Integrand size = 18, antiderivative size = 1096 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {5 b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d-e}}+\frac {b e \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {5 b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d-e}}+\frac {b e \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]
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Time = 2.90 (sec) , antiderivative size = 1096, normalized size of antiderivative = 1.00, number of steps used = 81, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6429, 5823, 5793, 5828, 745, 739, 212, 5827, 5680, 2221, 2317, 2438} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=-\frac {b \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} c}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}-\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}+\frac {b e \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {5 b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d-e}}+\frac {b e \text {arctanh}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \left (c^2 d-e\right )^{3/2}}+\frac {5 b \text {arctanh}\left (\frac {d c^2+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 d^{5/2} \sqrt {c^2 d-e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {csch}^{-1}(c x)} c}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {\sqrt {-d} e^{\text {csch}^{-1}(c x)} c}{\sqrt {e}+\sqrt {e-c^2 d}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]
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Rule 212
Rule 739
Rule 745
Rule 2221
Rule 2317
Rule 2438
Rule 5680
Rule 5793
Rule 5823
Rule 5827
Rule 5828
Rule 6429
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^4 \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \left (\frac {e^2 \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^3}-\frac {2 e \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{d^2 \left (e+d x^2\right )^2}+\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^3} \, dx,x,\frac {1}{x}\right )}{d^2} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(2 e) \text {Subst}\left (\int \left (-\frac {d \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2}-\frac {e^2 \text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}-d x\right )^3}-\frac {3 d \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d^3 \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}+d x\right )^3}-\frac {3 d \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {3 d \left (a+b \text {arcsinh}\left (\frac {x}{c}\right )\right )}{8 e^2 \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d^2} \\ & = \frac {3 \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 d}+\frac {3 \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{16 d}+\frac {3 \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{8 d}-\frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{2 d}-\frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{2 d}-\frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{d}-\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^2 \sqrt {e}}-\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^2 \sqrt {e}}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {-d}}-\frac {\sqrt {e} \text {Subst}\left (\int \frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^3} \, dx,x,\frac {1}{x}\right )}{8 \sqrt {-d}} \\ & = \frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2}+\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c d^2}-\frac {b \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c d^2}+\frac {3 \text {Subst}\left (\int \left (-\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{8 d}-\frac {\text {Subst}\left (\int \left (-\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \text {arcsinh}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{d}-\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^2 \sqrt {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^2 \sqrt {e}}-\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c (-d)^{3/2}}+\frac {\left (b \sqrt {e}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2 \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c (-d)^{3/2}} \\ & = -\frac {b c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {b c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}}}{16 (-d)^{3/2} \left (c^2 d-e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right )}{16 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}+\frac {5 \left (a+b \text {csch}^{-1}(c x)\right )}{16 d^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 c d^2}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{16 c d^2}-\frac {b \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {-d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{2 c d^2}+\frac {b \text {Subst}\left (\int \frac {1}{d^2-\frac {d e}{c^2}-x^2} \, dx,x,\frac {d-\frac {\sqrt {-d} \sqrt {e}}{c^2 x}}{\sqrt {1+\frac {1}{c^2 x^2}}}\right )}{2 c d^2}-\frac {3 \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 )}{16 d^2 \sqrt {e}}-\frac {3 \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 )}{16 d^2 \sqrt {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^2 \sqrt {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^2 \sqrt {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^2 \sqrt {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^2 \sqrt {e}}+\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^2 \sqrt {e}}+\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^2 \sqrt {e}}+\frac {(b e) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}-d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2 \left (c^2 d-e\right )}-\frac {(b e) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-d} \sqrt {e}+d x\right ) \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{16 c d^2 \left (c^2 d-e\right )} \\ & = \text {Too large to display} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.06 (sec) , antiderivative size = 2038, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]
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\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{\left (e \,x^{2}+d \right )^{3}}d x\]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
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