Integrand size = 20, antiderivative size = 501 \[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b f^3 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^4}+\frac {3 b f (d e-c f)^2 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{6 d^4}-\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b f^2 (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {4 b (d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {b^2 f^3 \log (c+d x)}{3 d^4}+\frac {3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}-\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4} \]
[Out]
Time = 0.70 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6457, 5577, 4275, 4267, 2317, 2438, 4269, 3556, 4270} \[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=-\frac {2 b f^2 (d e-c f) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {4 b (d e-c f)^3 \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (c+d x)^2 \sqrt {\frac {1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}-\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {3 b f (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} (d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )}{6 d^4}-\frac {b f^3 (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^4}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b^2 f^3 \log (c+d x)}{3 d^4}+\frac {b^2 f^2 x (d e-c f)}{d^3} \]
[In]
[Out]
Rule 2317
Rule 2438
Rule 3556
Rule 4267
Rule 4269
Rule 4270
Rule 4275
Rule 5577
Rule 6457
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int (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 )}{d^4} \\ & = \frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {b \text {Subst}\left (\int (a+b x) (d e-c f+f \text {csch}(x))^4 \, dx,x,\text {csch}^{-1}(c+d x)\right )}{2 d^4 f} \\ & = \frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {b \text {Subst}\left (\int \left (d^4 e^4 \left (1+\frac {c f \left (-4 d^3 e^3+6 c d^2 e^2 f-4 c^2 d e f^2+c^3 f^3\right )}{d^4 e^4}\right ) (a+b x)+4 d^3 e^3 f \left (1-\frac {c f \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )}{d^3 e^3}\right ) (a+b x) \text {csch}(x)+6 d^2 e^2 f^2 \left (1+\frac {c f (-2 d e+c f)}{d^2 e^2}\right ) (a+b x) \text {csch}^2(x)+4 d e f^3 \left (1-\frac {c f}{d e}\right ) (a+b x) \text {csch}^3(x)+f^4 (a+b x) \text {csch}^4(x)\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{2 d^4 f} \\ & = -\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {\left (b f^3\right ) \text {Subst}\left (\int (a+b x) \text {csch}^4(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{2 d^4}-\frac {\left (2 b f^2 (d e-c f)\right ) \text {Subst}\left (\int (a+b x) \text {csch}^3(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4}-\frac {\left (3 b f (d e-c f)^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4}-\frac {\left (2 b (d e-c f)^3\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4} \\ & = \frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}+\frac {3 b f (d e-c f)^2 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{6 d^4}-\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}+\frac {4 b (d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {\left (b f^3\right ) \text {Subst}\left (\int (a+b x) \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{3 d^4}+\frac {\left (b f^2 (d e-c f)\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4}-\frac {\left (3 b^2 f (d e-c f)^2\right ) \text {Subst}\left (\int \coth (x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {\left (2 b^2 (d e-c f)^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4}-\frac {\left (2 b^2 (d e-c f)^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4} \\ & = \frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b f^3 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^4}+\frac {3 b f (d e-c f)^2 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{6 d^4}-\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b f^2 (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {4 b (d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac {\left (b^2 f^3\right ) \text {Subst}\left (\int \coth (x) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{3 d^4}-\frac {\left (b^2 f^2 (d e-c f)\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {\left (b^2 f^2 (d e-c f)\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {\left (2 b^2 (d e-c f)^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {\left (2 b^2 (d e-c f)^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4} \\ & = \frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b f^3 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^4}+\frac {3 b f (d e-c f)^2 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{6 d^4}-\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b f^2 (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {4 b (d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {b^2 f^3 \log (c+d x)}{3 d^4}+\frac {3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {\left (b^2 f^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {\left (b^2 f^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4} \\ & = \frac {b^2 f^2 (d e-c f) x}{d^3}+\frac {b^2 f^3 (c+d x)^2}{12 d^4}-\frac {b f^3 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^4}+\frac {3 b f (d e-c f)^2 (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^2 (d e-c f) (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^4}+\frac {b f^3 (c+d x)^3 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{6 