Optimal. Leaf size=44 \[ -\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sin ^{-1}(a x)^2}{2 x^2} \]
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Rubi [A] time = 0.0798736, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4627, 4681, 29} \[ -\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sin ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4681
Rule 29
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^2}{x^3} \, dx &=-\frac{\sin ^{-1}(a x)^2}{2 x^2}+a \int \frac{\sin ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}-\frac{\sin ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac{1}{x} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}-\frac{\sin ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0254089, size = 44, normalized size = 1. \[ -\frac{a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\sin ^{-1}(a x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 43, normalized size = 1. \begin{align*} -{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac{a\arcsin \left ( ax \right ) }{x}\sqrt{-{a}^{2}{x}^{2}+1}}+{a}^{2}\ln \left ( ax \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62754, size = 54, normalized size = 1.23 \begin{align*} a^{2} \log \left (x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} a \arcsin \left (a x\right )}{x} - \frac{\arcsin \left (a x\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56778, size = 112, normalized size = 2.55 \begin{align*} \frac{2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1} a x \arcsin \left (a x\right ) - \arcsin \left (a x\right )^{2}}{2 \, x^{2}} \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{asin}^{2}{\left (a x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38224, size = 116, normalized size = 2.64 \begin{align*} \frac{1}{2} \,{\left ({\left (\frac{a^{4} x}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{x{\left | a \right |}}\right )} \arcsin \left (a x\right ) + a \log \left (a^{2} x^{2}\right )\right )} a - \frac{\arcsin \left (a x\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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