Optimal. Leaf size=180 \[ \frac{d \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f^2}-\frac{i d \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}-\frac{i d \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}+\frac{d \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f^2}+\frac{i \sinh (2 c+2 d x)}{2 a f (e+f x)}-\frac{\cosh (c+d x)}{a f (e+f x)} \]
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Rubi [A] time = 0.391807, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {5563, 3297, 3303, 3298, 3301, 5448, 12} \[ \frac{d \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (\frac{d e}{f}+d x\right )}{a f^2}-\frac{i d \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}-\frac{i d \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}+\frac{d \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f^2}+\frac{i \sinh (2 c+2 d x)}{2 a f (e+f x)}-\frac{\cosh (c+d x)}{a f (e+f x)} \]
Antiderivative was successfully verified.
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Rule 5563
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 5448
Rule 12
Rubi steps
\begin{align*} \int \frac{\cosh ^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx &=-\frac{i \int \frac{\cosh (c+d x) \sinh (c+d x)}{(e+f x)^2} \, dx}{a}+\frac{\int \frac{\cosh (c+d x)}{(e+f x)^2} \, dx}{a}\\ &=-\frac{\cosh (c+d x)}{a f (e+f x)}-\frac{i \int \frac{\sinh (2 c+2 d x)}{2 (e+f x)^2} \, dx}{a}+\frac{d \int \frac{\sinh (c+d x)}{e+f x} \, dx}{a f}\\ &=-\frac{\cosh (c+d x)}{a f (e+f x)}-\frac{i \int \frac{\sinh (2 c+2 d x)}{(e+f x)^2} \, dx}{2 a}+\frac{\left (d \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 f}+\frac{\left (d \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 f}\\ &=-\frac{\cosh (c+d x)}{a f (e+f x)}+\frac{d \text{Chi}\left (\frac{d e}{f}+d x\right ) \sinh \left (c-\frac{d e}{f}\right )}{a f^2}+\frac{i \sinh (2 c+2 d x)}{2 a f (e+f x)}+\frac{d \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f^2}-\frac{(i d) \int \frac{\cosh (2 c+2 d x)}{e+f x} \, dx}{a f}\\ &=-\frac{\cosh (c+d x)}{a f (e+f x)}+\frac{d \text{Chi}\left (\frac{d e}{f}+d x\right ) \sinh \left (c-\frac{d e}{f}\right )}{a f^2}+\frac{i \sinh (2 c+2 d x)}{2 a f (e+f x)}+\frac{d \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f^2}-\frac{\left (i d \cosh \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}{a f}-\frac{\left (i d \sinh \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}{a f}\\ &=-\frac{\cosh (c+d x)}{a f (e+f x)}-\frac{i d \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}+\frac{d \text{Chi}\left (\frac{d e}{f}+d x\right ) \sinh \left (c-\frac{d e}{f}\right )}{a f^2}+\frac{i \sinh (2 c+2 d x)}{2 a f (e+f x)}+\frac{d \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (\frac{d e}{f}+d x\right )}{a f^2}-\frac{i d \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d e}{f}+2 d x\right )}{a f^2}\\ \end{align*}
Mathematica [A] time = 0.712969, size = 212, normalized size = 1.18 \[ \frac{2 d (e+f x) \sinh \left (c-\frac{d e}{f}\right ) \text{Chi}\left (d \left (\frac{e}{f}+x\right )\right )-2 i d (e+f x) \cosh \left (2 c-\frac{2 d e}{f}\right ) \text{Chi}\left (\frac{2 d (e+f x)}{f}\right )-2 i d e \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d (e+f x)}{f}\right )-2 i d f x \sinh \left (2 c-\frac{2 d e}{f}\right ) \text{Shi}\left (\frac{2 d (e+f x)}{f}\right )+2 d e \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+2 d f x \cosh \left (c-\frac{d e}{f}\right ) \text{Shi}\left (d \left (\frac{e}{f}+x\right )\right )+i f \sinh (2 (c+d x))-2 f \cosh (c+d x)}{2 a f^2 (e+f x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 299, normalized size = 1.7 \begin{align*} -{\frac{d{{\rm e}^{-dx-c}}}{2\,af \left ( dfx+de \right ) }}+{\frac{d}{2\,a{f}^{2}}{{\rm e}^{-{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,dx+c-{\frac{cf-de}{f}} \right ) }-{\frac{d{{\rm e}^{dx+c}}}{2\,a{f}^{2}} \left ({\frac{de}{f}}+dx \right ) ^{-1}}-{\frac{d}{2\,a{f}^{2}}{{\rm e}^{{\frac{cf-de}{f}}}}{\it Ei} \left ( 1,-dx-c-{\frac{-cf+de}{f}} \right ) }+{\frac{{\frac{i}{4}}d{{\rm e}^{2\,dx+2\,c}}}{a{f}^{2}} \left ({\frac{de}{f}}+dx \right ) ^{-1}}+{\frac{{\frac{i}{2}}d}{a{f}^{2}}{{\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}}d{{\rm e}^{-2\,dx-2\,c}}}{af \left ( dfx+de \right ) }}+{\frac{{\frac{i}{2}}d}{a{f}^{2}}{{\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 = 2.21271, size = 510, normalized size = 2.83 \begin{align*} \frac{{\left (i \, f e^{\left (4 \, d x + 4 \, c\right )} - 2 \, f e^{\left (3 \, d x + 3 \, c\right )} +{\left ({\left (-2 i \, d f x - 2 i \, d e\right )}{\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 \,{\left (d f x + d e\right )}{\rm Ei}\left (-\frac{d f x + d e}{f}\right ) e^{\left (\frac{d e - c f}{f}\right )} + 2 \,{\left (d f x + d e\right )}{\rm Ei}\left (\frac{d f x + d e}{f}\right ) e^{\left (-\frac{d e - c f}{f}\right )} +{\left (-2 i \, d f x - 2 i \, d e\right )}{\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 )}\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, f e^{\left (d x + c\right )} - i \, f\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{4 \,{\left (a f^{3} x + a e f^{2}\right )}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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