d^4}-\frac {(d e-c f)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac {(e+f x)^4 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{4 f}-\frac {2 b f^2 (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {4 b (d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {b^2 f^3 \log (c+d x)}{3 d^4}+\frac {3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}-\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}+\frac {b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4}-\frac {2 b^2 (d e-c f)^3 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.06 (sec) , antiderivative size = 1487, normalized size of antiderivative = 2.97 \[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=a^2 e^3 x+\frac {3}{2} a^2 e^2 f x^2+a^2 e f^2 x^3+\frac {1}{4} a^2 f^3 x^4+\frac {1}{6} a b \left (3 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right ) \text {csch}^{-1}(c+d x)+\frac {f (c+d x) \sqrt {\frac {1+c^2+2 c d x+d^2 x^2}{(c+d x)^2}} \left (\left (-2+13 c^2\right ) f^2-2 c d f (15 e+2 f x)+d^2 \left (18 e^2+6 e f x+f^2 x^2\right )\right )-3 c \left (-4 d^3 e^3+6 c d^2 e^2 f-4 c^2 d e f^2+c^3 f^3\right ) \text {arcsinh}\left (\frac {1}{c+d x}\right )+6 \left (2 d^3 e^3-6 c d^2 e^2 f+\left (-1+6 c^2\right ) d e f^2+c \left (1-2 c^2\right ) f^3\right ) \log \left ((c+d x) \left (1+\sqrt {\frac {1+c^2+2 c d x+d^2 x^2}{(c+d x)^2}}\right )\right )}{d^4}\right )-\frac {b^2 e^3 \left (-\text {csch}^{-1}(c+d x) \left ((c+d x) \text {csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )}{d}-\frac {3 b^2 d e^2 f x \left (\frac {(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \text {csch}^{-1}(c+d x)}{d^2}+\frac {(c+d x)^2 \text {csch}^{-1}(c+d x)^2}{2 d^2}-\frac {c \text {csch}^{-1}(c+d x)^2 \coth \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 d^2}-\frac {\log \left (\frac {1}{c+d x}\right )}{d^2}-\frac {2 i c \left (i \text {csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )\right )}{d^2}+\frac {c \text {csch}^{-1}(c+d x)^2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 d^2}\right )}{(c+d x) \left (-1+\frac {c}{c+d x}\right )}-\frac {b^2 e f^2 \left (2 \left (-2+12 c \text {csch}^{-1}(c+d x)+\text {csch}^{-1}(c+d x)^2-6 c^2 \text {csch}^{-1}(c+d x)^2\right ) \coth \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+2 \text {csch}^{-1}(c+d x) \left (-1+3 c \text {csch}^{-1}(c+d x)\right ) \text {csch}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-\frac {\text {csch}^{-1}(c+d x)^2 \text {csch}^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 (c+d x)}-48 c \left (\log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+\log \left (\sqrt {1+\frac {1}{(c+d x)^2}}\right )\right )+8 \left (-1+6 c^2\right ) \left (\text {csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \text {csch}^{-1}(c+d x) \left (1+3 c \text {csch}^{-1}(c+d x)\right ) \text {sech}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-8 (c+d x)^3 \text {csch}^{-1}(c+d x)^2 \sinh ^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+2 \left (2+12 c \text {csch}^{-1}(c+d x)-\text {csch}^{-1}(c+d x)^2+6 c^2 \text {csch}^{-1}(c+d x)^2\right ) \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )}{8 d^3}-\frac {b^2 f^3 x^3 \left (-16 \left (2 \text {csch}^{-1}(c+d x)-18 c^2 \text {csch}^{-1}(c+d x)+6 c^3 \text {csch}^{-1}(c+d x)^2-3 c \left (-2+\text {csch}^{-1}(c+d x)^2\right )\right ) \coth \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+2 \left (2-24 c \text {csch}^{-1}(c+d x)-3 \text {csch}^{-1}(c+d x)^2+36 c^2 \text {csch}^{-1}(c+d x)^2\right ) \text {csch}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+3 \text {csch}^{-1}(c+d x)^2 \text {csch}^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-\frac {2 \text {csch}^{-1}(c+d x) \left (-1+6 c \text {csch}^{-1}(c+d x)\right ) \text {csch}^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{c+d x}-64 \left (-1+9 c^2\right ) \left (\log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+\log \left (\sqrt {1+\frac {1}{(c+d x)^2}}\right )\right )+192 c \left (-1+2 c^2\right ) \left (\text {csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \left (2+24 c \text {csch}^{-1}(c+d x)-3 \text {csch}^{-1}(c+d x)^2+36 c^2 \text {csch}^{-1}(c+d x)^2\right ) \text {sech}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+3 \text {csch}^{-1}(c+d x)^2 \text {sech}^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-32 (c+d x)^3 \text {csch}^{-1}(c+d x) \left (1+6 c \text {csch}^{-1}(c+d x)\right ) \sinh ^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+16 \left (-2 \text {csch}^{-1}(c+d x)+18 c^2 \text {csch}^{-1}(c+d x)+6 c^3 \text {csch}^{-1}(c+d x)^2-3 c \left (-2+\text {csch}^{-1}(c+d x)^2\right )\right ) \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )}{192 d (c+d x)^3 \left (-1+\frac {c}{c+d x}\right )^3} \]
[In]
[Out]
\[\int \left (f x +e \right )^{3} \left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}d x\]
[In]
[Out]
\[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{3} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{3}\, dx \]
[In]
[Out]
\[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{3} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
\[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{3} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int (e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int {\left (e+f\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2 \,d x \]
[In]
[Out]