b*x)",
"2*sqrt(a+b*x)/b",
NULL,
//132
"x/sqrt(a+b*x)",
"-2/3*(2*a-b*x)*sqrt(a+b*x)/b^2",
NULL,
//133
"x^2/sqrt(a+b*x)",
"2/15*(8*a^2-4*a*b*x+3*b^2*x^2)*sqrt(a+b*x)/b^3",
NULL,
//135
"1/x*1/sqrt(a+b*x)",
"1/sqrt(a)*log((sqrt(a+b*x)-sqrt(a))/(sqrt(a+b*x)+sqrt(a)))",
"a>0",
NULL,
//136
"1/x*1/sqrt(a+b*x)",
"2/sqrt(-a)*arctan(sqrt(-(a+b*x)/a))",
"a<0",
NULL,
//137
"1/x^2*1/sqrt(a+b*x)",
"-sqrt(a+b*x)/a/x-1/2*b/a*integral(1/x*1/sqrt(a+b*x),x)",
NULL,
//156
"sqrt(x^2+a)",
"1/2*(x*sqrt(x^2+a)+a*log(x+sqrt(x^2+a)))",
NULL,
//157
"1/sqrt(x^2+a)",
"log(x+sqrt(x^2+a))",
NULL,
//158
"1/x*1/sqrt(x^2+a)",
"arcsec(x/sqrt(-a))/sqrt(-a)",
"a<0",
NULL,
//159
"1/x*1/sqrt(x^2+a)",
"-1/sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x)",
"a>0",
NULL,
//160
"sqrt(x^2+a)/x",
"sqrt(x^2+a)-sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x)",
"a>0",
NULL,
//161
"sqrt(x^2+a)/x",
"sqrt(x^2+a)-sqrt(-a)*arcsec(x/sqrt(-a))",
"a<0",
NULL,
//162
"x/sqrt(x^2+a)",
"sqrt(x^2+a)",
NULL,
//163
"x*sqrt(x^2+a)",
"1/3*sqrt((x^2+a)^3)",
NULL,
//164 need an unexpanded version?
"sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2)",
"1/4*(x*sqrt((x^2+a^(1/3))^3)+3/2*a^(1/3)*x*sqrt(x^2+a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2+a^(1/3))))",
NULL,
// match doesn't work for the following
"sqrt(-a+x^6-3*a^(1/3)*x^4+3*a^(2/3)*x^2)",
"1/4*(x*sqrt((x^2-a^(1/3))^3)-3/2*a^(1/3)*x*sqrt(x^2-a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2-a^(1/3))))",
NULL,
//165
"1/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2)",
"x/a^(1/3)/sqrt(x^2+a^(1/3))",
NULL,
//166
"x/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2)",
"-1/sqrt(x^2+a^(1/3))",
NULL,
//167
"x*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2)",
"1/5*sqrt((x^2+a^(1/3))^5)",
NULL,
//168
"x^2*sqrt(x^2+a)",
"1/4*x*sqrt((x^2+a)^3)-1/8*a*x*sqrt(x^2+a)-1/8*a^2*log(x+sqrt(x^2+a))",
NULL,
//169
"x^3*sqrt(x^2+a)",
"(1/5*x^2-2/15*a)*sqrt((x^2+a)^3)",
NULL,
//170 like 169?
