Optimal. Leaf size=692 \[ -\frac{3 f \text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \sinh \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \cosh \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}+\frac{3 f \text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \cosh \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}+\frac{3 f \sinh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac{\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{1}{8 a^3 d (c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.76195, antiderivative size = 692, normalized size of antiderivative = 1., number of steps used = 60, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3728, 3297, 3303, 3298, 3301, 3313, 12, 5448, 5470} \[ -\frac{3 f \text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \sinh \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \cosh \left (4 e-\frac{4 c f}{d}\right )}{2 a^3 d^2}+\frac{3 f \text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \cosh \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}+\frac{3 f \sinh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{2 a^3 d^2}-\frac{3 f \cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac{\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac{3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{1}{8 a^3 d (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3728
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 3313
Rule 12
Rule 5448
Rule 5470
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx &=\int \left (\frac{1}{8 a^3 (c+d x)^2}-\frac{3 \cosh (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac{\cosh ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{3 \sinh (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{3 \cosh ^2(2 e+2 f x) \sinh (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac{\sinh ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac{3 \sinh (4 e+4 f x)}{8 a^3 (c+d x)^2}-\frac{3 \sinh (2 e+2 f x) \sinh (4 e+4 f x)}{16 a^3 (c+d x)^2}\right ) \, dx\\ &=-\frac{1}{8 a^3 d (c+d x)}-\frac{\int \frac{\cosh ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac{\int \frac{\sinh ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac{3 \int \frac{\sinh (2 e+2 f x) \sinh (4 e+4 f x)}{(c+d x)^2} \, dx}{16 a^3}-\frac{3 \int \frac{\cosh (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac{3 \int \frac{\cosh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac{3 \int \frac{\sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac{3 \int \frac{\cosh ^2(2 e+2 f x) \sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac{3 \int \frac{\sinh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac{3 \int \frac{\sinh (4 e+4 f x)}{(c+d x)^2} \, dx}{8 a^3}\\ &=-\frac{1}{8 a^3 d (c+d x)}+\frac{3 \cosh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{3 \int \left (-\frac{\cosh (2 e+2 f x)}{2 (c+d x)^2}+\frac{\cosh (6 e+6 f x)}{2 (c+d x)^2}\right ) \, dx}{16 a^3}+\frac{3 \int \left (\frac{\sinh (2 e+2 f x)}{4 (c+d x)^2}+\frac{\sinh (6 e+6 f x)}{4 (c+d x)^2}\right ) \, dx}{8 a^3}-\frac{(3 i f) \int \left (-\frac{i \sinh (2 e+2 f x)}{4 (c+d x)}-\frac{i \sinh (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{4 a^3 d}+\frac{(3 i f) \int -\frac{i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}-\frac{(3 i f) \int \frac{i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}+\frac{(3 f) \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}-\frac{(3 f) \int \left (\frac{\cosh (2 e+2 f x)}{4 (c+d x)}-\frac{\cosh (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{4 a^3 d}-\frac{(3 f) \int \frac{\sinh (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}-\frac{(3 f) \int \frac{\cosh (4 e+4 f x)}{c+d x} \, dx}{2 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}+\frac{3 \cosh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac{3 \int \frac{\cosh (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac{3 \int \frac{\cosh (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac{3 \int \frac{\sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac{3 \int \frac{\sinh (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac{(3 f) \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac{(3 f) \int \frac{\cosh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(3 f) \int \frac{\sinh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(3 f) \int \frac{\sinh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}+2 \frac{(3 f) \int \frac{\sinh (4 e+4 f x)}{c+d x} \, dx}{4 a^3 d}-\frac{\left (3 f \cosh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^3 d}+\frac{\left (3 f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}-\frac{\left (3 f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}-\frac{\left (3 f \sinh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{2 a^3 d}-\frac{\left (3 f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}+\frac{\left (3 f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{4 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}+\frac{9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac{3 