Integrand size = 31, antiderivative size = 76 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {\log (e+f x)}{a f}-\frac {i \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f} \]
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Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {5682, 31, 3384, 3379, 3382} \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {i \sinh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\log (e+f x)}{a f} \]
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Rule 31
Rule 3379
Rule 3382
Rule 3384
Rule 5682
Rubi steps \begin{align*} \text {integral}& = -\frac {i \int \frac {\sinh (c+d x)}{e+f x} \, dx}{a}+\frac {\int \frac {1}{e+f x} \, dx}{a} \\ & = \frac {\log (e+f x)}{a f}-\frac {\left (i \cosh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\sinh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}-\frac {\left (i \sinh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a} \\ & = \frac {\log (e+f x)}{a f}-\frac {i \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {\log (e+f x)-i \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right ) \sinh \left (c-\frac {d e}{f}\right )-i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )}{a f} \]
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Time = 12.95 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {\ln \left (f x +e \right )}{a f}+\frac {i {\mathrm e}^{\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (-d x -c -\frac {-c f +d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{-\frac {c f -d e}{f}} \operatorname {Ei}_{1}\left (d x +c -\frac {c f -d e}{f}\right )}{2 a f}\) | \(103\) |
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Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\frac {i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} - i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} + 2 \, \log \left (\frac {f x + e}{f}\right )}{2 \, a f} \]
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\[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\cosh ^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]
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Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {i \, e^{\left (-c + \frac {d e}{f}\right )} E_{1}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {i \, e^{\left (c - \frac {d e}{f}\right )} E_{1}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {\log \left (f x + e\right )}{a f} \]
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Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=-\frac {{\left (i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (2 \, c - \frac {d e}{f}\right )} - i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e}{f}\right )} - 2 \, e^{c} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-c\right )}}{2 \, a f} \]
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Timed out. \[ \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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