Optimal. Leaf size=95 \[ \frac{a^2}{12 x^2}-\frac{1}{3} a^4 \log (x)+\frac{a^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 x}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{6 x^3}-\frac{\cosh ^{-1}(a x)^2}{4 x^4} \]
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Rubi [A] time = 0.364883, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5662, 5748, 5724, 29, 30} \[ \frac{a^2}{12 x^2}-\frac{1}{3} a^4 \log (x)+\frac{a^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 x}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{6 x^3}-\frac{\cosh ^{-1}(a x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 5662
Rule 5748
Rule 5724
Rule 29
Rule 30
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^2}{x^5} \, dx &=-\frac{\cosh ^{-1}(a x)^2}{4 x^4}+\frac{1}{2} a \int \frac{\cosh ^{-1}(a x)}{x^4 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{6 x^3}-\frac{\cosh ^{-1}(a x)^2}{4 x^4}-\frac{1}{6} a^2 \int \frac{1}{x^3} \, dx+\frac{1}{3} a^3 \int \frac{\cosh ^{-1}(a x)}{x^2 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{a^2}{12 x^2}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{6 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 x}-\frac{\cosh ^{-1}(a x)^2}{4 x^4}-\frac{1}{3} a^4 \int \frac{1}{x} \, dx\\ &=\frac{a^2}{12 x^2}+\frac{a \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{6 x^3}+\frac{a^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 x}-\frac{\cosh ^{-1}(a x)^2}{4 x^4}-\frac{1}{3} a^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0776289, size = 69, normalized size = 0.73 \[ \frac{a^2 x^2-4 a^4 x^4 \log (x)+2 a x \sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2+1\right ) \cosh ^{-1}(a x)-3 \cosh ^{-1}(a x)^2}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 109, normalized size = 1.2 \begin{align*}{\frac{{a}^{4}{\rm arccosh} \left (ax\right )}{3}}+{\frac{{a}^{3}{\rm arccosh} \left (ax\right )}{3\,x}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{a{\rm arccosh} \left (ax\right )}{6\,{x}^{3}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{a}^{2}}{12\,{x}^{2}}}-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{4\,{x}^{4}}}-{\frac{{a}^{4}}{3}\ln \left ( 1+ \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 [A] time = 1.55523, size = 97, normalized size = 1.02 \begin{align*} -\frac{1}{12} \,{\left (4 \, a^{2} \log \left (x\right ) - \frac{1}{x^{2}}\right )} a^{2} + \frac{1}{6} \,{\left (\frac{2 \, \sqrt{a^{2} x^{2} - 1} a^{2}}{x} + \frac{\sqrt{a^{2} x^{2} - 1}}{x^{3}}\right )} a \operatorname{arcosh}\left (a x\right ) - \frac{\operatorname{arcosh}\left (a x\right )^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54507, size = 194, normalized size = 2.04 \begin{align*} -\frac{4 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} - 2 \,{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) + 3 \, \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{12 \, x^{4}} \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^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.62617, size = 198, normalized size = 2.08 \begin{align*} -\frac{1}{12} \,{\left (2 \, a^{3} \log \left (x^{2}\right ) - 4 \, a^{3} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) - \frac{8 \,{\left (3 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )} a^{2}{\left | a \right |} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3}} - \frac{2 \, a^{3} x^{2} + a}{x^{2}}\right )} a - \frac{\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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