//171
"x^2/sqrt(x^2+a)",
"1/2*x*sqrt(x^2+a)-1/2*a*log(x+sqrt(x^2+a))",
NULL,
//172
"x^3/sqrt(x^2+a)",
"1/3*sqrt((x^2+a)^3)-a*sqrt(x^2+a)",
NULL,
//173
"1/x^2*1/sqrt(x^2+a)",
"-sqrt(x^2+a)/a/x",
NULL,
//174
"1/x^3*1/sqrt(x^2+a)",
"-1/2*sqrt(x^2+a)/a/x^2+1/2*log((sqrt(a)+sqrt(x^2+a))/x)/a^(3/2)",
"a>0",
NULL,
//175
"1/x^3*1/sqrt(x^2-a)",
"1/2*sqrt(x^2-a)/a/x^2+1/2*1/(a^(3/2))*arcsec(x/(a^(1/2)))",
"a>0",
NULL,
//176+
"x^2*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2)",
"1/6*x*sqrt((x^2+a^(1/3))^5)"
"-1/24*a^(1/3)*x*sqrt((x^2+a^(1/3))^3)"
"-1/16*a^(2/3)*x*sqrt(x^2+a^(1/3))"
"-1/16*a*log(x+sqrt(x^2+a^(1/3)))",
"a>0",
NULL,
//176-
"x^2*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6)",
"1/6*x*sqrt((x^2-a^(1/3))^5)"
"+1/24*a^(1/3)*x*sqrt((x^2-a^(1/3))^3)"
"-1/16*a^(2/3)*x*sqrt(x^2-a^(1/3))"
"+1/16*a*log(x+sqrt(x^2-a^(1/3)))",
"a>0",
NULL,
//177+
"x^3*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2)",
"1/7*sqrt((x^2+a^(1/3))^7)-1/5*a^(1/3)*sqrt((x^2+a^(1/3))^5)",
"a>0",
NULL,
//177-
"x^3*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6)",
"1/7*sqrt((x^2-a^(1/3))^7)+1/5*a^(1/3)*sqrt((x^2-a^(1/3))^5)",
"a>0",
NULL,
//200+
"sqrt(a-x^2)",
"1/2*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(abs(a))))",
NULL,
//201 (seems to be handled somewhere else)
//202
"1/x*1/sqrt(a-x^2)",
"-1/sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x)",
"a>0",
NULL,
//203
"sqrt(a-x^2)/x",
"sqrt(a-x^2)-sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x)",
"a>0",
NULL,
//204
"x/sqrt(a-x^2)",
"-sqrt(a-x^2)",
NULL,
//205
"x*sqrt(a-x^2)",
"-1/3*sqrt((a-x^2)^3)",
NULL,
//210
"x^2*sqrt(a-x^2)",
"-x/4*sqrt((a-x^2)^3)+1/8*a*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(a)))",
"a>0",
NULL,
//211
"x^3*sqrt(a-x^2)",
"(-1/5*x^2-2/15*a)*sqrt((a-x^2)^3)",
"a>0",
NULL,
//214
"x^2/sqrt(a-x^2)",
"-x/2*sqrt(a-x^2)+a/2*arcsin(x/sqrt(a))",
"a>0",
NULL,
//215
"1/x^2*1/sqrt(a-x^2)",
"-sqrt(a-x^2)/a/x",
"a>0",
NULL,
//216
"sqrt(a-x^2)/x^2",
"-sqrt(a-x^2)/x-arcsin(x/sqrt(a))",
"a>0",
NULL,
//217
"sqrt(a-x^2)/x^3",
"-1/2*sqrt(a-x^2)/x^2+1/2*log((sqrt(a)+sqrt(a-x^2))/x)/sqrt(a)",
"a>0",
NULL,
//218
"sqrt(a-x^2)/x^4",
"-1/3*sqrt((a-x^2)^3)/a/x^3",
"a>0",
NULL,
// 290
"sin(a*x)",
"-cos(a*x)/a",
NULL,
// 291
"cos(a*x)",
"sin(a*x)/a",
NULL,
// 292
"tan(a*x)",
"-log(cos(a*x))/a",
NULL,
// 293
"1/tan(a*x)",
"log(sin(a*x))/a",
NULL,
// 294
"1/cos(a*x)",
"log(tan(pi/4+a*x/2))/a",
NULL,
// 295
"1/sin(a*x)",
"log(tan(a*x/2))/a",
NULL,
// 296
"sin(a*x)^2",
"x/2-sin(2*a*x)/(4*a)",
NULL,
// 297
"sin(a*x)^3",
"-cos(a*x)*(sin(a*x)^2+2)/(3*a)",
NULL,
// 298
"sin(a*x)^4",
