Optimal. Leaf size=145 \[ \frac{2 a^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{d}+\frac{a^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{2 d}+\frac{2 a^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d}+\frac{a^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 d}+\frac{3 a^2 \log (c+d x)}{2 d} \]
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Rubi [A] time = 0.341507, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3318, 3312, 3303, 3298, 3301} \[ \frac{2 a^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{d}+\frac{a^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{2 d}+\frac{2 a^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d}+\frac{a^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 d}+\frac{3 a^2 \log (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3312
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(a+a \cosh (e+f x))^2}{c+d x} \, dx &=\left (4 a^2\right ) \int \frac{\sin ^4\left (\frac{1}{2} (i e+\pi )+\frac{i f x}{2}\right )}{c+d x} \, dx\\ &=\left (4 a^2\right ) \int \left (\frac{3}{8 (c+d x)}+\frac{\cosh (e+f x)}{2 (c+d x)}+\frac{\cosh (2 e+2 f x)}{8 (c+d x)}\right ) \, dx\\ &=\frac{3 a^2 \log (c+d x)}{2 d}+\frac{1}{2} a^2 \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx+\left (2 a^2\right ) \int \frac{\cosh (e+f x)}{c+d x} \, dx\\ &=\frac{3 a^2 \log (c+d x)}{2 d}+\frac{1}{2} \left (a^2 \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx+\left (2 a^2 \cosh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx+\frac{1}{2} \left (a^2 \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx+\left (2 a^2 \sinh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac{2 a^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Chi}\left (\frac{c f}{d}+f x\right )}{d}+\frac{a^2 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{2 d}+\frac{3 a^2 \log (c+d x)}{2 d}+\frac{2 a^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d}+\frac{a^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.197252, size = 113, normalized size = 0.78 \[ \frac{a^2 \left (4 \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \cosh \left (e-\frac{c f}{d}\right )+\text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )+4 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+\sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+3 \log (c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 191, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}}{d}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{{a}^{2}}{d}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) }+{\frac{3\,{a}^{2}\ln \left ( dx+c \right ) }{2\,d}}-{\frac{{a}^{2}}{4\,d}{{\rm e}^{2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) }-{\frac{{a}^{2}}{4\,d}{{\rm e}^{-2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-2\,fx-2\,e-2\,{\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30309, size = 201, normalized size = 1.39 \begin{align*} -\frac{1}{4} \, a^{2}{\left (\frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{1}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{d} + \frac{e^{\left (2 \, e - \frac{2 \, c f}{d}\right )} E_{1}\left (-\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{d} - \frac{2 \, \log \left (d x + c\right )}{d}\right )} - a^{2}{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{d} + \frac{e^{\left (e - \frac{c f}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac{a^{2} \log \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14107, size = 470, normalized size = 3.24 \begin{align*} \frac{6 \, a^{2} \log \left (d x + c\right ) + 4 \,{\left (a^{2}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) + a^{2}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \cosh \left (-\frac{d e - c f}{d}\right ) +{\left (a^{2}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) + a^{2}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) - 4 \,{\left (a^{2}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) - a^{2}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{d}\right ) -{\left (a^{2}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) - a^{2}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int \frac{2 \cosh{\left (e + f x \right )}}{c + d x}\, dx + \int \frac{\cosh ^{2}{\left (e + f x \right )}}{c + d x}\, dx + \int \frac{1}{c + d x}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28038, size = 188, normalized size = 1.3 \begin{align*} \frac{a^{2}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} - 2 \, e\right )} + 4 \, a^{2}{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} + 4 \, a^{2}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} + a^{2}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac{2 \, c f}{d} + 2 \, e\right )} + 6 \, a^{2} \log \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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