Integrand size = 21, antiderivative size = 515 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d-e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2} \]
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Time = 0.82 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6439, 5823, 5821, 385, 211, 5827, 5680, 2221, 2317, 2438} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{2 d^2}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (\frac {d}{x^2}+e\right )}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}+1}}\right )}{2 d^2 \sqrt {c^2 d-e}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 d^2} \]
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Rule 211
Rule 385
Rule 2221
Rule 2317
Rule 2438
Rule 5680
Rule 5821
Rule 5823
Rule 5827
Rule 6439
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^3 \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (-\frac {e x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{d \left (e+d x^2\right )^2}+\frac {x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d}+\frac {e \text {Subst}\left (\int \frac {x \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{d} \\ & = -\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}-\frac {\text {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{c^2}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c d^2} \\ & = -\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 (-d)^{3/2}}+\frac {(b e) \text {Subst}\left (\int \frac {1}{e-\left (-d+\frac {e}{c^2}\right ) x^2} \, dx,x,\frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 c d^2} \\ & = -\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d-e}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 (-d)^{3/2}} \\ & = -\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d-e}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 (-d)^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 (-d)^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 (-d)^{3/2}} \\ & = -\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d-e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 d^2} \\ & = -\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d-e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 d^2}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 d^2} \\ & = -\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d-e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.41 (sec) , antiderivative size = 1428, normalized size of antiderivative = 2.77 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\frac {a}{2 d^2+2 d e x^2}+\frac {a \log (x)}{d^2}-\frac {a \log \left (d+e x^2\right )}{2 d^2}-\frac {b \left (\pi ^2-4 i \pi \text {csch}^{-1}(c x)-\frac {2 \sqrt {d} \text {csch}^{-1}(c x)}{\sqrt {d}-i \sqrt {e} x}-\frac {2 \sqrt {d} \text {csch}^{-1}(c x)}{\sqrt {d}+i \sqrt {e} x}-12 \text {csch}^{-1}(c x)^2+4 \text {arcsinh}\left (\frac {1}{c x}\right )+16 \arcsin \left (\frac {\sqrt {1+\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (c \sqrt {d}-\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c x)\right )\right )}{\sqrt {-c^2 d+e}}\right )-16 \arcsin \left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \arctan \left (\frac {\left (c \sqrt {d}+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c x)\right )\right )}{\sqrt {-c^2 d+e}}\right )-8 \text {csch}^{-1}(c x) \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )+2 i \pi \log \left (1-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \text {csch}^{-1}(c x) \log \left (1-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+8 i \arcsin \left (\frac {\sqrt {1+\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \pi \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \text {csch}^{-1}(c x) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+8 i \arcsin \left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \pi \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \text {csch}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-8 i \arcsin \left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i \pi \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \text {csch}^{-1}(c x) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-8 i \arcsin \left (\frac {\sqrt {1+\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+8 \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-2 i \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )-2 i \pi \log \left (\sqrt {e}+\frac {i \sqrt {d}}{x}\right )-\frac {2 \sqrt {e} \log \left (\frac {2 \sqrt {d} \sqrt {e} \left (i \sqrt {e}+c \left (c \sqrt {d}+i \sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d+e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d+e}}-\frac {2 \sqrt {e} \log \left (-\frac {2 \sqrt {d} \sqrt {e} \left (\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d+e} \sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{\sqrt {-c^2 d+e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d+e}}+4 \operatorname {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \operatorname {PolyLog}\left (2,\frac {i \left (-\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \operatorname {PolyLog}\left (2,-\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \operatorname {PolyLog}\left (2,\frac {i \left (\sqrt {e}+\sqrt {-c^2 d+e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+4 \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )\right )}{8 d^2} \]
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\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{x \left (e \,x^{2}+d \right )^{2}}d x\]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x \left (d + e x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^2} \,d x \]
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