Optimal. Leaf size=109 \[ \frac{\text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{\text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac{1}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]
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Rubi [A] time = 0.0804365, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5689, 74, 5694, 4182, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{\text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac{1}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 5689
Rule 74
Rule 5694
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac{a \int \frac{x}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{2 c^2}+\frac{\int \frac{\cosh ^{-1}(a x)}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac{1}{2 a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}-\frac{\operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac{1}{2 a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{\operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}-\frac{\operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac{1}{2 a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}\\ &=-\frac{1}{2 a c^2 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)}{2 c^2 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac{\text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac{\text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}\\ \end{align*}
Mathematica [A] time = 0.83427, size = 120, normalized size = 1.1 \[ \frac{2 \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )-\frac{2 \left (\cosh ^{-1}(a x) \left (\left (a^2 x^2-1\right ) \log \left (1-e^{\cosh ^{-1}(a x)}\right )+\left (1-a^2 x^2\right ) \log \left (e^{\cosh ^{-1}(a x)}+1\right )+a x\right )+\sqrt{\frac{a x-1}{a x+1}} (a x+1)\right )}{a^2 x^2-1}}{4 a c^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.078, size = 184, normalized size = 1.7 \begin{align*} -{\frac{x{\rm arccosh} \left (ax\right )}{ \left ( 2\,{a}^{2}{x}^{2}-2 \right ){c}^{2}}}-{\frac{1}{2\,a \left ({a}^{2}{x}^{2}-1 \right ){c}^{2}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\rm arccosh} \left (ax\right )}{2\,a{c}^{2}}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{1}{2\,a{c}^{2}}{\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{{\rm arccosh} \left (ax\right )}{2\,a{c}^{2}}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{1}{2\,a{c}^{2}}{\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \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 + 1\right )^{2} + 2 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) -{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 4 \, a x + 4 \,{\left (2 \, a x -{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) +{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right ) - 2 \,{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )}{16 \,{\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} + \frac{\log \left (a x - 1\right ) \log \left (\frac{1}{2} \, a x + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, a x + \frac{1}{2}\right )}{4 \, a c^{2}} - \frac{\log \left (a x + 1\right )}{8 \, a c^{2}} + \int -\frac{2 \, a x -{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) +{\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )}{4 \,{\left (a^{5} c^{2} x^{5} - 2 \, a^{3} c^{2} x^{3} + a c^{2} x +{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a x + 1} \sqrt{a x - 1}\right )}}\,{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 (a x\right )}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{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*} \frac{\int \frac{\operatorname{acosh}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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 )}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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