Optimal. Leaf size=27 \[ \frac {\log \left (\sin ^{-1}(a x)\right )}{2 a^3}-\frac {\text {Ci}\left (2 \sin ^{-1}(a x)\right )}{2 a^3} \]
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Rubi [A] time = 0.13, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 3302
Rule 3312
Rule 4723
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*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.81 \[ \frac {\log \left (\sin ^{-1}(a x)\right )-\text {Ci}\left (2 \sin ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 23, normalized size = 0.85 \[ -\frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{3}} + \frac {\log \left (\arcsin \left (a x\right )\right )}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.89 \[ -\frac {\Ci \left (2 \arcsin \left (a x \right )\right )}{2 a^{3}}+\frac {\ln \left (\arcsin \left (a x \right )\right )}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^2}{\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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