"3/8*x-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a)",
NULL,
// 302
"cos(a*x)^2",
"x/2+sin(2*a*x)/(4*a)",
NULL,
// 303
"cos(a*x)^3",
"sin(a*x)*(cos(a*x)^2+2)/(3*a)",
NULL,
// 304
"cos(a*x)^4",
"3/8*x+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a)",
NULL,
// 308
"1/sin(a*x)^2",
"-1/(a*tan(a*x))",
NULL,
// 312
"1/cos(a*x)^2",
"tan(a*x)/a",
NULL,
// 318
"sin(a*x)*cos(a*x)",
"sin(a*x)^2/(2*a)",
NULL,
// 320
"sin(a*x)^2*cos(a*x)^2",
"-sin(4*a*x)/(32*a)+x/8",
NULL,
// 326
"sin(a*x)/cos(a*x)^2",
"1/(a*cos(a*x))",
NULL,
// 327
"sin(a*x)^2/cos(a*x)",
"(log(tan(pi/4+a*x/2))-sin(a*x))/a",
NULL,
// 328
"cos(a*x)/sin(a*x)^2",
"-1/(a*sin(a*x))",
NULL,
// 329
"1/(sin(a*x)*cos(a*x))",
"log(tan(a*x))/a",
NULL,
// 330
"1/(sin(a*x)*cos(a*x)^2)",
"(1/cos(a*x)+log(tan(a*x/2)))/a",
NULL,
// 331
"1/(sin(a*x)^2*cos(a*x))",
"(log(tan(pi/4+a*x/2))-1/sin(a*x))/a",
NULL,
// 333
"1/(sin(a*x)^2*cos(a*x)^2)",
"-2/(a*tan(2*a*x))",
NULL,
// 335
"sin(a+b*x)",
"-cos(a+b*x)/b",
NULL,
// 336
"cos(a+b*x)",
"sin(a+b*x)/b",
NULL,
// 337+ (with the addition of b)
"1/(b+b*sin(a*x))",
"-tan(pi/4-a*x/2)/a/b",
NULL,
// 337- (with the addition of b)
"1/(b-b*sin(a*x))",
"tan(pi/4+a*x/2)/a/b",
NULL,
// 338 (with the addition of b)
"1/(b+b*cos(a*x))",
"tan(a*x/2)/a/b",
NULL,
// 339 (with the addition of b)
"1/(b-b*cos(a*x))",
"-1/tan(a*x/2)/a/b",
NULL,
// 340
"1/(a+b*sin(x))",
"1/sqrt(b^2-a^2)*log((a*tan(x/2)+b-sqrt(b^2-a^2))/(a*tan(x/2)+b+sqrt(b^2-a^2)))",
"test(a==b,0,1,1)",
NULL,
// 341
"1/(a+b*cos(x))",
"1/sqrt(b^2-a^2)*log((sqrt(b^2-a^2)*tan(x/2)+a+b)/(sqrt(b^2-a^2)*tan(x/2)-a-b))",
"test(a==b,0,1,1)",
NULL,
// 441
"arcsin(a*x)",
"x*arcsin(a*x)+sqrt(1-a^2*x^2)/a",
NULL,
// 442
"arccos(a*x)",
"x*arccos(a*x)+sqrt(1-a^2*x^2)/a",
NULL,
// 443
"arctan(a*x)",
"x*arctan(a*x)-1/2*log(1+a^2*x^2)/a",
NULL,
// 485 (with addition of a)
"log(a*x)",
"x*log(a*x)-x",
NULL,
// 486 (with addition of a)
"x*log(a*x)",
"x^2*log(a*x)/2-x^2/4",
NULL,
// 487 (with addition of a)
"x^2*log(a*x)",
"x^3*log(a*x)/3-1/9*x^3",
NULL,
// 489
"log(x)^2",
"x*log(x)^2-2*x*log(x)+2*x",
NULL,
// 493 (with addition of a)
"1/(x*log(a*x))",
"log(log(a*x))",
NULL,
// 499
"log(a*x+b)",
"(a*x+b)*log(a*x+b)/a-x",
NULL,
// 500
"log(a*x+b)/x^2",
"a/b*log(x)-(a*x+b)*log(a*x+b)/b/x",
NULL,
// 554
"sinh(x)",
"cosh(x)",
NULL,
// 555
"cosh(x)",
"sinh(x)",
NULL,
// 556
"tanh(x)",
"log(cosh(x))",
NULL,
// 560
"x*sinh(x)",
"x*cosh(x)-sinh(x)",
NULL,
// 562
"x*cosh(x)",
"x*sinh(x)-cosh(x)",
NULL,
// 566
"sinh(x)^2",
"sinh(2*x)/4-x/2",
NULL,
// 569
"tanh(x)^2",
"x-tanh(x)",
NULL,
// 572
"cosh(x)^2",
"sinh(2*x)/4+x/2",
NULL,
// ?
"x^3*exp(a*x^2)",
"exp(a*x^2)*(x^2/a-1/(a^2))/2",
NULL,
// ?
