Integrand size = 31, antiderivative size = 131 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \text {Chi}\left (\frac {2 d e}{f}+2 d x\right ) \sinh \left (2 c-\frac {2 d e}{f}\right )}{2 a f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f} \]
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Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {5682, 3384, 3379, 3382, 5556, 12} \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {i \sinh \left (2 c-\frac {2 d e}{f}\right ) \text {Chi}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f}+\frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f} \]
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5682
Rubi steps \begin{align*} \text {integral}& = -\frac {i \int \frac {\cosh (c+d x) \sinh (c+d x)}{e+f x} \, dx}{a}+\frac {\int \frac {\cosh (c+d x)}{e+f x} \, dx}{a} \\ & = -\frac {i \int \frac {\sinh (2 c+2 d x)}{2 (e+f x)} \, dx}{a}+\frac {\cosh \left (c-\frac {d e}{f}\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \int \frac {\sinh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a} \\ & = \frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \int \frac {\sinh (2 c+2 d x)}{e+f x} \, dx}{2 a} \\ & = \frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {\left (i \cosh \left (2 c-\frac {2 d e}{f}\right )\right ) \int \frac {\sinh \left (\frac {2 d e}{f}+2 d x\right )}{e+f x} \, dx}{2 a}-\frac {\left (i \sinh \left (2 c-\frac {2 d e}{f}\right )\right ) \int \frac {\cosh \left (\frac {2 d e}{f}+2 d x\right )}{e+f x} \, dx}{2 a} \\ & = \frac {\cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \text {Chi}\left (\frac {2 d e}{f}+2 d x\right ) \sinh \left (2 c-\frac {2 d e}{f}\right )}{2 a f}+\frac {\sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d e}{f}+2 d x\right )}{2 a f} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {2 \cosh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right )-i \left (\text {Chi}\left (\frac {2 d (e+f x)}{f}\right ) \sinh \left (2 c-\frac {2 d e}{f}\right )+2 i \sinh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )+\cosh \left (2 c-\frac {2 d e}{f}\right ) \text {Shi}\left (\frac {2 d (e+f x)}{f}\right )\right )}{2 a f} \]
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Time = 33.33 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.37
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (d x +c -\frac {c f -d e}{f}\right )}{2 a f}-\frac {{\mathrm e}^{\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (-d x -c -\frac {-c f +d e}{f}\right )}{2 a f}+\frac {i {\mathrm e}^{\frac {2 c f -2 d e}{f}} \operatorname {Ei}_{1}\left (-2 d x -2 c -\frac {2 \left (-c f +d e \right )}{f}\right )}{4 a f}-\frac {i {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{f}} \operatorname {Ei}_{1}\left (2 d x +2 c -\frac {2 \left (c f -d e \right )}{f}\right )}{4 a f}\) | \(180\) |
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Time = 0.24 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.97 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {i \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{f}\right )} + 2 \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} + 2 \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} - i \, {\rm Ei}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{f}\right )}}{4 \, a f} \]
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\[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\cosh ^{3}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]
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Exception generated. \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {{\left (i \, {\rm Ei}\left (\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (4 \, c - \frac {2 \, d e}{f}\right )} - 2 \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (3 \, c - \frac {d e}{f}\right )} - 2 \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (c + \frac {d e}{f}\right )} - i \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + d e\right )}}{f}\right ) e^{\left (\frac {2 \, d e}{f}\right )} + 3 i \, e^{\left (2 \, c\right )} \log \left (f x + e\right ) - 3 i \, e^{\left (2 \, c\right )} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-2 \, c\right )}}{4 \, a f} \]
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Timed out. \[ \int \frac {\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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