Optimal. Leaf size=131 \[ -\frac{i \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f}+\frac{\cosh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\sinh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f} \]
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Rubi [A] time = 0.324126, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {5563, 3303, 3298, 3301, 5448, 12} \[ -\frac{i \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f}+\frac{\cosh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f}+\frac{\sinh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f}-\frac{i \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{2 a f} \]
Antiderivative was successfully verified.
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Rule 5563
Rule 3303
Rule 3298
Rule 3301
Rule 5448
Rule 12
Rubi steps
\begin{align*} \int \frac{\cosh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=-\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*}
Mathematica [A] time = 0.418634, size = 112, normalized size = 0.85 \[ \frac{2 \cosh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (d \left (\frac{e}{f}+x\right )\right )-i \left (\sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d (e+f x)}{f}\right )+2 i \sinh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+\cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d (e+f x)}{f}\right )\right )}{2 a f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 180, normalized size = 1.4 \begin{align*} -{\frac{1}{2\,af}{{\rm e}^{-{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,dx+c-{\frac{cf-de}{f}} \right ) }-{\frac{1}{2\,af}{{\rm e}^{{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-dx-c-{\frac{-cf+de}{f}} \right ) }+{\frac{{\frac{i}{4}}}{af}{{\rm e}^{2\,{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-2\,dx-2\,c-2\,{\frac{-cf+de}{f}} \right ) }-{\frac{{\frac{i}{4}}}{af}{{\rm e}^{-2\,{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,2\,dx+2\,c-2\,{\frac{cf-de}{f}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12151, size = 258, normalized size = 1.97 \begin{align*} \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} \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 = 1.18696, size = 208, normalized size = 1.59 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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