Integrand size = 20, antiderivative size = 297 \[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {\text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d} \]
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Time = 0.58 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3809, 3384, 3379, 3382, 3393} \[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=-\frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {\text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 3809
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4 a^2 (c+d x)}+\frac {\cosh (2 e+2 f x)}{2 a^2 (c+d x)}+\frac {\cosh ^2(2 e+2 f x)}{4 a^2 (c+d x)}-\frac {\sinh (2 e+2 f x)}{2 a^2 (c+d x)}+\frac {\sinh ^2(2 e+2 f x)}{4 a^2 (c+d x)}-\frac {\sinh (4 e+4 f x)}{4 a^2 (c+d x)}\right ) \, dx \\ & = \frac {\log (c+d x)}{4 a^2 d}+\frac {\int \frac {\cosh ^2(2 e+2 f x)}{c+d x} \, dx}{4 a^2}+\frac {\int \frac {\sinh ^2(2 e+2 f x)}{c+d x} \, dx}{4 a^2}-\frac {\int \frac {\sinh (4 e+4 f x)}{c+d x} \, dx}{4 a^2}+\frac {\int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{2 a^2}-\frac {\int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{2 a^2} \\ & = \frac {\log (c+d x)}{4 a^2 d}-\frac {\int \left (\frac {1}{2 (c+d x)}-\frac {\cosh (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{4 a^2}+\frac {\int \left (\frac {1}{2 (c+d x)}+\frac {\cosh (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{4 a^2}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^2}+\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^2}-\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^2}-\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{4 a^2}-\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^2}+\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^2} \\ & = \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^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 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^2 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^2 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^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \frac {\int \frac {\cosh (4 e+4 f x)}{c+d x} \, dx}{8 a^2} \\ & = \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^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 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^2 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^2 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^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \left (\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^2}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^2}\right ) \\ & = \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^2 d}+\frac {\log (c+d x)}{4 a^2 d}-\frac {\text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 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^2 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^2 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^2 d}-\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+2 \left (\frac {\cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^2 d}+\frac {\sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^2 d}\right ) \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=\frac {\left (\cosh \left (2 e-\frac {2 c f}{d}\right )-\sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \left (2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right )+\cosh \left (2 e-\frac {2 c f}{d}\right ) \log (f (c+d x))+\text {Chi}\left (\frac {4 f (c+d x)}{d}\right ) \left (\cosh \left (2 e-\frac {2 c f}{d}\right )-\sinh \left (2 e-\frac {2 c f}{d}\right )\right )+\log (f (c+d x)) \sinh \left (2 e-\frac {2 c f}{d}\right )-2 \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )+\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )\right )}{4 a^2 d} \]
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Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.36
method | result | size |
risch | \(\frac {\ln \left (d x +c \right )}{4 a^{2} d}-\frac {{\mathrm e}^{\frac {4 c f -4 d e}{d}} \operatorname {Ei}_{1}\left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{4 a^{2} d}-\frac {{\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 a^{2} d}\) | \(106\) |
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Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=\frac {2 \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + \log \left (d x + c\right )}{4 \, a^{2} d} \]
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\[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=\frac {\int \frac {1}{c \tanh ^{2}{\left (e + f x \right )} + 2 c \tanh {\left (e + f x \right )} + c + d x \tanh ^{2}{\left (e + f x \right )} + 2 d x \tanh {\left (e + f x \right )} + d x}\, dx}{a^{2}} \]
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Time = 0.71 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.27 \[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=-\frac {e^{\left (-4 \, e + \frac {4 \, c f}{d}\right )} E_{1}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{4 \, a^{2} d} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, a^{2} d} + \frac {\log \left (d x + c\right )}{4 \, a^{2} d} \]
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Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.25 \[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=\frac {{\left (2 \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e + \frac {2 \, c f}{d}\right )} + {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {4 \, c f}{d}\right )} + e^{\left (4 \, e\right )} \log \left (d x + c\right )\right )} e^{\left (-4 \, e\right )}}{4 \, a^{2} d} \]
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Timed out. \[ \int \frac {1}{(c+d x) (a+a \tanh (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right )} \,d x \]
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