Integrand size = 31, antiderivative size = 231 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d} \]
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Time = 0.18 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {5682, 3377, 2718, 5554, 3392, 32, 2715, 8} \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {6 f^3 \cosh (c+d x)}{a d^4}+\frac {3 i f^3 \sinh (c+d x) \cosh (c+d x)}{8 a d^4}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d} \]
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Rule 8
Rule 32
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 5554
Rule 5682
Rubi steps \begin{align*} \text {integral}& = -\frac {i \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \cosh (c+d x) \, dx}{a} \\ & = \frac {(e+f x)^3 \sinh (c+d x)}{a d}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}+\frac {(3 i f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d} \\ & = -\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac {(3 i f) \int (e+f x)^2 \, dx}{4 a d}+\frac {\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\frac {\left (3 i f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 a d^3} \\ & = -\frac {i (e+f x)^3}{4 a d}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d}-\frac {\left (3 i f^3\right ) \int 1 \, dx}{8 a d^3}-\frac {\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3} \\ & = -\frac {3 i f^3 x}{8 a d^3}-\frac {i (e+f x)^3}{4 a d}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d}+\frac {3 i f^3 \cosh (c+d x) \sinh (c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^2}-\frac {3 i f^2 (e+f x) \sinh ^2(c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \sinh ^2(c+d x)}{2 a d} \\ \end{align*}
Time = 1.86 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.58 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {-96 f \left (2 f^2+d^2 (e+f x)^2\right ) \cosh (c+d x)-4 i d (e+f x) \left (3 f^2+2 d^2 (e+f x)^2\right ) \cosh (2 (c+d x))+4 \left (8 d (e+f x) \left (6 f^2+d^2 (e+f x)^2\right )+3 i f \left (f^2+2 d^2 (e+f x)^2\right ) \cosh (c+d x)\right ) \sinh (c+d x)}{32 a d^4} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (213 ) = 426\).
Time = 34.93 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.86
method | result | size |
risch | \(-\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x -6 d^{2} f^{3} x^{2}+4 e^{3} d^{3}-12 d^{2} e \,f^{2} x -6 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d -3 f^{3}\right ) {\mathrm e}^{2 d x +2 c}}{32 d^{4} a}+\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+e^{3} d^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d -6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}-\frac {\left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+e^{3} d^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d +6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}-\frac {i \left (4 d^{3} x^{3} f^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x +6 d^{2} f^{3} x^{2}+4 e^{3} d^{3}+12 d^{2} e \,f^{2} x +6 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d +3 f^{3}\right ) {\mathrm e}^{-2 d x -2 c}}{32 d^{4} a}\) | \(429\) |
derivativedivides | \(-\frac {3 i d e \,f^{2} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )-\frac {3 i c \,d^{2} e^{2} f \cosh \left (d x +c \right )^{2}}{2}+3 i c^{2} f^{3} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-6 i c d e \,f^{2} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )+i f^{3} \left (\frac {\left (d x +c \right )^{3} \cosh \left (d x +c \right )^{2}}{2}-\frac {3 \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{4}-\frac {3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}\right )+\frac {3 i c^{2} d e \,f^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {i c^{3} f^{3} \cosh \left (d x +c \right )^{2}}{2}-3 i c \,f^{3} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )+3 i