Optimal. Leaf size=296 \[ -\frac{i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{i a^2 \sqrt{a x-1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{i a^2 \sqrt{a x-1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 x^2}-\frac{a^2 \sqrt{a x-1} \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )}{\sqrt{1-a x}}+\frac{a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{a \sqrt{a x-1} \cosh ^{-1}(a x)}{x \sqrt{1-a x}} \]
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Rubi [A] time = 0.722817, antiderivative size = 398, normalized size of antiderivative = 1.34, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5798, 5748, 5761, 4180, 2531, 2282, 6589, 5662, 92, 205} \[ -\frac{i a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{a^2 \sqrt{a x-1} \sqrt{a x+1} \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )}{\sqrt{1-a^2 x^2}}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5748
Rule 5761
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5662
Rule 92
Rule 205
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{x^3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt{1-a^2 x^2}}-\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{x^2} \, dx}{\sqrt{1-a^2 x^2}}+\frac{\left (a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^2}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 \sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{\left (a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{1-a^2 x^2}}-\frac{\left (a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (a^3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{1}{a+a x^2} \, dx,x,\sqrt{-1+a x} \sqrt{1+a x}\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \tan ^{-1}\left (\sqrt{-1+a x} \sqrt{1+a x}\right )}{\sqrt{1-a^2 x^2}}-\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \tan ^{-1}\left (\sqrt{-1+a x} \sqrt{1+a x}\right )}{\sqrt{1-a^2 x^2}}-\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{\left (i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \tan ^{-1}\left (\sqrt{-1+a x} \sqrt{1+a x}\right )}{\sqrt{1-a^2 x^2}}-\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.981897, size = 233, normalized size = 0.79 \[ \frac{i a^2 \sqrt{-(a x-1) (a x+1)} \left (2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(a x)}\right )+\frac{i \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^2}{a^2 x^2}+\frac{2 i \cosh ^{-1}(a x)}{a x}+\cosh ^{-1}(a x)^2 \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )\right )\right )}{2 \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.163, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{{x}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, 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 (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{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{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )^{2}}{a^{2} x^{5} - x^{3}}, 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 (a x \right )}}{x^{3} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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