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac{3 f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Chi}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac{3 f \text{Chi}\left (\frac{2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{3 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac{3 f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac{3 f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac{(3 f) \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac{(3 f) \int \frac{\sinh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac{(9 f) \int \frac{\cosh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac{(9 f) \int \frac{\sinh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \cosh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \cosh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \sinh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \sinh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}+2 \left (\frac{\left (3 f \cosh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^3 d}+\frac{\left (3 f \sinh \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^3 d}\right )-\frac{\left (3 f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (3 f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}+\frac{9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac{9 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{16 a^3 d^2}-\frac{3 f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Chi}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac{3 f \cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Chi}\left (\frac{6 c f}{d}+6 f x\right )}{16 a^3 d^2}-\frac{3 f \text{Chi}\left (\frac{6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac{6 c f}{d}\right )}{16 a^3 d^2}-\frac{15 f \text{Chi}\left (\frac{2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{16 a^3 d^2}-\frac{15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{15 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{16 a^3 d^2}+\frac{9 f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{16 a^3 d^2}-\frac{3 f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}+2 \left (\frac{3 f \text{Chi}\left (\frac{4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )-\frac{3 f \cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (\frac{6 c f}{d}+6 f x\right )}{16 a^3 d^2}+\frac{3 f \sinh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (\frac{6 c f}{d}+6 f x\right )}{16 a^3 d^2}+\frac{\left (9 f \cosh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (9 f \cosh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}-\frac{\left (9 f \sinh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (9 f \sinh \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}+\frac{\left (3 f \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{16 a^3 d}\\ &=-\frac{1}{8 a^3 d (c+d x)}+\frac{9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac{3 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac{3 f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Chi}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac{3 f \cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Chi}\left (\frac{6 c f}{d}+6 f x\right )}{4 a^3 d^2}-\frac{3 f \text{Chi}\left (\frac{6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac{6 c f}{d}\right )}{4 a^3 d^2}-\frac{3 f \text{Chi}\left (\frac{2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{4 a^3 d^2}-\frac{15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac{3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac{\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac{3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac{3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac{3 f \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac{3 f \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac{3 f \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{2 a^3 d^2}+2 \left (\frac{3 f \text{Chi}\left (\frac{4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{4 a^3 d^2}+\frac{3 f \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (\frac{4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )-\frac{3 f \cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (\frac{6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac{3 f \sinh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (\frac{6 c f}{d}+6 f x\right )}{4 a^3 d^2}\\ \end{align*}