d^{2} e^{2} f \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )+\frac {i d^{3} e^{3} \cosh \left (d x +c \right )^{2}}{2}+\sinh \left (d x +c \right ) c^{3} f^{3}-3 \sinh \left (d x +c \right ) c^{2} d e \,f^{2}-3 c^{2} f^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 \sinh \left (d x +c \right ) c \,d^{2} e^{2} f +6 c d e \,f^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 c \,f^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-\sinh \left (d x +c \right ) d^{3} e^{3}-3 d^{2} e^{2} f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-f^{3} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4} a}\) | \(726\) |
default | \(-\frac {3 i d e \,f^{2} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )-\frac {3 i c \,d^{2} e^{2} f \cosh \left (d x +c \right )^{2}}{2}+3 i c^{2} f^{3} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )-6 i c d e \,f^{2} \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )+i f^{3} \left (\frac {\left (d x +c \right )^{3} \cosh \left (d x +c \right )^{2}}{2}-\frac {3 \left (d x +c \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3}}{4}+\frac {3 \left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{4}-\frac {3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{8}-\frac {3 d x}{8}-\frac {3 c}{8}\right )+\frac {3 i c^{2} d e \,f^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {i c^{3} f^{3} \cosh \left (d x +c \right )^{2}}{2}-3 i c \,f^{3} \left (\frac {\left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}+\frac {\cosh \left (d x +c \right )^{2}}{4}\right )+3 i d^{2} e^{2} f \left (\frac {\left (d x +c \right ) \cosh \left (d x +c \right )^{2}}{2}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{4}-\frac {d x}{4}-\frac {c}{4}\right )+\frac {i d^{3} e^{3} \cosh \left (d x +c \right )^{2}}{2}+\sinh \left (d x +c \right ) c^{3} f^{3}-3 \sinh \left (d x +c \right ) c^{2} d e \,f^{2}-3 c^{2} f^{3} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 \sinh \left (d x +c \right ) c \,d^{2} e^{2} f +6 c d e \,f^{2} \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )+3 c \,f^{3} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-\sinh \left (d x +c \right ) d^{3} e^{3}-3 d^{2} e^{2} f \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )-3 d e \,f^{2} \left (\left (d x +c \right )^{2} \sinh \left (d x +c \right )-2 \left (d x +c \right ) \cosh \left (d x +c \right )+2 \sinh \left (d x +c \right )\right )-f^{3} \left (\left (d x +c \right )^{3} \sinh \left (d x +c \right )-3 \left (d x +c \right )^{2} \cosh \left (d x +c \right )+6 \left (d x +c \right ) \sinh \left (d x +c \right )-6 \cosh \left (d x +c \right )\right )}{d^{4} a}\) | \(726\) |
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Time = 0.25 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.75 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {{\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} - 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} - 3 i \, f^{3} - 6 \, {\left (2 i \, d^{3} e f^{2} + i \, d^{2} f^{3}\right )} x^{2} - 6 \, {\left (2 i \, d^{3} e^{2} f + 2 i \, d^{2} e f^{2} + i \, d f^{3}\right )} x + {\left (-4 i \, d^{3} f^{3} x^{3} - 4 i \, d^{3} e^{3} + 6 i \, d^{2} e^{2} f - 6 i \, d e f^{2} + 3 i \, f^{3} - 6 \, {\left (2 i \, d^{3} e f^{2} - i \, d^{2} f^{3}\right )} x^{2} - 6 \, {\left (2 i \, d^{3} e^{2} f - 2 i \, d^{2} e f^{2} + i \, d f^{3}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, {\left (d^{3} f^{3} x^{3} + d^{3} e^{3} - 3 \, d^{2} e^{2} f + 6 \, d e f^{2} - 6 \, f^{3} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - 16 \, {\left (d^{3} f^{3} x^{3} + d^{3} e^{3} + 3 \, d^{2} e^{2} f + 6 \, d e f^{2} + 6 \, f^{3} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x\right )} e^{\left (d x + c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1040 vs. \(2 (214) = 428\).
