Optimal. Leaf size=124 \[ -\frac{a \sqrt{a x-1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{x}+\frac{a \sqrt{a x-1} \cosh ^{-1}(a x)^2}{\sqrt{1-a x}}-\frac{2 a \sqrt{a x-1} \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt{1-a x}} \]
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Rubi [A] time = 0.444493, antiderivative size = 174, normalized size of antiderivative = 1.4, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {5798, 5724, 5660, 3718, 2190, 2279, 2391} \[ -\frac{a \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5724
Rule 5660
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^2}{x^2 \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^2 \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}{x \sqrt{1-a^2 x^2}}-\frac{\left (2 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)}{x} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}}-\frac{\left (2 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \tanh (x) \, 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)^2}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}}-\frac{\left (4 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, 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)^2}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}}-\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (2 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 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)^2}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}}-\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}+\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \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)^2}{\sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}}-\frac{2 a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}-\frac{a \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.450159, size = 111, normalized size = 0.9 \[ \frac{a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (\text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x) \left (\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-2 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )\right )\right )}{\sqrt{-(a x-1) (a x+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.151, size = 241, normalized size = 1.9 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{x \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}-\sqrt{ax+1}\sqrt{ax-1}ax-1 \right ) }-2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}a}{{a}^{2}{x}^{2}-1}}+2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\rm arccosh} \left (ax\right )\ln \left ( 1+ \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) a}{{a}^{2}{x}^{2}-1}}+{\frac{a}{{a}^{2}{x}^{2}-1}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\it polylog} \left ( 2,- \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) ^{2} \right ) } \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*} \frac{{\left (a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2}}{\sqrt{a x + 1} \sqrt{-a x + 1} x} - \int \frac{2 \,{\left (a^{3} x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}{{\left (\sqrt{a x + 1} a x^{2} +{\left (a x + 1\right )} \sqrt{a x - 1} x\right )} \sqrt{-a x + 1}}\,{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^{4} - x^{2}}, 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^{2} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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