"x^3*exp(a*x^2+b)",
"exp(a*x^2)*exp(b)*(x^2/a-1/(a^2))/2",
NULL,
NULL,
};
void
scan_integrals(int flag)
{
int h, i, k;
k = tos;
i = 0;
while (integrals[i]) {
h = tos;
while (integrals[i]) {
scan(integrals[i++]);
if (flag) {
push(a);
push(yya);
subst();
push(b);
push(yyb);
subst();
push(x);
push(yyx);
subst();
}
}
list(tos - h);
i++;
}
list(tos - k);
table = pop();
}
void
print_integrals(void)
{
int n = 1;
save();
scan_integrals(0);
p1 = table;
while (iscons(p1)) {
p2 = car(p1);
printf("integral %d:\n", n++);
push(car(p2));
//eval();
p3 = pop();
print(p3);
p2 = cdr(p2);
printf("solution:\n");
push(car(p2));
//eval();
p3 = pop();
print(p3);
p2 = cdr(p2);
if (iscons(p2))
printf("caveats:\n");
while (iscons(p2)) {
print(car(p2));
p2 = cdr(p2);
}
p1 = cdr(p1);
printf("\n");
}
restore();
}
static char *s[] = {
"#1",
"integral(A,X)",
"A*X",
"#4",
"integral(A+B,X)",
"A*X+B*X",
"#9",
"integral(1/X,X)",
"log(X)",
"#11",
"integral(exp(X),X)",
"exp(X)",
"#12",
"integral(exp(A*X),X)-exp(A*X)/A",
"0",
"#14",
"integral(log(X),X)-X*log(X)+X",
"0",
"#15",
"integral(3^X*log(3),X)",
"3^X",
"#16",
"integral(1/(3+x^2),x)-3^(-1/2)*arctan(3^(-1/2)*x)",
"0",
"#17",
"integral(1/(a-x^2),x)-a^(-1/2)*arctanh(a^(-1/2)*x)",
"0",
"#19",
"integral(1/sqrt(a-x^2),x)-arcsin(a^(-1/2)*x)",
"0",
"#20",
"integral(1/sqrt(a+x^2),x)",
"log(x+(a+x^2)^(1/2))",
"#27",
"integral(1/(a+b*x),x)",
"log(a+b*x)/b",
"#28",
"integral(1/(A+B*X)^2,X)+1/B*1/(A+B*X)",
"0",
"#29",
"integral(1/(a+b*x)^3,x)+1/2*1/b*(a+b*x)^(-2)",
"0",
"#30",
"integral(X/(A+B*X),X)+A*B^(-2)*log(A+B*X)-X/B",
"0",
"#31",
"integral(X/(A+B*X)^2,X)-1/B^2*(log(A+B*X)+A/(A+B*X))",
"0",
"#33",
"integral(X^2/(A+B*X),X)-1/B^2*(1/2*(A+B*X)^2-2*A*(A+B*X)+A^2*log(A+B*X))",
"0",
"#34",
"integral(X^2/(A+B*X)^2,X)-1/B^3*(A+B*X-2*A*log(A+B*X)-A^2/(A+B*X))",
"0",
"#35",
"integral(X^2/(A+B*X)^3,X)-1/B^3*(log(A+B*X)+2*A/(A+B*X)-1/2*A^2/(A+B*X)^2)",
"0",
"#37",
"integral(1/X*1/(A+B*X),X)+1/A*log((A+B*X)/X)",
"0",
"#38",
"integral(1/X*1/(A+B*X)^2,X)-1/A*1/(A+B*X)+1/A^2*log((A+B*X)/X)",
"0",
"#39",
"integral(1/X*1/(A+B*X)^3,X)-1/A^3*(1/2*((2*A+B*X)/(A+B*X))^2+log(X/(A+B*X)))",
"0",
"#40",
"integral(1/X^2*1/(A+B*X),X)+1/(A*X)-B/A^2*log((A+B*X)/X)",
"0",
"#41",
"integral(1/X^3*1/(A+B*X),X)-(2*B*X-A)/(2*A^2*X^2)-B^2/A^3*log(X/(A+B*X))",
"0",
"#42",
"integral(1/X^2*1/(A+B*X)^2,X)+(A+2*B*X)/(A^2*X*(A+B*X))-2*B/A^3*log((A+B*X)/X)",
"0",