Mathematica [A] time = 3.33111, size = 796, normalized size = 1.15 \[ \frac{\text{csch}^3(e+f x) \left (\cosh \left (\frac{3 c f}{d}\right )+\sinh \left (\frac{3 c f}{d}\right )\right ) \left (3 d \cosh \left (e+f \left (x-\frac{3 c}{d}\right )\right )-d \cosh \left (3 \left (e+f \left (x-\frac{c}{d}\right )\right )\right )+d \cosh \left (3 \left (e+f \left (\frac{c}{d}+x\right )\right )\right )-3 d \cosh \left (e+f \left (\frac{3 c}{d}+x\right )\right )+6 c f \cosh \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Chi}\left (\frac{6 f (c+d x)}{d}\right )+6 d f x \cosh \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Chi}\left (\frac{6 f (c+d x)}{d}\right )+6 f (c+d x) \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac{c f}{d}+3 f x\right )+\sinh \left (e-\frac{c f}{d}+3 f x\right )\right )+3 d \sinh \left (e+f \left (x-\frac{3 c}{d}\right )\right )-d \sinh \left (3 \left (e+f \left (x-\frac{c}{d}\right )\right )\right )-d \sinh \left (3 \left (e+f \left (\frac{c}{d}+x\right )\right )\right )+3 d \sinh \left (e+f \left (\frac{3 c}{d}+x\right )\right )-6 c f \text{Chi}\left (\frac{6 f (c+d x)}{d}\right ) \sinh \left (3 e-\frac{3 f (c+d x)}{d}\right )-6 d f x \text{Chi}\left (\frac{6 f (c+d x)}{d}\right ) \sinh \left (3 e-\frac{3 f (c+d x)}{d}\right )+12 f (c+d x) \text{Chi}\left (\frac{4 f (c+d x)}{d}\right ) \left (\sinh \left (e-\frac{f (c+3 d x)}{d}\right )-\cosh \left (e-\frac{f (c+3 d x)}{d}\right )\right )-6 c f \cosh \left (e-\frac{c f}{d}+3 f x\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-6 d f x \cosh \left (e-\frac{c f}{d}+3 f x\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-6 c f \sinh \left (e-\frac{c f}{d}+3 f x\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-6 d f x \sinh \left (e-\frac{c f}{d}+3 f x\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+12 c f \cosh \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Shi}\left (\frac{4 f (c+d x)}{d}\right )+12 d f x \cosh \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Shi}\left (\frac{4 f (c+d x)}{d}\right )-12 c f \sinh \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Shi}\left (\frac{4 f (c+d x)}{d}\right )-12 d f x \sinh \left (e-\frac{f (c+3 d x)}{d}\right ) \text{Shi}\left (\frac{4 f (c+d x)}{d}\right )-6 c f \cosh \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Shi}\left (\frac{6 f (c+d x)}{d}\right )-6 d f x \cosh \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Shi}\left (\frac{6 f (c+d x)}{d}\right )+6 c f \sinh \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Shi}\left (\frac{6 f (c+d x)}{d}\right )+6 d f x \sinh \left (3 e-\frac{3 f (c+d x)}{d}\right ) \text{Shi}\left (\frac{6 f (c+d x)}{d}\right )\right )}{8 a^3 d^2 (c+d x) (\coth (e+f x)+1)^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.408, size = 239, normalized size = 0.4 \begin{align*} -{\frac{1}{8\,{a}^{3}d \left ( dx+c \right ) }}+{\frac{f{{\rm e}^{-6\,fx-6\,e}}}{8\,{a}^{3}d \left ( dfx+cf \right ) }}-{\frac{3\,f}{4\,{a}^{3}{d}^{2}}{{\rm e}^{6\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,6\,fx+6\,e+6\,{\frac{cf-de}{d}} \right ) }-{\frac{3\,f{{\rm e}^{-4\,fx-4\,e}}}{8\,{a}^{3}d \left ( dfx+cf \right ) }}+{\frac{3\,f}{2\,{a}^{3}{d}^{2}}{{\rm e}^{4\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,4\,fx+4\,e+4\,{\frac{cf-de}{d}} \right ) }+{\frac{3\,f{{\rm e}^{-2\,fx-2\,e}}}{8\,{a}^{3}d \left ( dfx+cf \right ) }}-{\frac{3\,f}{4\,{a}^{3}{d}^{2}}{{\rm e}^{2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 23.9139, size = 189, normalized size = 0.27 \begin{align*} -\frac{1}{8 \,{\left (a^{3} d^{2} x + a^{3} c d\right )}} + \frac{e^{\left (-6 \, e + \frac{6 \, c f}{d}\right )} E_{2}\left (\frac{6 \,{\left (d x + c\right )} f}{d}\right )}{8 \,{\left (d x + c\right )} a^{3} d} - \frac{3 \, e^{\left (-4 \, e + \frac{4 \, c f}{d}\right )} E_{2}\left (\frac{4 \,{\left (d x + c\right )} f}{d}\right )}{8 \,{\left (d x + c\right )} a^{3} d} + \frac{3 \, e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{2}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{8 \,{\left (d x + c\right )} a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33885, size = 2685, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c^{2} \coth ^{3}{\left (e + f x \right )} + 3 c^{2} \coth ^{2}{\left (e + f x \right )} + 3 c^{2} \coth{\left (e + f x \right )} + c^{2} + 2 c d x \coth ^{3}{\left (e + f x \right )} + 6 c d x \coth ^{2}{\left (e + f x \right )} + 6 c d x \coth{\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth ^{3}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth ^{2}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth{\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.38177, size = 358, normalized size = 0.52 \begin{align*} \frac{6 \, d f x{\rm Ei}\left (-\frac{6 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{6 \, c f}{d} - 6 \, e\right )} - 12 \, d f x{\rm Ei}\left (-\frac{4 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{4 \, c f}{d} - 4 \, e\right )} + 6 \, d f x{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} - 2 \, e\right )} + 6 \, c f{\rm Ei}\left (-\frac{6 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{6 \, c f}{d} - 6 \, e\right )} - 12 \, c f{\rm Ei}\left (-\frac{4 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{4 \, c f}{d} - 4 \, e\right )} + 6 \, c f{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} - 2 \, e\right )} + 3 \, d e^{\left (-2 \, f x - 2 \, e\right )} - 3 \, d e^{\left (-4 \, f x - 4 \, e\right )} + d e^{\left (-6 \, f x - 6 \, e\right )}}{8 \,{\left (a^{3} d^{3} x + a^{3} c d^{2}\right )}} - \frac{1}{8 \,{\left (d x + c\right )} a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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