Optimal. Leaf size=85 \[ -\frac{a^2}{12 x^2}+\frac{a^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 x}-\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{6 x^3}-\frac{1}{3} a^4 \log (x)-\frac{\sinh ^{-1}(a x)^2}{4 x^4} \]
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Rubi [A] time = 0.137922, antiderivative size = 85, 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 = {5661, 5747, 5723, 29, 30} \[ -\frac{a^2}{12 x^2}+\frac{a^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 x}-\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{6 x^3}-\frac{1}{3} a^4 \log (x)-\frac{\sinh ^{-1}(a x)^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 5747
Rule 5723
Rule 29
Rule 30
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)^2}{x^5} \, dx &=-\frac{\sinh ^{-1}(a x)^2}{4 x^4}+\frac{1}{2} a \int \frac{\sinh ^{-1}(a x)}{x^4 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{6 x^3}-\frac{\sinh ^{-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{\sinh ^{-1}(a x)}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a^2}{12 x^2}-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{6 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 x}-\frac{\sinh ^{-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^2 x^2} \sinh ^{-1}(a x)}{6 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{3 x}-\frac{\sinh ^{-1}(a x)^2}{4 x^4}-\frac{1}{3} a^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0592826, size = 64, normalized size = 0.75 \[ -\frac{a^2 x^2+4 a^4 x^4 \log (x)-2 a x \sqrt{a^2 x^2+1} \left (2 a^2 x^2-1\right ) \sinh ^{-1}(a x)+3 \sinh ^{-1}(a x)^2}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 99, normalized size = 1.2 \begin{align*}{\frac{{a}^{4}{\it Arcsinh} \left ( ax \right ) }{3}}+{\frac{{a}^{3}{\it Arcsinh} \left ( ax \right ) }{3\,x}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{a{\it Arcsinh} \left ( ax \right ) }{6\,{x}^{3}}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}}{12\,{x}^{2}}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{4\,{x}^{4}}}-{\frac{{a}^{4}}{3}\ln \left ( \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22541, size = 96, normalized size = 1.13 \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{arsinh}\left (a x\right ) - \frac{\operatorname{arsinh}\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 = 2.16209, size = 194, normalized size = 2.28 \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{asinh}^{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 [B] time = 1.59414, size = 200, normalized size = 2.35 \begin{align*} -\frac{1}{12} \,{\left (2 \, a^{3} \log \left (x^{2}\right ) - 4 \, a^{3} \log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\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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