"#60",
"integral(1/(2+3*X^2),X)-1/sqrt(6)*arctan(1/2*X*sqrt(6))",
"0",
"integral(1/(-2-3*X^2),X)-1/sqrt(6)*arctan(-1/2*X*sqrt(6))",
"0",
"#61",
"integral(1/(2-3*X^2),X)-1/2*1/sqrt(6)*log((2+X*sqrt(6))/(2-X*sqrt(6)))",
"0",
"integral(1/(-2+3*X^2),X)-1/2*1/sqrt(6)*log((-2+X*sqrt(6))/(-2-X*sqrt(6)))",
"0",
"#63",
"integral(X/(A+B*X^2),X)-1/2*1/B*log(A+B*X^2)",
"0",
"#64",
"integral(X^2/(A+B*X^2),X)-X/B+A/B*integral(1/(A+B*X^2),X)",
"0",
"#65",
"integral(1/(A+B*X^2)^2,X)-X/(2*A*(A+B*X^2))-1/2*1/A*integral(1/(A+B*X^2),X)",
"0",
"#70",
"integral(1/X*1/(A+B*X^2),X)-1/2*1/A*log(X^2/(A+B*X^2))",
"0",
"#71",
"integral(1/X^2*1/(A+B*X^2),X)+1/(A*X)+B/A*integral(1/(A+B*X^2),X)",
"0",
"#74",
"integral(1/(A+B*X^3),X)-1/3*1/A*(A/B)^(1/3)*(1/2*log(((A/B)^(1/3)+X)^3/(A+B*X^3))+sqrt(3)*arctan((2*X-(A/B)^(1/3))*(A/B)^(-1/3)/sqrt(3)))",
"0",
"#76",
"integral(X^2/(A+B*X^3),X)-1/3*1/B*log(A+B*X^3)",
"0",
"#77",
"integral(1/(2+3*X^4),X)-1/2*1/2*(2/3/4)^(1/4)*(1/2*log((X^2+2*(2/3/4)^(1/4)*X+2*(2/3/4)^(1/2))/(X^2-2*(2/3/4)^(1/4)*X+2*(2/3/4)^(1/2)))+arctan(2*(2/3/4)^(1/4)*X/(2*(2/3/4)^(1/2)-X^2)))",
"0",
"#78",
"integral(1/(2-3*X^4),X)-1/2*(2/3)^(1/4)/2*(1/2*log((X+(2/3)^(1/4))/(X-(2/3)^(1/4)))+arctan(X*(2/3)^(-1/4)))",
"0",
"#79",
"integral(X/(2+3*X^4),X)-1/2*1/3*1/sqrt(2/3)*arctan(X^2/sqrt(2/3))",
"0",
"#80",
"integral(X/(2-3*X^4),X)+1/4*1/3*sqrt(3/2)*log((X^2-sqrt(2/3))/(X^2+sqrt(2/3)))",
"0",
"#81",
"integral(X^2/(2+3*X^4),X)-1/4*1/3*(2/3/4)^(-1/4)*(1/2*log((X^2-2*(2/3/4)^(1/4)*X+2*sqrt(2/3/4))/(X^2+2*(2/3/4)^(1/4)*X+2*sqrt(2/3/4)))+arctan(2*(2/3/4)^(1/4)*X/(2*sqrt(2/3/4)-X^2)))",
"0",
"#82",
"integral(X^2/(2-3*X^4),X)+1/4*1/3*(2/3)^(-1/4)*(log((X-(2/3)^(1/4))/(X+(2/3)^(1/4)))+2*arctan(X*(2/3)^(-1/4)))",
"0",
"#83",
"integral(X^3/(A+B*X^4),X)-1/4*1/B*log(A+B*X^4)",
"0",
"#124",
"integral(sqrt(A+B*X),X)-2/3/B*sqrt((A+B*X)^3)",
"0",
"#125",
"integral(X*sqrt(A+B*X),X)+2*(2*A-3*B*X)*sqrt((A+B*X)^3)/15*B^(-2)",
"0",
"#126",
"integral(X^2*sqrt(A+B*X),X)-2*(8*A^2-12*A*B*X+15*B^2*X^2)*sqrt((A+B*X)^3)/105*B^(-3)",
"0",
"#128",
"integral(sqrt(A+B*X)/X,X)-2*sqrt(A+B*X)-A*integral(1/X*1/sqrt(A+B*X),X)",
"0",
"#129",
"integral(sqrt(A+B*X)/X^2,X)+sqrt(A+B*X)/X-B/2*integral(1/X*1/sqrt(A+B*X),X)",
"0",
"#131",
"integral(1/sqrt(A+B*X),X)-2*sqrt(A+B*X)/B",
"0",
"#132",
"integral(X/sqrt(A+B*X),X)+2/3*(2*A-B*X)*sqrt(A+B*X)/B^2",
"0",
"#133",
