A substitution
y = cx, c constant
(%i3) integrate(exp(x)/(2 + 3*exp(2*x)), x); x 3 %e atan(-------) sqrt(6) (%o3) ------------- sqrt(6)
When constants are used, Maxima may ask you about possible values of these constants. In the following example, the integral depends from the sign of the product a*b:
(%i4) integrate (exp(2*x)/(a + b*exp(4*x)), x); Is a b positive or negative? positive; 2 x b %e atan(---------) sqrt(a b) (%o4) --------------- 2 sqrt(a b) (%i5) integrate (exp(2*x)/(a + b*exp(4*x)), x); Is a b positive or negative? negative; 2 x 2 b %e - 2 sqrt(- a b) log(-------------------------) 2 x 2 b %e + 2 sqrt(- a b) (%o5) ------------------------------ 4 sqrt(- a b)
When assumptions are used to inform Maxima about possible values of symbolic constants, no questions will be asked:
(%i6) assume(a>0, b<0); (%o6) [a > 0, b < 0] (%i7) integrate (exp(2*x)/(a + b*exp(4*x)), x); 2 x 2 b %e - 2 sqrt(a) sqrt(- b) log(-------------------------------) 2 x 2 b %e + 2 sqrt(a) sqrt(- b) (%o7) ------------------------------------ 4 sqrt(a) sqrt(- b)
A substitution
y = xk
(%i10) integrate (x^3*sin(x^2), x); 2 2 2 sin(x ) - x cos(x ) (%o10) -------------------- 2 (%i11) integrate (x^7/(x^12 + 1), x); 4 2 x - 1 8 4 atan(--------) 4 log(x - x + 1) sqrt(3) log(x + 1) (%o11) ---------------- + -------------- - ----------- 24 4 sqrt(3) 12
The substitution
1/k
/a*x + b\
y = | ------- |
\c*x + d/(%i12) integrate (x*(x + 1)^(1/2), x); 5/2 3/2 2 (x + 1) 2 (x + 1) (%o12) ------------ - ------------ 5 3 (%i13) integrate (((x + 1)/(2*x + 3))^(1/2), x); x + 1 4 sqrt(-------) - 2 sqrt(2) 2 x + 3 log(---------------------------) x + 1 x + 1 4 sqrt(-------) + 2 sqrt(2) sqrt(-------) 2 x + 3 2 x + 3 (%o13) 2 (-------------------------------- - -------------) 8 sqrt(2) 8 (x + 1) --------- - 4 2 x + 3
This is a decision procedure for expressions of the form
A*xr*(c1 + c2*xq)p
(%i14) integrate (x^(1/2)*(1 + x)^(5/2), x); x + 1 x + 1 5 log(sqrt(-----) + 1) 5 log(sqrt(-----) - 1) x x (%o14) - ---------------------- + ---------------------- 128 128 x + 1 7/2 x + 1 5/2 x + 1 3/2 x + 1 15 (-----) + 73 (-----) - 55 (-----) + 15 sqrt(-----) x x x x + --------------------------------------------------------------- 4 3 2 192 (x + 1) 768 (x + 1) 1152 (x + 1) 768 (x + 1) ------------ - ------------ + ------------- - ----------- + 192 4 3 2 x x x x (%i15) ratsimp(%); x + 1 x + 1 4 x + 1 7/2 (%o15) - (15 log(sqrt(-----) + 1) - 15 log(sqrt(-----) - 1) - 30 x (-----) x x x 4 x + 1 5/2 4 x + 1 3/2 4 x + 1 - 146 x (-----) + 110 x (-----) - 30 x sqrt(-----))/384 x x x (%i16) integrate ( x^4*(1 - x^2)^(-5/2) ,x); 2 x 2 x (%o16) asin(x) + x (----------- - -------------) - -------------- 2 3/2 2 3/2 2 (1 - x ) 3 (1 - x ) 3 sqrt(1 - x )
