Optimal. Leaf size=27 \[ \frac{\log \left (\sin ^{-1}(a x)\right )}{2 a^3}-\frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^3} \]
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Rubi [A] time = 0.134613, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4723, 3312, 3302} \[ \frac{\log \left (\sin ^{-1}(a x)\right )}{2 a^3}-\frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 4723
Rule 3312
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin ^2(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cos (2 x)}{2 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac{\log \left (\sin ^{-1}(a x)\right )}{2 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac{\text{Ci}\left (2 \sin ^{-1}(a x)\right )}{2 a^3}+\frac{\log \left (\sin ^{-1}(a x)\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0135606, size = 22, normalized size = 0.81 \[ \frac{\log \left (\sin ^{-1}(a x)\right )-\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 24, normalized size = 0.9 \begin{align*} -{\frac{{\it Ci} \left ( 2\,\arcsin \left ( ax \right ) \right ) }{2\,{a}^{3}}}+{\frac{\ln \left ( \arcsin \left ( ax \right ) \right ) }{2\,{a}^{3}}} \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{x^{2}}{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\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{\sqrt{-a^{2} x^{2} + 1} x^{2}}{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}, 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{x^{2}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{asin}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31592, size = 31, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{3}} + \frac{\log \left (\arcsin \left (a x\right )\right )}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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