Optimal. Leaf size=157 \[ -\frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}+\frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}+\frac{\sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}-\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\log (c+d x)}{2 a d} \]
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Rubi [A] time = 0.294951, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3726, 3303, 3298, 3301} \[ -\frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}+\frac{\text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}+\frac{\sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}-\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{2 a d}+\frac{\log (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3726
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+a \tanh (e+f x))} \, dx &=\frac{\log (c+d x)}{2 a d}+\frac{\int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{2 a}-\frac{\int \frac{\sinh (2 e+2 f x)}{c+d x} \, dx}{2 a}\\ &=\frac{\log (c+d x)}{2 a d}+\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}-\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}-\frac{\sinh \left (2 e-\frac{2 c f}{d}\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}+\frac{\sinh \left (2 e-\frac{2 c f}{d}\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{2 a}\\ &=\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{2 a d}+\frac{\log (c+d x)}{2 a d}-\frac{\text{Chi}\left (\frac{2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{2 a d}-\frac{\cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{2 a d}+\frac{\sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{2 a d}\\ \end{align*}
Mathematica [A] time = 0.248775, size = 122, normalized size = 0.78 \[ \frac{\text{sech}(e+f x) (\sinh (f x)+\cosh (f x)) \left (\text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac{2 c f}{d}\right )-\sinh \left (e-\frac{2 c f}{d}\right )\right )+\text{Shi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\sinh \left (e-\frac{2 c f}{d}\right )-\cosh \left (e-\frac{2 c f}{d}\right )\right )+(\sinh (e)+\cosh (e)) \log (f (c+d x))\right )}{2 a d (\tanh (e+f x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.157, size = 61, normalized size = 0.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{2\,da}}-{\frac{1}{2\,da}{{\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 = 1.74156, size = 65, normalized size = 0.41 \begin{align*} -\frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{1}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{2 \, a d} + \frac{\log \left (d x + c\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18156, size = 165, normalized size = 1.05 \begin{align*} \frac{{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) +{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + \log \left (d x + c\right )}{2 \, a 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*} \frac{\int \frac{1}{c \tanh{\left (e + f x \right )} + c + d x \tanh{\left (e + f x \right )} + d x}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20022, size = 66, normalized size = 0.42 \begin{align*} \frac{{\left ({\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d}\right )} + e^{\left (2 \, e\right )} \log \left (d x + c\right )\right )} e^{\left (-2 \, e\right )}}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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