Integrand size = 29, antiderivative size = 175 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {3 i e x}{2 a}+\frac {3 i f x^2}{4 a}+\frac {(e+f x) \cosh (c+d x)}{a d}+\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \sinh (c+d x)}{a d^2}-\frac {i (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
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Time = 0.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5676, 3391, 3377, 2717, 3399, 4269, 3556} \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i f \sinh ^2(c+d x)}{4 a d^2}-\frac {f \sinh (c+d x)}{a d^2}+\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{a d^2}+\frac {(e+f x) \cosh (c+d x)}{a d}-\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {3 i e x}{2 a}+\frac {3 i f x^2}{4 a} \]
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Rule 2717
Rule 3377
Rule 3391
Rule 3399
Rule 3556
Rule 4269
Rule 5676
Rubi steps \begin{align*} \text {integral}& = i \int \frac {(e+f x) \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x) \sinh ^2(c+d x) \, dx}{a} \\ & = -\frac {i (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f \sinh ^2(c+d x)}{4 a d^2}+\frac {i \int (e+f x) \, dx}{2 a}+\frac {\int (e+f x) \sinh (c+d x) \, dx}{a}-\int \frac {(e+f x) \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {i e x}{2 a}+\frac {i f x^2}{4 a}+\frac {(e+f x) \cosh (c+d x)}{a d}-\frac {i (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f \sinh ^2(c+d x)}{4 a d^2}-i \int \frac {e+f x}{a+i a \sinh (c+d x)} \, dx+\frac {i \int (e+f x) \, dx}{a}-\frac {f \int \cosh (c+d x) \, dx}{a d} \\ & = \frac {3 i e x}{2 a}+\frac {3 i f x^2}{4 a}+\frac {(e+f x) \cosh (c+d x)}{a d}-\frac {f \sinh (c+d x)}{a d^2}-\frac {i (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f \sinh ^2(c+d x)}{4 a d^2}-\frac {i \int (e+f x) \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a} \\ & = \frac {3 i e x}{2 a}+\frac {3 i f x^2}{4 a}+\frac {(e+f x) \cosh (c+d x)}{a d}-\frac {f \sinh (c+d x)}{a d^2}-\frac {i (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(i f) \int \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d} \\ & = \frac {3 i e x}{2 a}+\frac {3 i f x^2}{4 a}+\frac {(e+f x) \cosh (c+d x)}{a d}+\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \sinh (c+d x)}{a d^2}-\frac {i (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \\ \end{align*}
Time = 3.20 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.86 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right ) \left (-8 i d (e+f x) \cosh (c+d x)+f \cosh (2 (c+d x))+2 \left (6 c d e-4 i c f-3 c^2 f+6 d^2 e x-4 i d f x+3 d^2 f x^2+8 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 f \log (\cosh (c+d x))+4 i f \sinh (c+d x)-d (e+f x) \sinh (2 (c+d x))\right )\right )+\sinh \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cosh (c+d x)+i \left (f \cosh (2 (c+d x))+2 \left (8 i d e+6 c d e-4 i c f-3 c^2 f+6 d^2 e x+4 i d f x+3 d^2 f x^2+8 i f \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+4 f \log (\cosh (c+d x))+4 i f \sinh (c+d x)-d (e+f x) \sinh (2 (c+d x))\right )\right )\right )\right )}{8 a d^2 (-i+\sinh (c+d x))} \]
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Time = 2.03 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {3 i f \,x^{2}}{4 a}+\frac {3 i e x}{2 a}-\frac {i \left (2 d f x +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{2}}+\frac {\left (d f x +d e -f \right ) {\mathrm e}^{d x +c}}{2 a \,d^{2}}+\frac {\left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{2 a \,d^{2}}+\frac {i \left (2 d f x +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{2}}-\frac {2 i f x}{a d}-\frac {2 i f c}{a \,d^{2}}+\frac {2 f x +2 e}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {2 i f \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{2}}\) | \(197\) |
parallelrisch | \(\frac {32 f \left (i \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )+\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (1-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 f \left (i \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )+\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\left (-6-12 d^{2} x^{2}+\left (16+24 i\right ) x d \right ) f +56 e \left (-\frac {3 d x}{7}+i\right ) d \right ) \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (-6 i-12 i d^{2} x^{2}+\left (24+16 i\right ) x d \right ) f -24 i d^{2} e x -8 d e \right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (6 i d f x +6 i d e +7 f \right ) \cosh \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\left (2 i d f x +2 i d e -f \right ) \cosh \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\left (\left (-6 d x -7 i\right ) f -6 d e \right ) \sinh \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-\left (\left (-2 d x +i\right ) f -2 d e \right ) \sinh \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{16 \left (i \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) d^{2} a}\) | \(295\) |
