Integrand size = 20, antiderivative size = 1024 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=-\frac {b d^2 f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {b^2 d^2 f \log \left (f+\frac {d e-c f}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {b^2 d^2 f^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {b^2 d^2 f^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \]
[Out]
Time = 1.62 (sec) , antiderivative size = 1024, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {6457, 5577, 4276, 3403, 2296, 2221, 2317, 2438, 3405, 2747, 31} \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2 d^2}{2 f (d e-c f)^2}-\frac {b f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right ) d^2}{(d e-c f) \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}-\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right ) d^2}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+\frac {b f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}+\frac {2 b \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right ) d^2}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}-\frac {b f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+1\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}+\frac {b^2 f \log \left (f+\frac {d e-c f}{c+d x}\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right ) d^2}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}+\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}+\frac {2 b^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right ) d^2}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right ) d^2}{(d e-c f)^2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )^{3/2}}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2} \]
[In]
[Out]
Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
Rule 4276
Rule 5577
Rule 6457
Rubi steps \begin{align*} \text {integral}& = -\left (d^2 \text {Subst}\left (\int \frac {(a+b x)^2 \coth (x) \text {csch}(x)}{(d e-c f+f \text {csch}(x))^3} \, dx,x,\text {csch}^{-1}(c+d x)\right )\right ) \\ & = -\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {a+b x}{(d e-c f+f \text {csch}(x))^2} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{f} \\ & = -\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \left (\frac {a+b x}{(d e-c f)^2}+\frac {2 f (a+b x)}{(d e-c f)^2 \left (-f-d e \left (1-\frac {c f}{d e}\right ) \sinh (x)\right )}+\frac {f^2 (a+b x)}{(d e-c f)^2 \left (f+d e \left (1-\frac {c f}{d e}\right ) \sinh (x)\right )^2}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{f} \\ & = \frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {\left (2 b d^2\right ) \text {Subst}\left (\int \frac {a+b x}{-f-d e \left (1-\frac {c f}{d e}\right ) \sinh (x)} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2}+\frac {\left (b d^2 f\right ) \text {Subst}\left (\int \frac {a+b x}{\left (f+d e \left (1-\frac {c f}{d e}\right ) \sinh (x)\right )^2} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2} \\ & = -\frac {b d^2 f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {\left (4 b d^2\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{-2 e^x f+d e \left (1-\frac {c f}{d e}\right )-d e e^{2 x} \left (1-\frac {c f}{d e}\right )} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2}+\frac {\left (b d^2 f^2\right ) \text {Subst}\left (\int \frac {a+b x}{f+d e \left (1-\frac {c f}{d e}\right ) \sinh (x)} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (b^2 d^2 f\right ) \text {Subst}\left (\int \frac {\cosh (x)}{f+d e \left (1-\frac {c f}{d e}\right ) \sinh (x)} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ & = -\frac {b d^2 f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {\left (b^2 d^2 f\right ) \text {Subst}\left (\int \frac {1}{f+x} \, dx,x,\frac {d e \left (1-\frac {c f}{d e}\right )}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (2 b d^2 f^2\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 e^x f-d e \left (1-\frac {c f}{d e}\right )+d e e^{2 x} \left (1-\frac {c f}{d e}\right )} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (4 b d^2\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{-2 f-2 d e e^x \left (1-\frac {c f}{d e}\right )-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {\left (4 b d^2\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{-2 f-2 d e e^x \left (1-\frac {c f}{d e}\right )+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \\ & = -\frac {b d^2 f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}-\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {b^2 d^2 f \log \left (\frac {e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {\left (2 b d^2 f^2\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac {c f}{d e}\right )-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {\left (2 b d^2 f^2\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac {c f}{d e}\right )+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {\left (2 b^2 d^2\right ) \text {Subst}\left (\int \log \left (1-\frac {2 d e e^x \left (1-\frac {c f}{d e}\right )}{-2 f-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {\left (2 b^2 d^2\right ) \text {Subst}\left (\int \log \left (1-\frac {2 d e e^x \left (1-\frac {c f}{d e}\right )}{-2 f+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \\ & = -\frac {b d^2 f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {b^2 d^2 f \log \left (\frac {e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (b^2 d^2 f^2\right ) \text {Subst}\left (\int \log \left (1+\frac {2 d e e^x \left (1-\frac {c f}{d e}\right )}{2 f-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {\left (b^2 d^2 f^2\right ) \text {Subst}\left (\int \log \left (1+\frac {2 d e e^x \left (1-\frac {c f}{d e}\right )}{2 f+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {\left (2 b^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 d e \left (1-\frac {c f}{d e}\right ) x}{-2 f-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {\left (2 b^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 d e \left (1-\frac {c f}{d e}\right ) x}{-2 f+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \\ & = -\frac {b d^2 f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {b^2 d^2 f \log \left (\frac {e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {\left (b^2 d^2 f^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 d e \left (1-\frac {c f}{d e}\right ) x}{2 f-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {\left (b^2 d^2 f^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 d e \left (1-\frac {c f}{d e}\right ) x}{2 f+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}} \\ & = -\frac {b d^2 f \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{(d e-c f) \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {d^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (d e-c f)^2}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f (e+f x)^2}+\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {b d^2 f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {2 b d^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {b^2 d^2 f \log \left (\frac {e+f x}{c+d x}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {b^2 d^2 f^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}-\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {b^2 d^2 f^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^{3/2}}+\frac {2 b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f)^2 \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.37 (sec) , antiderivative size = 8350, normalized size of antiderivative = 8.15 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}}{\left (f x +e \right )^{3}}d x\]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{{\left (e+f\,x\right )}^3} \,d x \]
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