Integrand size = 17, antiderivative size = 195 \[ \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx=-\frac {3 \cosh (a-3 c+(b-3 d) x)}{32 (b-3 d)}-\frac {9 \cosh (a-c+(b-d) x)}{32 (b-d)}+\frac {\cosh (3 (a-c)+3 (b-d) x)}{96 (b-d)}+\frac {3 \cosh (3 a-c+(3 b-d) x)}{32 (3 b-d)}-\frac {9 \cosh (a+c+(b+d) x)}{32 (b+d)}+\frac {\cosh (3 (a+c)+3 (b+d) x)}{96 (b+d)}+\frac {3 \cosh (3 a+c+(3 b+d) x)}{32 (3 b+d)}-\frac {3 \cosh (a+3 c+(b+3 d) x)}{32 (b+3 d)} \]
-3/32*cosh(a-3*c+(b-3*d)*x)/(b-3*d)-9/32*cosh(a-c+(b-d)*x)/(b-d)+1/96*cosh (3*a-3*c+3*(b-d)*x)/(b-d)+3/32*cosh(3*a-c+(3*b-d)*x)/(3*b-d)-9/32*cosh(a+c +(b+d)*x)/(b+d)+1/96*cosh(3*a+3*c+3*(b+d)*x)/(b+d)+3/32*cosh(3*a+c+(3*b+d) *x)/(3*b+d)-3/32*cosh(a+3*c+(b+3*d)*x)/(b+3*d)
Time = 1.04 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.90 \[ \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx=\frac {1}{96} \left (-\frac {9 \cosh (a-3 c+b x-3 d x)}{b-3 d}-\frac {27 \cosh (a-c+b x-d x)}{b-d}+\frac {\cosh (3 (a-c+b x-d x))}{b-d}+\frac {9 \cosh (3 a-c+3 b x-d x)}{3 b-d}+\frac {9 \cosh (3 a+c+3 b x+d x)}{3 b+d}-\frac {9 \cosh (a+3 c+b x+3 d x)}{b+3 d}-\frac {27 \cosh (a+c+(b+d) x)}{b+d}+\frac {\cosh (3 (a+c+(b+d) x))}{b+d}\right ) \]
((-9*Cosh[a - 3*c + b*x - 3*d*x])/(b - 3*d) - (27*Cosh[a - c + b*x - d*x]) /(b - d) + Cosh[3*(a - c + b*x - d*x)]/(b - d) + (9*Cosh[3*a - c + 3*b*x - d*x])/(3*b - d) + (9*Cosh[3*a + c + 3*b*x + d*x])/(3*b + d) - (9*Cosh[a + 3*c + b*x + 3*d*x])/(b + 3*d) - (27*Cosh[a + c + (b + d)*x])/(b + d) + Co sh[3*(a + c + (b + d)*x)]/(b + d))/96
Time = 0.40 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6152, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sinh ^3(a+b x) \cosh ^3(c+d x) \, dx\) |
| \(\Big \downarrow \) 6152 |
| \(\displaystyle \int \left (-\frac {3}{32} \sinh (a+x (b-3 d)-3 c)-\frac {9}{32} \sinh (a+x (b-d)-c)+\frac {1}{32} \sinh (3 (a-c)+3 x (b-d))+\frac {3}{32} \sinh (3 a+x (3 b-d)-c)-\frac {9}{32} \sinh (a+x (b+d)+c)+\frac {1}{32} \sinh (3 (a+c)+3 x (b+d))+\frac {3}{32} \sinh (3 a+x (3 b+d)+c)-\frac {3}{32} \sinh (a+x (b+3 d)+3 c)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \cosh (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac {9 \cosh (a+x (b-d)-c)}{32 (b-d)}+\frac {\cosh (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac {3 \cosh (3 a+x (3 b-d)-c)}{32 (3 b-d)}-\frac {9 \cosh (a+x (b+d)+c)}{32 (b+d)}+\frac {\cosh (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac {3 \cosh (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac {3 \cosh (a+x (b+3 d)+3 c)}{32 (b+3 d)}\) |
(-3*Cosh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) - (9*Cosh[a - c + (b - d)* x])/(32*(b - d)) + Cosh[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Cosh[3* a - c + (3*b - d)*x])/(32*(3*b - d)) - (9*Cosh[a + c + (b + d)*x])/(32*(b + d)) + Cosh[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) + (3*Cosh[3*a + c + (3* b + d)*x])/(32*(3*b + d)) - (3*Cosh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))
3.2.87.3.1 Defintions of rubi rules used
Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v ]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x ]))
Time = 11.36 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {3 \cosh \left (a -3 c +\left (b -3 d \right ) x \right )}{32 \left (b -3 d \right )}-\frac {9 \cosh \left (a -c +\left (b -d \right ) x \right )}{32 \left (b -d \right )}-\frac {9 \cosh \left (a +c +\left (b +d \right ) x \right )}{32 \left (b +d \right )}-\frac {3 \cosh \left (a +3 c +\left (b +3 d \right ) x \right )}{32 \left (b +3 d \right )}+\frac {\cosh \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{96 b -96 d}+\frac {3 \cosh \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}+\frac {3 \cosh \left (3 a +c +\left (3 b +d \right ) x \right )}{32 \left (3 b +d \right )}+\frac {\cosh \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{96 b +96 d}\) | \(190\) |
