Integrand size = 20, antiderivative size = 211 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=-\frac {f}{2 a d^2 (c+d x)}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}+\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+a \tanh (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \tanh (e+f x))} \]
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Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3806, 3805, 3384, 3379, 3382} \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=-\frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a d^3}+\frac {f}{d^2 (c+d x) (a \tanh (e+f x)+a)}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3805
Rule 3806
Rubi steps \begin{align*} \text {integral}& = -\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \tanh (e+f x))}-\frac {f \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))} \, dx}{d} \\ & = -\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \tanh (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \tanh (e+f x))}+\frac {f^2 \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{a d^2}-\frac {f^2 \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{a d^2} \\ & = -\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+a \tanh (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \tanh (e+f x))}+\frac {\left (f^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}{a d^2}-\frac {\left (f^2 \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^2}-\frac {\left (f^2 \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^2}+\frac {\left (f^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}{a d^2} \\ & = -\frac {f}{2 a d^2 (c+d x)}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}+\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+a \tanh (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \tanh (e+f x))} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=-\frac {\text {sech}(e+f x) \left (\cosh \left (\frac {c f}{d}\right )+\sinh \left (\frac {c f}{d}\right )\right ) \left (d \left (d \cosh \left (e+f \left (-\frac {c}{d}+x\right )\right )+(d-2 c f-2 d f x) \cosh \left (e+f \left (\frac {c}{d}+x\right )\right )+d \sinh \left (e+f \left (-\frac {c}{d}+x\right )\right )-d \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )+2 c f \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )+2 d f x \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+4 f^2 (c+d x)^2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (-\cosh \left (e-\frac {f (c+d x)}{d}\right )+\sinh \left (e-\frac {f (c+d x)}{d}\right )\right )+4 f^2 (c+d x)^2 \left (\cosh \left (e-\frac {f (c+d x)}{d}\right )-\sinh \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{4 a d^3 (c+d x)^2 (1+\tanh (e+f x))} \]
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Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {1}{4 d a \left (d x +c \right )^{2}}+\frac {f^{3} {\mathrm e}^{-2 f x -2 e} x}{2 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} {\mathrm e}^{-2 f x -2 e} c}{2 a \,d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} {\mathrm e}^{-2 f x -2 e}}{4 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a \,d^{3}}\) | \(211\) |
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Time = 0.25 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\frac {2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (d^{2} f x + c d f + 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - d^{2}\right )} \cosh \left (f x + e\right ) - {\left (d^{2} f x + c d f - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )\right )} \sinh \left (f x + e\right )}{2 \, {\left ({\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \cosh \left (f x + e\right ) + {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \sinh \left (f x + e\right )\right )}} \]
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\[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\frac {\int \frac {1}{c^{3} \tanh {\left (e + f x \right )} + c^{3} + 3 c^{2} d x \tanh {\left (e + f x \right )} + 3 c^{2} d x + 3 c d^{2} x^{2} \tanh {\left (e + f x \right )} + 3 c d^{2} x^{2} + d^{3} x^{3} \tanh {\left (e + f x \right )} + d^{3} x^{3}}\, dx}{a} \]
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Time = 0.65 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.32 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=-\frac {1}{4 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )}^{2} a d} \]
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Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\frac {4 \, d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 8 \, c d f^{2} x {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 4 \, c^{2} f^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 2 \, d^{2} f x e^{\left (-2 \, f x\right )} + 2 \, c d f e^{\left (-2 \, f x\right )} - d^{2} e^{\left (-2 \, f x\right )} - d^{2} e^{\left (2 \, e\right )}}{4 \, {\left (a d^{5} x^{2} e^{\left (2 \, e\right )} + 2 \, a c d^{4} x e^{\left (2 \, e\right )} + a c^{2} d^{3} e^{\left (2 \, e\right )}\right )}} \]
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Timed out. \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3} \,d x \]
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