(%i13) 'integrate (x^4/((1 - x^2)^(5/2)), x); / 4 [ x (%o13) I ----------- dx ] 2 5/2 / (1 - x ) (%i14) ev (%, integrate); 2 x 2 x (%o14) asin(x) + x (----------- - -------------) - -------------- 2 3/2 2 3/2 2 (1 - x ) 3 (1 - x ) 3 sqrt(1 - x ) (%i16) 'integrate (((a^2 + b^2 - b^2*y^2)^(1/2))/(1 - y^2), y); / 2 2 2 2 [ sqrt(- b y + b + a ) (%o16) I ----------------------- dy ] 2 / 1 - y (%i17) ev(%, integrate); 2 2 2 2 2 2 2 2 2 2 2 a sqrt(- b y + b + a ) 2 a 2 2 a sqrt(- b y + b + a ) 2 a 2 a log(--------------------------- + ------------ + b ) a log(--------------------------- + ------------ - b ) abs(2 y + 2) abs(2 y + 2) abs(2 y - 2) abs(2 y - 2) (%o17) - ------------------------------------------------------ + ------------------------------------------------------ 2 2 2 b y + b asin(----------------) 4 2 2 sqrt(b + a b )
(%i8) integrate (sin(2*x)*cos(x), x); cos(3 x) cos(x) (%o8) - -------- - ------ 6 2 (%i9) integrate(sin(x)^2, x); sin(2 x) x - -------- 2 (%o9) ------------ 2 (%i10) integrate (sec(t)^2/(1 + sec(t)^2 - 3*tan(t)), t); (%o10) log(tan(t) - 2) - log(tan(t) - 1) (%i11) integrate (1/(1 + cos(x)), x); sin(x) (%o11) ---------- cos(x) + 1
(%i18) 'integrate (x*exp(x)/(x + 1)^2, x); / x [ x %e (%o18) I -------- dx ] 2 / (x + 1) (%i19) ev(%, integrate); x %e (%o19) ----- x + 1 (%i20) 'integrate ( (2*x^6 + 5*x^4 + x^3 + 4*x^2 + 1)/((x^2 + 1)^2)*exp(x^2), x); 2 / 6 4 3 2 x [ (2 x + 5 x + x + 4 x + 1) %e (%o20) I ---------------------------------- dx ] 2 2 / (x + 1) (%i21) ev(%, integrate); 2 3 x (2 x + 2 x + 1) %e (%o21) --------------------- 2 2 x + 2 (%i22) integrate (exp(x^2), x); sqrt(%pi) %i erf(%i x) - ---------------------- 2
Hermite reduction is used to compute the rational part of the integral. The transcendental part can be computed for integrands with special denominators, if that is not possible, an unevaluated integral is answered.
integrate (x/(x^3 + 1), x);
2 x - 1
2 atan(-------)
log(x - x + 1) sqrt(3) log(x + 1)
--------------- + ------------- - ----------
6 sqrt(3) 3
Occasionally, Maxima gives an answer that contains an integral:
integrate((x^8+7*x^6+42*x^4+48*x^2+30)/(x^10+8*x^8+19*x^6+9*x^4+27) ,x)
/ 2
[ x + 1 x
I ----------- dx + -------------
] 4 2 4 2
/ x - x + 1 x + 6 x + 9
(%i25) integrate (x^2*asin(x), x); 2 2 2 x sqrt(1 - x ) 2 sqrt(1 - x ) 3 - --------------- - -------------- x asin(x) 3 3 (%o25) ---------- - ---------------------------------- 3 3 (%i26) integrate (x* log(x), x); 2 2 x log(x) x (%o26) --------- - -- 2 4
(%i1) 'integrate (log(x)/(log(x) + 1)^2, x); / [ log(x) (%o1) I ------------- dx ] 2 / (log(x) + 1) (%i2) ev(%, integrate); x (%o2) ---------- log(x) + 1
(%i3) integrate (x*(cos(x) + sin(x)), x); (%o3) x sin(x) + sin(x) - x cos(x) + cos(x) (%i4) integrate ((x + exp(x))/exp(x), x); - x (%o4) (- x - 1) %e + x