Time = 0.60 (sec) , antiderivative size = 1040, normalized size of antiderivative = 4.50 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\begin {cases} \frac {\left (\left (- 2048 a^{3} d^{15} e^{3} e^{2 c} - 6144 a^{3} d^{15} e^{2} f x e^{2 c} - 6144 a^{3} d^{15} e f^{2} x^{2} e^{2 c} - 2048 a^{3} d^{15} f^{3} x^{3} e^{2 c} - 6144 a^{3} d^{14} e^{2} f e^{2 c} - 12288 a^{3} d^{14} e f^{2} x e^{2 c} - 6144 a^{3} d^{14} f^{3} x^{2} e^{2 c} - 12288 a^{3} d^{13} e f^{2} e^{2 c} - 12288 a^{3} d^{13} f^{3} x e^{2 c} - 12288 a^{3} d^{12} f^{3} e^{2 c}\right ) e^{- d x} + \left (2048 a^{3} d^{15} e^{3} e^{4 c} + 6144 a^{3} d^{15} e^{2} f x e^{4 c} + 6144 a^{3} d^{15} e f^{2} x^{2} e^{4 c} + 2048 a^{3} d^{15} f^{3} x^{3} e^{4 c} - 6144 a^{3} d^{14} e^{2} f e^{4 c} - 12288 a^{3} d^{14} e f^{2} x e^{4 c} - 6144 a^{3} d^{14} f^{3} x^{2} e^{4 c} + 12288 a^{3} d^{13} e f^{2} e^{4 c} + 12288 a^{3} d^{13} f^{3} x e^{4 c} - 12288 a^{3} d^{12} f^{3} e^{4 c}\right ) e^{d x} + \left (- 512 i a^{3} d^{15} e^{3} e^{c} - 1536 i a^{3} d^{15} e^{2} f x e^{c} - 1536 i a^{3} d^{15} e f^{2} x^{2} e^{c} - 512 i a^{3} d^{15} f^{3} x^{3} e^{c} - 768 i a^{3} d^{14} e^{2} f e^{c} - 1536 i a^{3} d^{14} e f^{2} x e^{c} - 768 i a^{3} d^{14} f^{3} x^{2} e^{c} - 768 i a^{3} d^{13} e f^{2} e^{c} - 768 i a^{3} d^{13} f^{3} x e^{c} - 384 i a^{3} d^{12} f^{3} e^{c}\right ) e^{- 2 d x} + \left (- 512 i a^{3} d^{15} e^{3} e^{5 c} - 1536 i a^{3} d^{15} e^{2} f x e^{5 c} - 1536 i a^{3} d^{15} e f^{2} x^{2} e^{5 c} - 512 i a^{3} d^{15} f^{3} x^{3} e^{5 c} + 768 i a^{3} d^{14} e^{2} f e^{5 c} + 1536 i a^{3} d^{14} e f^{2} x e^{5 c} + 768 i a^{3} d^{14} f^{3} x^{2} e^{5 c} - 768 i a^{3} d^{13} e f^{2} e^{5 c} - 768 i a^{3} d^{13} f^{3} x e^{5 c} + 384 i a^{3} d^{12} f^{3} e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{4096 a^{4} d^{16}} & \text {for}\: a^{4} d^{16} e^{3 c} \neq 0 \\\frac {x^{4} \left (- i f^{3} e^{4 c} + 2 f^{3} e^{3 c} + 2 f^{3} e^{c} + i f^{3}\right ) e^{- 2 c}}{16 a} + \frac {x^{3} \left (- i e f^{2} e^{4 c} + 2 e f^{2} e^{3 c} + 2 e f^{2} e^{c} + i e f^{2}\right ) e^{- 2 c}}{4 a} + \frac {x^{2} \left (- 3 i e^{2} f e^{4 c} + 6 e^{2} f e^{3 c} + 6 e^{2} f e^{c} + 3 i e^{2} f\right ) e^{- 2 c}}{8 a} + \frac {x \left (- i e^{3} e^{4 c} + 2 e^{3} e^{3 c} + 2 e^{3} e^{c} + i e^{3}\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (207) = 414\).