| parallelrisch | \(\frac {9 \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b +d \right ) \left (b -\frac {d}{3}\right ) \cosh \left (\left (3 b -3 d \right ) x +3 a -3 c \right )+27 \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b -d \right ) \left (b +d \right ) \cosh \left (3 a -c +\left (3 b -d \right ) x \right )+9 \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b -d \right ) \left (b -\frac {d}{3}\right ) \cosh \left (\left (3 b +3 d \right ) x +3 a +3 c \right )-81 \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -d \right ) \left (b +d \right ) \left (b -\frac {d}{3}\right ) \cosh \left (a -3 c +\left (b -3 d \right ) x \right )-243 \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b +d \right ) \left (b -\frac {d}{3}\right ) \cosh \left (a -c +\left (b -d \right ) x \right )-81 \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b -d \right ) \left (b +d \right ) \left (b -\frac {d}{3}\right ) \cosh \left (a +3 c +\left (b +3 d \right ) x \right )+27 \left (b +3 d \right ) \left (b -3 d \right ) \left (b -d \right ) \left (b +d \right ) \left (b -\frac {d}{3}\right ) \cosh \left (3 a +c +\left (3 b +d \right ) x \right )-243 \left (b +3 d \right ) \left (b +\frac {d}{3}\right ) \left (b -3 d \right ) \left (b -d \right ) \left (b -\frac {d}{3}\right ) \cosh \left (a +c +\left (b +d \right ) x \right )+576 b^{5}-3904 b^{3} d^{2}}{864 b^{6}-8736 b^{4} d^{2}+8736 b^{2} d^{4}-864 d^{6}}\) | \(361\) |
| risch | \(\frac {\left (b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-9 b \,d^{2} {\mathrm e}^{6 b x +6 a}+9 d^{3} {\mathrm e}^{6 b x +6 a}-9 b^{3} {\mathrm e}^{4 b x +4 a}+27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+9 b \,d^{2} {\mathrm e}^{4 b x +4 a}-27 d^{3} {\mathrm e}^{4 b x +4 a}-9 b^{3} {\mathrm e}^{2 b x +2 a}-27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+9 b \,d^{2} {\mathrm e}^{2 b x +2 a}+27 d^{3} {\mathrm e}^{2 b x +2 a}+b^{3}+b^{2} d -9 b \,d^{2}-9 d^{3}\right ) {\mathrm e}^{-3 b x +3 d x -3 a +3 c}}{192 \left (b +d \right ) \left (b +3 d \right ) \left (b -d \right ) \left (b -3 d \right )}+\frac {3 \left (3 b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}+d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}+27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}-3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}-27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}+3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}+b^{2} d -3 b \,d^{2}-d^{3}\right ) {\mathrm e}^{-3 b x +d x -3 a +c}}{64 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}+\frac {3 \left (3 b^{3} {\mathrm e}^{6 b x +6 a}+b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}-d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}-27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}+3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}+27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}-3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}-b^{2} d -3 b \,d^{2}+d^{3}\right ) {\mathrm e}^{-3 b x -d x -3 a -c}}{64 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}+\frac {\left (b^{3} {\mathrm e}^{6 b x +6 a}+b^{2} d \,{\mathrm e}^{6 b x +6 a}-9 b \,d^{2} {\mathrm e}^{6 b x +6 a}-9 d^{3} {\mathrm e}^{6 b x +6 a}-9 b^{3} {\mathrm e}^{4 b x +4 a}-27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+9 b \,d^{2} {\mathrm e}^{4 b x +4 a}+27 d^{3} {\mathrm e}^{4 b x +4 a}-9 b^{3} {\mathrm e}^{2 b x +2 a}+27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+9 b \,d^{2} {\mathrm e}^{2 b x +2 a}-27 d^{3} {\mathrm e}^{2 b x +2 a}+b^{3}-b^{2} d -9 b \,d^{2}+9 d^{3}\right ) {\mathrm e}^{-3 b x -3 d x -3 a -3 c}}{192 \left (b +d \right ) \left (b +3 d \right ) \left (b -d \right ) \left (b -3 d \right )}\) | \(954\) |
-3/32*cosh(a-3*c+(b-3*d)*x)/(b-3*d)-9/32*cosh(a-c+(b-d)*x)/(b-d)-9/32*cosh (a+c+(b+d)*x)/(b+d)-3/32*cosh(a+3*c+(b+3*d)*x)/(b+3*d)+1/32/(3*b-3*d)*cosh ((3*b-3*d)*x+3*a-3*c)+3/32*cosh(3*a-c+(3*b-d)*x)/(3*b-d)+3/32*cosh(3*a+c+( 3*b+d)*x)/(3*b+d)+1/32/(3*b+3*d)*cosh((3*b+3*d)*x+3*a+3*c)
Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (179) = 358\).