"integral(X^2/sqrt(A+B*X),X)-2/15*(8*A^2-4*A*B*X+3*B^2*X^2)*sqrt(A+B*X)/B^3",
"0",
"#134",
"integral(1/X*1/sqrt(2+B*X),X)-1/sqrt(2)*log((sqrt(2+B*X)-sqrt(2))/(sqrt(2+B*X)+sqrt(2)))",
"0",
"#136",
"integral(1/X*1/sqrt(-2+B*X),X)-2/sqrt(2)*arctan(sqrt((-2+B*X)/2))",
"0",
"#137",
"integral(1/X^2*1/sqrt(A+B*X),X)+sqrt(A+B*X)/A/X+1/2*B/A*integral(1/X*1/sqrt(A+B*X),X)",
"0",
"#156",
"integral(sqrt(X^2+A),X)-1/2*(X*sqrt(X^2+A)+A*log(X+sqrt(X^2+A)))",
"0",
"#157",
"integral(1/sqrt(X^2+A),X)-log(X+sqrt(X^2+A))",
"0",
"#158",
"integral(1/X*1/sqrt(X^2-2),X)-arcsec(X/sqrt(2))/sqrt(2)",
"0",
"#159",
"integral(1/X*1/sqrt(X^2+2),X)+1/sqrt(2)*log((sqrt(2)+sqrt(X^2+2))/X)",
"0",
"#160",
"integral(sqrt(X^2+2)/X,X)-sqrt(X^2+2)+sqrt(2)*log((sqrt(2)+sqrt(X^2+2))/X)",
"0",
"#161",
"integral(sqrt(X^2-2)/X,X)-sqrt(X^2-2)+sqrt(2)*arcsec(X/sqrt(2))",
"0",
"#162",
"integral(X/sqrt(X^2+A),X)-sqrt(X^2+A)",
"0",
"#163",
"integral(X*sqrt(X^2+A),X)-1/3*sqrt((X^2+A)^3)",
"0",
"#164",
"integral(sqrt((X^2+A)^3),X)-1/4*(X*sqrt((X^2+A)^3)+3/2*A*X*sqrt(X^2+A)+3/2*A^2*log(X+sqrt(X^2+A)))",
"0",
// "integral(sqrt((X^2-A)^3),X)-1/4*(X*sqrt((X^2-A)^3)-3/2*A*X*sqrt(X^2-A)+3/2*A^2*log(X+sqrt(X^2-A)))",
// "0",
"#165",
"integral(1/sqrt((X^2+A)^3),X)-X/A/sqrt(X^2+A)",
"0",
"#166",
"integral(X/sqrt((X^2+A)^3),X)+1/sqrt(X^2+A)",
"0",
"#167",
"integral(X*sqrt((X^2+A)^3),X)-1/5*sqrt((X^2+A)^5)",
"0",
"#168",
"integral(X^2*sqrt(X^2+A),X)-1/4*X*sqrt((X^2+A)^3)+1/8*A*X*sqrt(X^2+A)+1/8*A^2*log(X+sqrt(X^2+A))",
"0",
"#169",
"integral(X^3*sqrt(X^2+A),X)-(1/5*X^2-2/15*A)*sqrt((X^2+A)^3)",
"0",
"#171",
"integral(X^2/sqrt(X^2+A),X)-1/2*X*sqrt(X^2+A)+1/2*A*log(X+sqrt(X^2+A))",
"0",
"#172",
"integral(X^3/sqrt(X^2+A),X)-1/3*sqrt((X^2+A)^3)+A*sqrt(X^2+A)",
"0",
"#173",
"integral(1/X^2*1/sqrt(X^2+A),X)+sqrt(X^2+A)/A/X",
"0",
"#174",
"integral(1/X^3*1/sqrt(X^2+2),X)+1/2*sqrt(X^2+2)/2/X^2-1/2*log((sqrt(2)+sqrt(X^2+2))/X)/(sqrt(2)^3)",
"0",
"#175",
"integral(1/X^3*1/sqrt(X^2-2),X)-1/2*sqrt(X^2-2)/2/X^2-1/2*1/(2^(3/2))*arcsec(X/(2^(1/2)))",
"0",
"#176+",
"integral(X^2*sqrt((X^2+2^2)^3),X)"
"-1/6*X*sqrt((X^2+2^2)^5)"
"+1/24*(2^2)*X*sqrt((X^2+2^2)^3)"
"+1/16*(2^4)X*sqrt(X^2+2^2)"
"+1/16*(2^6)*log(X+sqrt(X^2+2^2))",
"0",
"#176-",
"integral(X^2*sqrt((X^2-2^2)^3),X)"
"-1/6*X*sqrt((X^2-2^2)^5)"
"-1/24*(2^2)*X*sqrt((X^2-2^2)^3)"
"+1/16*(2^4)X*sqrt(X^2-2^2)"
"-1/16*(2^6)*log(X+sqrt(X^2-2^2))",
"0",
"#177+",
"integral(X^3*sqrt((X^2+7^2)^3),X)"
