Optimal. Leaf size=756 \[ \frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{4 e^{5/2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}-\frac {3 b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 e^2 \sqrt {c^2 d-e}}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 e^2 \sqrt {c^2 d-e}}+\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c e^2} \]
[Out]
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Rubi [A] time = 2.47, antiderivative size = 756, normalized size of antiderivative = 1.00, number of steps used = 51, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6304, 5791, 5661, 266, 63, 208, 5706, 5801, 725, 206, 5799, 5561, 2190, 2279, 2391} \[ -\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{4 e^{5/2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 e^2 \sqrt {c^2 d-e}}+\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 e^2 \sqrt {c^2 d-e}}+\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 206
Rule 208
Rule 266
Rule 725
Rule 2190
Rule 2279
Rule 2391
Rule 5561
Rule 5661
Rule 5706
Rule 5791
Rule 5799
Rule 5801
Rule 6304
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{x^2 \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{e^2 x^2}-\frac {d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{e \left (e+d x^2\right )^2}-\frac {d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{x^2} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{e}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c e^2}+\frac {d \operatorname {Subst}\left (\int \left (\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {d \operatorname {Subst}\left (\int \left (-\frac {d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}-d x\right )^2}-\frac {d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{4 e \left (\sqrt {-d} \sqrt {e}+d x\right )^2}-\frac {d \left (a+b \sinh ^{-1}\left (\frac {x}{c}\right )\right )}{2 e \left (-d e-d^2 x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e}\\ &=\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {d \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^{5/2}}+\frac {d \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 e^{5/2}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c e^2}-\frac {d^2 \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}-d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e^2}-\frac {d^2 \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\left (\sqrt {-d} \sqrt {e}+d x\right )^2} \, dx,x,\frac {1}{x}\right )}{4 e^2}-\frac {d^2 \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{-d e-d^2 x^2} \, dx,x,\frac {1}{x}\right )}{2 e^2}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {d \operatorname {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 e^{5/2}}+\frac {d \operatorname {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 e^{5/2}}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{-c^2+c^2 x^2} \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )}{e^2}+\frac {(b d) \operatorname {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 )}{4 c e^2}-\frac {(b d) \operatorname {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 )}{4 c e^2}-\frac {d^2 \operatorname {Subst}\left (\int \left (-\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}-\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 d \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{2 e^2}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {d \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{4 e^{5/2}}+\frac {d \operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{4 e^{5/2}}+\frac {d \operatorname {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 e^{5/2}}+\frac {d \operatorname {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 e^{5/2}}+\frac {d \operatorname {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 e^{5/2}}+\frac {d \operatorname {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 e^{5/2}}-\frac {(b d) \operatorname {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 )}{4 c e^2}+\frac {(b d) \operatorname {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 )}{4 c e^2}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {\sqrt {-d} \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 e^{5/2}}-\frac {\sqrt {-d} \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 e^{5/2}}+\frac {\sqrt {-d} \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 e^{5/2}}-\frac {\sqrt {-d} \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 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}+\frac {d \operatorname {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 )}{4 e^{5/2}}+\frac {d \operatorname {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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {\sqrt {-d} \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 e^{5/2}}-\frac {\sqrt {-d} \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 e^{5/2}}+\frac {\sqrt {-d} \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 e^{5/2}}-\frac {\sqrt {-d} \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 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 e^{5/2}}+\frac {d \operatorname {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 )}{4 e^{5/2}}+\frac {d \operatorname {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 )}{4 e^{5/2}}+\frac {d \operatorname {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 )}{4 e^{5/2}}+\frac {d \operatorname {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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}-\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}+\frac {\left (b \sqrt {-d}\right ) \operatorname {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 )}{4 e^{5/2}}\\ &=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {b \sqrt {d} \tanh ^{-1}\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 )}{4 \sqrt {c^2 d-e} e^2}+\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \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 )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}}\\ \end {align*}
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Mathematica [C] time = 6.17, size = 1583, normalized size = 2.09 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \operatorname {arcsch}\left (c x\right ) + a x^{4}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 15.13, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {d x}{e^{3} x^{2} + d e^{2}} - \frac {3 \, d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {2 \, x}{e^{2}}\right )} + b \int \frac {x^{4} \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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