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Time = 0.25 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.31 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 \, d f x + 2 \, d e + {\left (-2 i \, d f x - 2 i \, d e + i \, f\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (6 \, d f x + 6 \, d e - 7 \, f\right )} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, {\left (-3 i \, d^{2} f x^{2} + 2 i \, d e + 2 \, {\left (-3 i \, d^{2} e + 5 i \, d f\right )} x - 2 i \, f\right )} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, {\left (3 \, d^{2} f x^{2} + 10 \, d e + 2 \, {\left (3 \, d^{2} e + d f\right )} x + 2 \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-6 i \, d f x - 6 i \, d e - 7 i \, f\right )} e^{\left (d x + c\right )} - 32 \, {\left (-i \, f e^{\left (3 \, d x + 3 \, c\right )} - f e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + f}{16 \, {\left (a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \]
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Time = 0.41 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.26 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 e + 2 f x}{a d e^{c} e^{d x} - i a d} + \begin {cases} \frac {\left (\left (512 a^{3} d^{7} e e^{2 c} + 512 a^{3} d^{7} f x e^{2 c} + 512 a^{3} d^{6} f e^{2 c}\right ) e^{- d x} + \left (512 a^{3} d^{7} e e^{4 c} + 512 a^{3} d^{7} f x e^{4 c} - 512 a^{3} d^{6} f e^{4 c}\right ) e^{d x} + \left (128 i a^{3} d^{7} e e^{c} + 128 i a^{3} d^{7} f x e^{c} + 64 i a^{3} d^{6} f e^{c}\right ) e^{- 2 d x} + \left (- 128 i a^{3} d^{7} e e^{5 c} - 128 i a^{3} d^{7} f x e^{5 c} + 64 i a^{3} d^{6} f e^{5 c}\right ) e^{2 d x}\right ) e^{- 3 c}}{1024 a^{4} d^{8}} & \text {for}\: a^{4} d^{8} e^{3 c} \neq 0 \\\frac {x^{2} \left (- i f e^{4 c} + 2 f e^{3 c} - 2 f e^{c} - i f\right ) e^{- 2 c}}{8 a} + \frac {x \left (- i e e^{4 c} + 2 e e^{3 c} - 2 e e^{c} - i e\right ) e^{- 2 c}}{4 a} & \text {otherwise} \end {cases} + \frac {3 i f x^{2}}{4 a} + \frac {x \left (3 i d e - 4 i f\right )}{2 a d} + \frac {2 i f \log {\left (e^{d x} - i e^{- c} \right )}}{a d^{2}} \]
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Exception generated. \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (145) = 290\).
Time = 0.28 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.95 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {12 i \, d^{2} f x^{2} e^{\left (3 \, d x + 3 \, c\right )} + 12 \, d^{2} f x^{2} e^{\left (2 \, d x + 2 \, c\right )} + 24 i \, d^{2} e x e^{\left (3 \, d x + 3 \, c\right )} + 24 \, d^{2} e x e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, d f x e^{\left (5 \, d x + 5 \, c\right )} + 6 \, d f x e^{\left (4 \, d x + 4 \, c\right )} - 40 i \, d f x e^{\left (3 \, d x + 3 \, c\right )} + 8 \, d f x e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, d f x e^{\left (d x + c\right )} + 2 \, d f x - 2 i \, d e e^{\left (5 \, d x + 5 \, c\right )} + 6 \, d e e^{\left (4 \, d x + 4 \, c\right )} - 8 i \, d e e^{\left (3 \, d x + 3 \, c\right )} + 40 \, d e e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, d e e^{\left (d x + c\right )} + 32 i \, f e^{\left (3 \, d x + 3 \, c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 32 \, f e^{\left (2 \, d x + 2 \, c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 2 \, d e + i \, f e^{\left (5 \, d x + 5 \, c\right )} - 7 \, f e^{\left (4 \, d x + 4 \, c\right )} + 8 i \, f e^{\left (3 \, d x + 3 \, c\right )} + 8 \, f e^{\left (2 \, d x + 2 \, c\right )} - 7 i \, f e^{\left (d x + c\right )} + f}{16 \, {\left (a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \]
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Time = 1.46 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x) \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx={\mathrm {e}}^{-c-d\,x}\,\left (\frac {f+d\,e}{2\,a\,d^2}+\frac {f\,x}{2\,a\,d}\right )+{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (\frac {\left (f+2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}+\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (\frac {\left (f-2\,d\,e\right )\,1{}\mathrm {i}}{16\,a\,d^2}-\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )-{\mathrm {e}}^{c+d\,x}\,\left (\frac {f-d\,e}{2\,a\,d^2}-\frac {f\,x}{2\,a\,d}\right )+\frac {f\,x^2\,3{}\mathrm {i}}{4\,a}+\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}-\frac {x\,\left (4\,f-3\,d\,e\right )\,1{}\mathrm {i}}{2\,a\,d}+\frac {f\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-\mathrm {i}\right )\,2{}\mathrm {i}}{a\,d^2} \]
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