"-1/7*sqrt((X^2+7^2)^7)"
"+1/5*(7^2)*sqrt((X^2+7^2)^5)",
"0",
"#177-",
"integral(X^3*sqrt((X^2-7^2)^3),X)"
"-1/7*sqrt((X^2-7^2)^7)"
"-1/5*(7^2)*sqrt((X^2-7^2)^5)",
"0",
"#200",
"integral(sqrt(7-X^2),X)-1/2*(X*sqrt(7-X^2)+7*arcsin(X/sqrt(7)))",
"0",
"#201",
"integral(1/sqrt(7-X^2),X)-arcsin(X/sqrt(7))",
"0",
"#202",
"integral(1/X*1/sqrt(7-X^2),X)+1/sqrt(7)*log((sqrt(7)+sqrt(7-X^2))/X)",
"0",
"#203",
"integral(sqrt(7-X^2)/X,X)"
"-sqrt(7-X^2)+sqrt(7)*log((sqrt(7)+sqrt(7-X^2))/X)",
"0",
"#204",
"integral(X/sqrt(A-X^2),X)"
"+sqrt(A-X^2)",
"0",
"#205",
"integral(X*sqrt(A-X^2),X)"
"+1/3*sqrt((A-X^2)^3)",
"0",
"#210",
"integral(X^2*sqrt(7-X^2),X)"
"+1/4*X*sqrt((7-X^2)^3)"
"-7/8*(X*sqrt(7-X^2)+7*arcsin(X/sqrt(7)))",
"0",
"#211",
"integral(X^3*sqrt(7-X^2),X)"
"-(-1/5*X^2-2/15*7)*sqrt((7-X^2)^3)",
"0",
"#214",
"integral(X^2/sqrt(7-X^2),X)"
"+X/2*sqrt(7-X^2)"
"-7/2*arcsin(X/sqrt(7))",
"0",
"#215",
"integral(1/X^2*1/sqrt(7-X^2),X)"
"+sqrt(7-X^2)/7/X",
"0",
"#216",
"integral(sqrt(7-X^2)/X^2,X)"
"+sqrt(7-X^2)/X"
"+arcsin(X/sqrt(7))",
"0",
"#217",
"integral(sqrt(7-X^2)/X^3,X)"
"+1/2*sqrt(7-X^2)/X^2"
"-1/2*log((sqrt(7)+sqrt(7-X^2))/X)/sqrt(7)",
"0",
"#218",
"integral(sqrt(7-X^2)/X^4,X)"
"+1/3*sqrt((7-X^2)^3)/7/X^3",
"0",
"#290",
"integral(sin(A*X),X)+cos(A*X)/A",
"0",
"#291",
"integral(cos(A*X),X)-sin(A*X)/A",
"0",
"#292",
"integral(tan(A*X),X)+log(cos(A*X))/A",
"0",
"#293",
"integral(1/tan(A*X),X)-log(sin(A*X))/A",
"0",
"#294",
"integral(1/cos(A*X),X)-log(tan(pi/4+A*X/2))/A",
"0",
"#295",
"integral(1/sin(A*X),X)-log(tan(A*X/2))/A",
"0",
"#296",
"integral(sin(A*X)^2,X)-X/2+sin(2*A*X)/(4*A)",
"0",
"#297",
"integral(sin(A*X)^3,X)+cos(A*X)*(sin(A*X)^2+2)/(3*A)",
"0",
"#298",
"integral(sin(A*X)^4,X)-3/8*X+sin(2*A*X)/(4*A)-sin(4*A*X)/(32*A)",
"0",
"#302",
"integral(cos(A*X)^2,X)-X/2-sin(2*A*X)/(4*A)",
"0",
"#303",
"integral(cos(A*X)^3,X)-sin(A*X)*(cos(A*X)^2+2)/(3*A)",
"0",
"#304",
"integral(cos(A*X)^4,X)-3/8*X-sin(2*A*X)/(4*A)-sin(4*A*X)/(32*A)",
"0",
"#308",
"integral((1/sin(A*X))^2,X)+1/A*1/tan(A*X)",
"0",
"#312",
"integral((1/cos(A*X))^2,X)-tan(A*X)/A",
"0",
"#318",
"integral(sin(A*X)*cos(A*X),X)-sin(A*X)^2/(2*A)",
"0",
"#320",
"integral(sin(A*X)^2*cos(A*X)^2,X)+sin(4*A*X)/(32*A)-X/8",
"0",
"#326",
"integral(sin(A*X)/cos(A*X)/cos(A*X),X)-1/(A*cos(A*X))",
"0",
"#327",
"integral(sin(A*X)^2/cos(A*X),X)+sin(A*X)/A-log(tan(pi/4+A*X/2))/A",
"0",
"#328",
