Optimal. Leaf size=272 \[ \frac{2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e}-\frac{2 \text{PolyLog}\left (3,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}-\frac{2 \text{PolyLog}\left (3,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}+1\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}+1\right )}{e}-\frac{\cosh ^{-1}(c x)^3}{3 e} \]
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Rubi [A] time = 0.42594, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5800, 5562, 2190, 2531, 2282, 6589} \[ \frac{2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e}-\frac{2 \text{PolyLog}\left (3,-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}-\frac{2 \text{PolyLog}\left (3,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}+1\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}+1\right )}{e}-\frac{\cosh ^{-1}(c x)^3}{3 e} \]
Antiderivative was successfully verified.
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Rule 5800
Rule 5562
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(c x)^2}{d+e x} \, dx &=\operatorname{Subst}\left (\int \frac{x^2 \sinh (x)}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac{\cosh ^{-1}(c x)^3}{3 e}+\operatorname{Subst}\left (\int \frac{e^x x^2}{c d-\sqrt{c^2 d^2-e^2}+e e^x} \, dx,x,\cosh ^{-1}(c x)\right )+\operatorname{Subst}\left (\int \frac{e^x x^2}{c d+\sqrt{c^2 d^2-e^2}+e e^x} \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac{\cosh ^{-1}(c x)^3}{3 e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}-\frac{2 \operatorname{Subst}\left (\int x \log \left (1+\frac{e e^x}{c d-\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}-\frac{2 \operatorname{Subst}\left (\int x \log \left (1+\frac{e e^x}{c d+\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac{\cosh ^{-1}(c x)^3}{3 e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}-\frac{2 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e e^x}{c d-\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}-\frac{2 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{e e^x}{c d+\sqrt{c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac{\cosh ^{-1}(c x)^3}{3 e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}-\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{e x}{-c d+\sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e}-\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{e x}{c d+\sqrt{c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e}\\ &=-\frac{\cosh ^{-1}(c x)^3}{3 e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{\cosh ^{-1}(c x)^2 \log \left (1+\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}+\frac{2 \cosh ^{-1}(c x) \text{Li}_2\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}-\frac{2 \text{Li}_3\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right )}{e}-\frac{2 \text{Li}_3\left (-\frac{e e^{\cosh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.157307, size = 252, normalized size = 0.93 \[ -\frac{-6 \cosh ^{-1}(c x) \text{PolyLog}\left (2,\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )-6 \cosh ^{-1}(c x) \text{PolyLog}\left (2,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )+6 \text{PolyLog}\left (3,\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right )+6 \text{PolyLog}\left (3,-\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right )-3 \cosh ^{-1}(c x)^2 \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}+1\right )-3 \cosh ^{-1}(c x)^2 \log \left (\frac{e e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}+1\right )+\cosh ^{-1}(c x)^3}{3 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{ex+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (c x\right )^{2}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcosh}\left (c x\right )^{2}}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{2}{\left (c x \right )}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (c x\right )^{2}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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