Optimal. Leaf size=167 \[ -\frac{i a^2 \sqrt{a x-1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a x}}+\frac{i a^2 \sqrt{a x-1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{2 x^2}+\frac{a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a x}}+\frac{a \sqrt{a x-1}}{2 x \sqrt{1-a x}} \]
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Rubi [A] time = 0.471289, antiderivative size = 230, normalized size of antiderivative = 1.38, number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {5798, 5748, 5761, 4180, 2279, 2391, 30} \[ -\frac{i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{a \sqrt{a x-1} \sqrt{a x+1}}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (a x+1) \cosh ^{-1}(a x)}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5748
Rule 5761
Rule 4180
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)}{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)}{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 x^2 \sqrt{1-a^2 x^2}}-\frac{\left (a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{1}{x^2} \, dx}{2 \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)}{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}}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{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 \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \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 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{1-a^2 x^2}}+\frac{\left (i a^2 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 \sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \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 \frac{\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{2 \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{\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x}}{2 x \sqrt{1-a^2 x^2}}-\frac{(1-a x) (1+a x) \cosh ^{-1}(a x)}{2 x^2 \sqrt{1-a^2 x^2}}+\frac{a^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \tan ^{-1}\left (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}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}+\frac{i a^2 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.29017, size = 234, normalized size = 1.4 \[ \frac{(a x+1) \left (-i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )-i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )+i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+a x \sqrt{\frac{a x-1}{a x+1}}+a x \cosh ^{-1}(a x)-\cosh ^{-1}(a x)\right )}{2 x^2 \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.206, size = 349, normalized size = 2.1 \begin{align*} -{\frac{1}{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ){x}^{2}} \left ({a}^{2}{x}^{2}{\rm arccosh} \left (ax\right )+\sqrt{ax+1}\sqrt{ax-1}ax-{\rm arccosh} \left (ax\right ) \right ) \sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{i{\rm arccosh} \left (ax\right ){a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }-{\frac{i{\rm arccosh} \left (ax\right ){a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}\ln \left ( 1-i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }+{\frac{i{a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\it dilog} \left ( 1+i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \right ) }-{\frac{i{a}^{2}}{2\,{a}^{2}{x}^{2}-2}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}{\it dilog} \left ( 1-i \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) \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*} \int \frac{\operatorname{arcosh}\left (a x\right )}{\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 )}{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}{\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 )}{\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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