Optimal. Leaf size=631 \[ -\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{e^3}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\frac {2 d \log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {b d \sqrt {\frac {1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 e^{5/2} \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}}-\frac {b d \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}-\frac {b d \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{e^3}-\frac {b d \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{e^3}-\frac {b d \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{e^3}-\frac {b d \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {b x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{2 c e^2} \]
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Rubi [A] time = 1.55, antiderivative size = 611, normalized size of antiderivative = 0.97, number of steps used = 32, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6303, 5792, 5662, 95, 5660, 3718, 2190, 2279, 2391, 5788, 519, 377, 208, 5800, 5562} \[ -\frac {b d \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}-\frac {b d \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}+\frac {b d \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}+1\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {c^2 d+e}+\sqrt {e}}+1\right )}{e^3}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}+\frac {2 d \log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {b d \sqrt {\frac {1}{c^2 x^2}-1} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}-1}}\right )}{2 e^{5/2} \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1} \sqrt {c^2 d+e}}-\frac {b x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{2 c e^2} \]
Warning: Unable to verify antiderivative.
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Rule 95
Rule 208
Rule 377
Rule 519
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5562
Rule 5660
Rule 5662
Rule 5788
Rule 5792
Rule 5800
Rule 6303
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{x^3 \left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{e^2 x^3}-\frac {2 d \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{e^3 x}+\frac {d^2 x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{e^2 \left (e+d x^2\right )^2}+\frac {2 d^2 x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{e^3 \left (e+d x^2\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {(2 d) \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{x^3} \, dx,x,\frac {1}{x}\right )}{e^2}-\frac {d^2 \operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{\left (e+d x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{e^2}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}+\frac {(2 d) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{e^3}-\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}\left (\frac {x}{c}\right )\right )}{2 d \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}}} \, dx,x,\frac {1}{x}\right )}{2 c e^2}-\frac {(b d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+\frac {x}{c}} \sqrt {1+\frac {x}{c}} \left (e+d x^2\right )} \, dx,x,\frac {1}{x}\right )}{2 c e^2}\\ &=-\frac {b \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}-\frac {(-d)^{3/2} \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {(-d)^{3/2} \operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {(4 d) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(c x)\right )}{e^3}-\frac {\left (b d \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \operatorname {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 e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=-\frac {b \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )^2}{b e^3}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {(-d)^{3/2} \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{e^3}+\frac {(-d)^{3/2} \operatorname {Subst}\left (\int \frac {(a+b x) \sinh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \cosh (x)} \, dx,x,\text {sech}^{-1}(c x)\right )}{e^3}-\frac {(2 b d) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{e^3}-\frac {\left (b d \sqrt {-1+\frac {1}{c^2 x^2}}\right ) \operatorname {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 e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}\\ &=-\frac {b \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {b d \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {(-d)^{3/2} \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 {sech}^{-1}(c x)\right )}{e^3}-\frac {(-d)^{3/2} \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 {sech}^{-1}(c x)\right )}{e^3}+\frac {(-d)^{3/2} \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 {sech}^{-1}(c x)\right )}{e^3}+\frac {(-d)^{3/2} \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 {sech}^{-1}(c x)\right )}{e^3}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}\\ &=-\frac {b \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {b d \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {b d \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)\right )}{e^3}\\ &=-\frac {b \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {b d \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {b d \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)}\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)}\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)}\right )}{e^3}+\frac {(b d) \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 {sech}^{-1}(c x)}\right )}{e^3}\\ &=-\frac {b \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {x^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 e^2}-\frac {b d \sqrt {-1+\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {-1+\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {2 d \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}-\frac {b d \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}-\frac {b d \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{e^3}+\frac {b d \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )}{e^3}\\ \end {align*}
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Mathematica [C] time = 5.62, size = 1278, normalized size = 2.03 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{5} \operatorname {arsech}\left (c x\right ) + a x^{5}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.63, size = 870, normalized size = 1.38 \[ \frac {a \,x^{2}}{2 e^{2}}-\frac {c^{2} a \,d^{2}}{2 e^{3} \left (c^{2} x^{2} e +c^{2} d \right )}-\frac {a d \ln \left (c^{2} x^{2} e +c^{2} d \right )}{e^{3}}-\frac {c b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x^{3}}{2 \left (c^{2} x^{2} e +c^{2} d \right ) e}-\frac {c b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x d}{2 \left (c^{2} x^{2} e +c^{2} d \right ) e^{2}}+\frac {c^{2} b \,\mathrm {arcsech}\left (c x \right ) x^{4}}{2 \left (c^{2} x^{2} e +c^{2} d \right ) e}+\frac {c^{2} b \,\mathrm {arcsech}\left (c x \right ) d \,x^{2}}{\left (c^{2} x^{2} e +c^{2} d \right ) e^{2}}+\frac {b \,x^{2}}{2 \left (c^{2} x^{2} e +c^{2} d \right ) e}+\frac {b d}{2 \left (c^{2} x^{2} e +c^{2} d \right ) e^{2}}+\frac {b \sqrt {e \left (c^{2} d +e \right )}\, d \arctanh \left (\frac {2 c^{2} d \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}+2 c^{2} d +4 e}{4 \sqrt {c^{2} d e +e^{2}}}\right )}{2 e^{3} \left (c^{2} d +e \right )}-\frac {c^{2} b \,d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}}+\frac {2 b d \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 b d \,\mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 b d \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}+\frac {2 b d \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{e^{3}}-\frac {b d \left (\munderset {\textit {\_R1} =\RootOf \left (c^{2} d \,\textit {\_Z}^{4}+\left (2 c^{2} d +4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\left (\textit {\_R1}^{2} c^{2} d +c^{2} d +4 e \right ) \left (\mathrm {arcsech}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {1}{c x}-\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d +c^{2} d +2 e}\right )}{2 e^{3}} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {d^{2}}{e^{4} x^{2} + d e^{3}} - \frac {x^{2}}{e^{2}} + \frac {2 \, d \log \left (e x^{2} + d\right )}{e^{3}}\right )} + b \int \frac {x^{5} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]
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 \[ \int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
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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