Optimal. Leaf size=76 \[ -\frac{i \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\log (e+f x)}{a f} \]
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Rubi [A] time = 0.204152, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {5563, 31, 3303, 3298, 3301} \[ -\frac{i \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\log (e+f x)}{a f} \]
Antiderivative was successfully verified.
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Rule 5563
Rule 31
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=-\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*}
Mathematica [A] time = 0.320749, size = 62, normalized size = 0.82 \[ \frac{-i \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (d \left (\frac{e}{f}+x\right )\right )-i \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+\log (e+f x)}{a f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 103, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( fx+e \right ) }{af}}+{\frac{{\frac{i}{2}}}{af}{{\rm e}^{{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-dx-c-{\frac{-cf+de}{f}} \right ) }-{\frac{{\frac{i}{2}}}{af}{{\rm e}^{-{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,dx+c-{\frac{cf-de}{f}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51494, size = 103, normalized size = 1.36 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18993, size = 159, normalized size = 2.09 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18928, size = 109, normalized size = 1.43 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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