Time = 0.27 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.74 \[ \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx=\frac {{\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )^{3} - 9 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right )^{3} - {\left ({\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left (81 \, b^{4} d - 90 \, b^{2} d^{3} + 9 \, d^{5} - {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} + 27 \, {\left (b^{5} - 10 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right )^{2} + 3 \, {\left (3 \, {\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left ({\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )^{3} - 9 \, {\left (9 \, b^{5} - 10 \, b^{3} d^{2} + b d^{4}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 27 \, {\left ({\left (b^{5} - 10 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{5} - 82 \, b^{3} d^{2} + 9 \, b d^{4}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) - 3 \, {\left ({\left (3 \, b^{4} d - 30 \, b^{2} d^{3} + 27 \, d^{5} + {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )^{3} - 3 \, {\left (27 \, b^{4} d - 246 \, b^{2} d^{3} + 27 \, d^{5} - 3 \, {\left (b^{4} d - 10 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} + {\left (81 \, b^{4} d - 90 \, b^{2} d^{3} + 9 \, d^{5} - {\left (9 \, b^{4} d - 82 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{48 \, {\left ({\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{6} - 91 \, b^{4} d^{2} + 91 \, b^{2} d^{4} - 9 \, d^{6}\right )} \sinh \left (b x + a\right )^{4}\right )}} \]
1/48*(((9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)^3 - 9*(9*b^5 - 10*b^3* d^2 + b*d^4)*cosh(b*x + a))*cosh(d*x + c)^3 - ((9*b^4*d - 82*b^2*d^3 + 9*d ^5)*sinh(b*x + a)^3 - 3*(81*b^4*d - 90*b^2*d^3 + 9*d^5 - (9*b^4*d - 82*b^2 *d^3 + 9*d^5)*cosh(b*x + a)^2)*sinh(b*x + a))*sinh(d*x + c)^3 + 3*((9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)*cosh(d*x + c)^3 + 27*(b^5 - 10*b^3*d ^2 + 9*b*d^4)*cosh(b*x + a)*cosh(d*x + c))*sinh(b*x + a)^2 + 3*(3*(9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^2 + ((9*b ^5 - 82*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)^3 - 9*(9*b^5 - 10*b^3*d^2 + b*d^4 )*cosh(b*x + a))*cosh(d*x + c))*sinh(d*x + c)^2 + 27*((b^5 - 10*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)^3 - (9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cosh(b*x + a))*c osh(d*x + c) - 3*((3*b^4*d - 30*b^2*d^3 + 27*d^5 + (9*b^4*d - 82*b^2*d^3 + 9*d^5)*cosh(d*x + c)^2)*sinh(b*x + a)^3 - 3*(27*b^4*d - 246*b^2*d^3 + 27* d^5 - 3*(b^4*d - 10*b^2*d^3 + 9*d^5)*cosh(b*x + a)^2 + (81*b^4*d - 90*b^2* d^3 + 9*d^5 - (9*b^4*d - 82*b^2*d^3 + 9*d^5)*cosh(b*x + a)^2)*cosh(d*x + c )^2)*sinh(b*x + a))*sinh(d*x + c))/((9*b^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d ^6)*cosh(b*x + a)^4 - 2*(9*b^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d^6)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (9*b^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d^6)*sinh(b *x + a)^4)
Leaf count of result is larger than twice the leaf count of optimal. 3580 vs. \(2 (172) = 344\).
