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 = 0.29, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 3298
Rule 3301
Rule 3303
Rule 3726
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*}
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Mathematica [A] time = 0.27, 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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fricas [A] time = 0.42, size = 73, normalized size = 0.46 \[ \frac {{\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + \log \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 49, normalized size = 0.31 \[ \frac {{\left ({\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + e^{\left (2 \, e\right )} \log \left (d x + c\right )\right )} e^{\left (-2 \, e\right )}}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 61, normalized size = 0.39 \[ \frac {\ln \left (d x +c \right )}{2 d a}-\frac {{\mathrm e}^{\frac {2 c f -2 d e}{d}} \Ei \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 48, normalized size = 0.31 \[ -\frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, a d} + \frac {\log \left (d x + c\right )}{2 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{c \tanh {\left (e + f x \right )} + c + d x \tanh {\left (e + f x \right )} + d x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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