Integrand size = 20, antiderivative size = 159 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=-\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}+\frac {f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^2}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {1}{d (c+d x) (a+a \tanh (e+f x))} \]
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Time = 0.18 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3805, 3384, 3379, 3382} \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=\frac {f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{a d^2}-\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^2}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3805
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{d (c+d x) (a+a \tanh (e+f x))}-\frac {f \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{a d}+\frac {f \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{a d} \\ & = -\frac {1}{d (c+d x) (a+a \tanh (e+f x))}-\frac {\left (f \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}{a d}+\frac {\left (f \cosh \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}{a d}+\frac {\left (f \sinh \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}{a d}-\frac {\left (f \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}{a d} \\ & = -\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}+\frac {f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^2}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^2}-\frac {1}{d (c+d x) (a+a \tanh (e+f x))} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=-\frac {\text {sech}(e+f x) \left (\cosh \left (\frac {c f}{d}\right )+\sinh \left (\frac {c f}{d}\right )\right ) \left (d \left (\cosh \left (e+f \left (-\frac {c}{d}+x\right )\right )+\cosh \left (e+f \left (\frac {c}{d}+x\right )\right )+\sinh \left (e+f \left (-\frac {c}{d}+x\right )\right )-\sinh \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac {f (c+d x)}{d}\right )-\sinh \left (e-\frac {f (c+d x)}{d}\right )\right )+2 f (c+d x) \left (-\cosh \left (e-\frac {f (c+d x)}{d}\right )+\sinh \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 a d^2 (c+d x) (1+\tanh (e+f x))} \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {1}{2 d a \left (d x +c \right )}-\frac {f \,{\mathrm e}^{-2 f x -2 e}}{2 a d \left (d x f +c f \right )}+\frac {f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a \,d^{2}}\) | \(90\) |
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Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=-\frac {{\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left ({\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + d\right )} \cosh \left (f x + e\right ) + {\left ({\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )\right )} \sinh \left (f x + e\right )}{{\left (a d^{3} x + a c d^{2}\right )} \cosh \left (f x + e\right ) + {\left (a d^{3} x + a c d^{2}\right )} \sinh \left (f x + e\right )} \]
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\[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=\frac {\int \frac {1}{c^{2} \tanh {\left (e + f x \right )} + c^{2} + 2 c d x \tanh {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \tanh {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a} \]
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Time = 0.49 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.35 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=-\frac {1}{2 \, {\left (a d^{2} x + a c d\right )}} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )} a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (159) = 318\).
Time = 0.30 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.01 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=-\frac {{\left (2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 2 \, d e f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, c f^{3} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + d f^{2} e^{\left (-\frac {2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} + d f^{2}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} a d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - a d^{5} e + a c d^{4} f\right )} f} \]
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Timed out. \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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