Time = 16.87 (sec) , antiderivative size = 3580, normalized size of antiderivative = 18.36 \[ \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx=\text {Too large to display} \]
Piecewise((x*sinh(a)**3*cosh(c)**3, Eq(b, 0) & Eq(d, 0)), (9*x*sinh(a - 3* d*x)**3*sinh(c + d*x)**2*cosh(c + d*x)/32 + 3*x*sinh(a - 3*d*x)**3*cosh(c + d*x)**3/32 + 3*x*sinh(a - 3*d*x)**2*sinh(c + d*x)**3*cosh(a - 3*d*x)/32 + 9*x*sinh(a - 3*d*x)**2*sinh(c + d*x)*cosh(a - 3*d*x)*cosh(c + d*x)**2/32 - 9*x*sinh(a - 3*d*x)*sinh(c + d*x)**2*cosh(a - 3*d*x)**2*cosh(c + d*x)/3 2 - 3*x*sinh(a - 3*d*x)*cosh(a - 3*d*x)**2*cosh(c + d*x)**3/32 - 3*x*sinh( c + d*x)**3*cosh(a - 3*d*x)**3/32 - 9*x*sinh(c + d*x)*cosh(a - 3*d*x)**3*c osh(c + d*x)**2/32 + sinh(a - 3*d*x)**3*sinh(c + d*x)**3/(12*d) - 13*sinh( a - 3*d*x)**3*sinh(c + d*x)*cosh(c + d*x)**2/(320*d) + 3*sinh(a - 3*d*x)** 2*sinh(c + d*x)**2*cosh(a - 3*d*x)*cosh(c + d*x)/(20*d) - 101*sinh(a - 3*d *x)**2*cosh(a - 3*d*x)*cosh(c + d*x)**3/(320*d) - 27*sinh(a - 3*d*x)*sinh( c + d*x)**3*cosh(a - 3*d*x)**2/(320*d) - 51*sinh(c + d*x)**2*cosh(a - 3*d* x)**3*cosh(c + d*x)/(320*d) + cosh(a - 3*d*x)**3*cosh(c + d*x)**3/(5*d), E q(b, -3*d)), (-3*x*sinh(a - d*x)**3*sinh(c + d*x)**2*cosh(c + d*x)/16 + 5* x*sinh(a - d*x)**3*cosh(c + d*x)**3/16 - 3*x*sinh(a - d*x)**2*sinh(c + d*x )**3*cosh(a - d*x)/16 + 9*x*sinh(a - d*x)**2*sinh(c + d*x)*cosh(a - d*x)*c osh(c + d*x)**2/16 + 9*x*sinh(a - d*x)*sinh(c + d*x)**2*cosh(a - d*x)**2*c osh(c + d*x)/16 - 3*x*sinh(a - d*x)*cosh(a - d*x)**2*cosh(c + d*x)**3/16 + 5*x*sinh(c + d*x)**3*cosh(a - d*x)**3/16 - 3*x*sinh(c + d*x)*cosh(a - d*x )**3*cosh(c + d*x)**2/16 - sinh(a - d*x)**3*sinh(c + d*x)**3/(48*d) + s...
Exception generated. \[ \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(-(3*d)/b>0)', see `assume?` for more detai
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (179) = 358\).
