Optimal. Leaf size=763 \[ -\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{\sqrt {-d} \sqrt {e}} \]
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Rubi [A] time = 1.31, antiderivative size = 763, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5707, 5800, 5562, 2190, 2531, 2282, 6589} \[ -\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {PolyLog}\left (3,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {PolyLog}\left (3,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {PolyLog}\left (3,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {PolyLog}\left (3,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 \sqrt {-d} \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 5562
Rule 5707
Rule 5800
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{d+e x^2} \, dx &=\int \left (\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {\int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^2 \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^2 \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-d}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)^2}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-d}}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {Subst}\left (\int (a+b x) \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \operatorname {Subst}\left (\int (a+b x) \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}-\frac {b \operatorname {Subst}\left (\int (a+b x) \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \operatorname {Subst}\left (\int (a+b x) \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-d} \sqrt {e}}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {e} x}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-d} \sqrt {e}}\\ &=\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \text {Li}_3\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \text {Li}_3\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{\sqrt {-d} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 623, normalized size = 0.82 \[ \frac {2 b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )-2 b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d c^2-e}-c \sqrt {-d}}\right )-2 b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )+2 b \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )-\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )+\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}-c \sqrt {-d}}+1\right )+\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )-\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )-2 b^2 \text {Li}_3\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )+2 b^2 \text {Li}_3\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d c^2-e}-c \sqrt {-d}}\right )+2 b^2 \text {Li}_3\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )-2 b^2 \text {Li}_3\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 \sqrt {-d} \sqrt {e}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}{e x^{2} + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{e \,x^{2}+d}\, dx \]
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 \[ \frac {a^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}} + \int \frac {b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{e x^{2} + d} + \frac {2 \, a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e x^{2} + d}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{e\,x^2+d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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