Time = 0.29 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.68 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {{\left (4 i \, d^{3} f^{3} x^{3} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{3} f^{3} x^{3} e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{3} f^{3} x^{3} e^{\left (d x + c\right )} + 4 i \, d^{3} f^{3} x^{3} + 12 i \, d^{3} e f^{2} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, d^{3} e f^{2} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{3} e f^{2} x^{2} e^{\left (d x + c\right )} + 12 i \, d^{3} e f^{2} x^{2} + 12 i \, d^{3} e^{2} f x e^{\left (4 \, d x + 4 \, c\right )} - 6 i \, d^{2} f^{3} x^{2} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, d^{3} e^{2} f x e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{2} f^{3} x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{3} e^{2} f x e^{\left (d x + c\right )} + 48 \, d^{2} f^{3} x^{2} e^{\left (d x + c\right )} + 12 i \, d^{3} e^{2} f x + 6 i \, d^{2} f^{3} x^{2} + 4 i \, d^{3} e^{3} e^{\left (4 \, d x + 4 \, c\right )} - 12 i \, d^{2} e f^{2} x e^{\left (4 \, d x + 4 \, c\right )} - 16 \, d^{3} e^{3} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, d^{2} e f^{2} x e^{\left (3 \, d x + 3 \, c\right )} + 16 \, d^{3} e^{3} e^{\left (d x + c\right )} + 96 \, d^{2} e f^{2} x e^{\left (d x + c\right )} + 4 i \, d^{3} e^{3} + 12 i \, d^{2} e f^{2} x - 6 i \, d^{2} e^{2} f e^{\left (4 \, d x + 4 \, c\right )} + 6 i \, d f^{3} x e^{\left (4 \, d x + 4 \, c\right )} + 48 \, d^{2} e^{2} f e^{\left (3 \, d x + 3 \, c\right )} - 96 \, d f^{3} x e^{\left (3 \, d x + 3 \, c\right )} + 48 \, d^{2} e^{2} f e^{\left (d x + c\right )} + 96 \, d f^{3} x e^{\left (d x + c\right )} + 6 i \, d^{2} e^{2} f + 6 i \, d f^{3} x + 6 i \, d e f^{2} e^{\left (4 \, d x + 4 \, c\right )} - 96 \, d e f^{2} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, d e f^{2} e^{\left (d x + c\right )} + 6 i \, d e f^{2} - 3 i \, f^{3} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, f^{3} e^{\left (3 \, d x + 3 \, c\right )} + 96 \, f^{3} e^{\left (d x + c\right )} + 3 i \, f^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32 \, a d^{4}} \]
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Time = 1.87 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.94 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-{\mathrm {e}}^{c+d\,x}\,\left (\frac {-d^3\,e^3+3\,d^2\,e^2\,f-6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}-\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f-d\,e\right )}{2\,a\,d^2}-\frac {3\,f\,x\,\left (d^2\,e^2-2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right )-{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (4\,d^3\,e^3+6\,d^2\,e^2\,f+6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}+\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}+\frac {f\,x\,\left (2\,d^2\,e^2+2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f+2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (-4\,d^3\,e^3+6\,d^2\,e^2\,f-6\,d\,e\,f^2+3\,f^3\right )\,1{}\mathrm {i}}{32\,a\,d^4}-\frac {f^3\,x^3\,1{}\mathrm {i}}{8\,a\,d}-\frac {f\,x\,\left (2\,d^2\,e^2-2\,d\,e\,f+f^2\right )\,3{}\mathrm {i}}{16\,a\,d^3}+\frac {f^2\,x^2\,\left (f-2\,d\,e\right )\,3{}\mathrm {i}}{16\,a\,d^2}\right )-{\mathrm {e}}^{-c-d\,x}\,\left (\frac {d^3\,e^3+3\,d^2\,e^2\,f+6\,d\,e\,f^2+6\,f^3}{2\,a\,d^4}+\frac {f^3\,x^3}{2\,a\,d}+\frac {3\,f^2\,x^2\,\left (f+d\,e\right )}{2\,a\,d^2}+\frac {3\,f\,x\,\left (d^2\,e^2+2\,d\,e\,f+2\,f^2\right )}{2\,a\,d^3}\right ) \]
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