Time = 0.30 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.91 \[ \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}}{192 \, {\left (b + d\right )}} + \frac {3 \, e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{64 \, {\left (3 \, b + d\right )}} + \frac {3 \, e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{64 \, {\left (3 \, b - d\right )}} + \frac {e^{\left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}}{192 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{64 \, {\left (b + 3 \, d\right )}} - \frac {9 \, e^{\left (b x + d x + a + c\right )}}{64 \, {\left (b + d\right )}} - \frac {9 \, e^{\left (b x - d x + a - c\right )}}{64 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{64 \, {\left (b - 3 \, d\right )}} - \frac {3 \, e^{\left (-b x + 3 \, d x - a + 3 \, c\right )}}{64 \, {\left (b - 3 \, d\right )}} - \frac {9 \, e^{\left (-b x + d x - a + c\right )}}{64 \, {\left (b - d\right )}} - \frac {9 \, e^{\left (-b x - d x - a - c\right )}}{64 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (-b x - 3 \, d x - a - 3 \, c\right )}}{64 \, {\left (b + 3 \, d\right )}} + \frac {e^{\left (-3 \, b x + 3 \, d x - 3 \, a + 3 \, c\right )}}{192 \, {\left (b - d\right )}} + \frac {3 \, e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{64 \, {\left (3 \, b - d\right )}} + \frac {3 \, e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{64 \, {\left (3 \, b + d\right )}} + \frac {e^{\left (-3 \, b x - 3 \, d x - 3 \, a - 3 \, c\right )}}{192 \, {\left (b + d\right )}} \]
1/192*e^(3*b*x + 3*d*x + 3*a + 3*c)/(b + d) + 3/64*e^(3*b*x + d*x + 3*a + c)/(3*b + d) + 3/64*e^(3*b*x - d*x + 3*a - c)/(3*b - d) + 1/192*e^(3*b*x - 3*d*x + 3*a - 3*c)/(b - d) - 3/64*e^(b*x + 3*d*x + a + 3*c)/(b + 3*d) - 9 /64*e^(b*x + d*x + a + c)/(b + d) - 9/64*e^(b*x - d*x + a - c)/(b - d) - 3 /64*e^(b*x - 3*d*x + a - 3*c)/(b - 3*d) - 3/64*e^(-b*x + 3*d*x - a + 3*c)/ (b - 3*d) - 9/64*e^(-b*x + d*x - a + c)/(b - d) - 9/64*e^(-b*x - d*x - a - c)/(b + d) - 3/64*e^(-b*x - 3*d*x - a - 3*c)/(b + 3*d) + 1/192*e^(-3*b*x + 3*d*x - 3*a + 3*c)/(b - d) + 3/64*e^(-3*b*x + d*x - 3*a + c)/(3*b - d) + 3/64*e^(-3*b*x - d*x - 3*a - c)/(3*b + d) + 1/192*e^(-3*b*x - 3*d*x - 3*a - 3*c)/(b + d)
Time = 2.80 (sec) , antiderivative size = 908, normalized size of antiderivative = 4.66 \[ \int \cosh ^3(c+d x) \sinh ^3(a+b x) \, dx=-{\mathrm {e}}^{3\,a+c+3\,b\,x+d\,x}\,\left (\frac {-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )-{\mathrm {e}}^{3\,a-c+3\,b\,x-d\,x}\,\left (\frac {-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )-{\mathrm {e}}^{3\,a-3\,c+3\,b\,x-3\,d\,x}\,\left (\frac {-b^3-b^2\,d+9\,b\,d^2+9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-b^3+b^2\,d+9\,b\,d^2-9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right )-{\mathrm {e}}^{3\,a+3\,c+3\,b\,x+3\,d\,x}\,\left (\frac {-b^3+b^2\,d+9\,b\,d^2-9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-b^3-b^2\,d+9\,b\,d^2+9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right ) \]
- exp(3*a + c + 3*b*x + d*x)*((9*b*d^2 + 3*b^2*d - 9*b^3 - 3*d^3)/(576*b^4 + 64*d^4 - 640*b^2*d^2) + (exp(- 6*a - 6*b*x)*(9*b*d^2 - 3*b^2*d - 9*b^3 + 3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- 2*a - 2*b*x)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- 4* a - 4*b*x)*(9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(576*b^4 + 64*d^4 - 640* b^2*d^2)) - exp(3*a - c + 3*b*x - d*x)*((9*b*d^2 - 3*b^2*d - 9*b^3 + 3*d^3 )/(576*b^4 + 64*d^4 - 640*b^2*d^2) + (exp(- 6*a - 6*b*x)*(9*b*d^2 + 3*b^2* d - 9*b^3 - 3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- 2*a - 2*b*x) *(9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- 4*a - 4*b*x)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 9*d^3))/(576*b^4 + 64* d^4 - 640*b^2*d^2)) - exp(3*a - 3*c + 3*b*x - 3*d*x)*((9*b*d^2 - b^2*d - b ^3 + 9*d^3)/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) + (exp(- 6*a - 6*b*x)*(9*b *d^2 + b^2*d - b^3 - 9*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- 2*a - 2*b*x)*(9*b*d^2 - 27*b^2*d - 9*b^3 + 27*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- 4*a - 4*b*x)*(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3)) /(192*b^4 + 1728*d^4 - 1920*b^2*d^2)) - exp(3*a + 3*c + 3*b*x + 3*d*x)*((9 *b*d^2 + b^2*d - b^3 - 9*d^3)/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) + (exp(- 6*a - 6*b*x)*(9*b*d^2 - b^2*d - b^3 + 9*d^3))/(192*b^4 + 1728*d^4 - 1920* b^2*d^2) - (exp(- 2*a - 2*b*x)*(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3))/(192 *b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- 4*a - 4*b*x)*(9*b*d^2 - 27*b^2...