"integral(cos(A*X)/sin(A*X)^2,X)+1/(A*sin(A*X))",
"0",
"#329",
"integral(1/sin(A*X)/cos(A*X),X)-log(tan(A*X))/A",
"0",
"#330",
"integral(1/sin(A*X)/cos(A*X)^2,X)-(1/cos(A*X)+log(tan(A*X/2)))/A",
"0",
"#332",
"integral(1/sin(A*X)^2/cos(A*X),X)-(log(tan(pi/4+A*X/2))-1/sin(A*X))/A",
"0",
"#333",
"integral(1/sin(A*X)^2/cos(A*X)^2,X)+2/(A*tan(2*A*X))",
"0",
"#335",
"integral(sin(A+B*X),X)+cos(A+B*X)/B",
"0",
"#336",
"integral(cos(A+B*X),X)-sin(A+B*X)/B",
"0",
"#337+",
"integral(1/(1+sin(A*X)),X)+tan(pi/4-A*X/2)/A",
"0",
"#337b+",
"integral(1/(B+B*sin(A*X)),X)+tan(pi/4-A*X/2)/A/B",
"0",
"#337-",
"integral(1/(1-sin(A*X)),X)-tan(pi/4+A*X/2)/A",
"0",
"#337b-",
"integral(1/(B-B*sin(A*X)),X)-tan(pi/4+A*X/2)/A/B",
"0",
"#338",
"integral(1/(1+cos(A*X)),X)-tan(A*X/2)/A",
"0",
"#339",
"integral(1/(1-cos(A*X)),X)+1/(A*tan(A*X/2))",
"0",
"#340",
"integral(1/(A+B*sin(X)),X)-1/sqrt(B^2-A^2)*log((A*tan(X/2)+B-sqrt(B^2-A^2))/(A*tan(X/2)+B+sqrt(B^2-A^2)))",
"0",
"#341",
"integral(1/(A+B*cos(X)),X)-1/sqrt(B^2-A^2)*log((sqrt(B^2-A^2)*tan(X/2)+A+B)/(sqrt(B^2-A^2)*tan(X/2)-A-B))",
"0",
"#441",
"integral(arcsin(A*X),X)-X*arcsin(A*X)-sqrt(1-A^2*X^2)/A",
"0",
"#442",
"integral(arccos(A*X),X)-X*arccos(A*X)-sqrt(1-A^2*X^2)/A",
"0",
"#443",
"integral(arctan(A*X),X)-X*arctan(A*X)+log(1+A^2*X^2)/(2*A)",
"0",
"#485",
"integral(log(X),X)-X*log(X)+X",
"0",
"#485a",
"integral(log(A*X),X)-X*log(A*X)+X",
"0",
"#486",
"integral(X*log(X),X)-1/2*X^2*log(X)+1/4*X^2",
"0",
"#486a",
"integral(X*log(A*X),X)-1/2*X^2*log(A*X)+1/4*X^2",
"0",
"#487",
"integral(X^2*log(A*X),X)-1/3*X^3*log(A*X)+X^3/9",
"0",
"#489",
"integral(log(X)^2,X)-X*log(X)^2+2*X*log(X)-2*X",
"0",
"#493",
"integral(1/X*1/log(A*X),X)-log(log(A*X))",
"0",
"#499",
"integral(log(A*X+B),X)-(A*X+B)*log(A*X+B)/A+X",
"0",
"#500",
"integral(log(A*X+B)/X^2,X)-A*log(X)/B+(A*X+B)*log(A*X+B)/B/X",
"0",
"#554",
"integral(sinh(X),X)-cosh(X)",
"0",
"#555",
"integral(cosh(X),X)-sinh(X)",
"0",
"#556",
"integral(tanh(X),X)-log(cosh(X))",
"0",
"#560",
"integral(X*sinh(X),X)-X*cosh(X)+sinh(X)",
"0",
"#562",
"integral(X*cosh(X),X)-X*sinh(X)+cosh(X)",
"0",
"#566",
"integral(sinh(X)^2,X)-sinh(2*X)/4+X/2",
"0",
"#569",
"integral(tanh(X)^2,X)-X+tanh(X)",
"0",
"#572",
"integral(cosh(X)^2,X)-sinh(2*X)/4-X/2",
"0",
};
void
test_integral(void)
{
test(__FILE__, s, sizeof s / sizeof (char *));
}
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