Optimal. Leaf size=27 \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2} x^2 \sin ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0152655, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6715, 4619, 261} \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2} x^2 \sin ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 6715
Rule 4619
Rule 261
Rubi steps
\begin{align*} \int x \sin ^{-1}\left (x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sin ^{-1}(x) \, dx,x,x^2\right )\\ &=\frac{1}{2} x^2 \sin ^{-1}\left (x^2\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{1-x^4}}{2}+\frac{1}{2} x^2 \sin ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0044751, size = 24, normalized size = 0.89 \[ \frac{1}{2} \left (\sqrt{1-x^4}+x^2 \sin ^{-1}\left (x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}\arcsin \left ({x}^{2} \right ) }{2}}+{\frac{1}{2}\sqrt{-{x}^{4}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41565, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05932, size = 57, normalized size = 2.11 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.17522, size = 19, normalized size = 0.7 \begin{align*} \frac{x^{2} \operatorname{asin}{\left (x^{2} \right )}}{2} + \frac{\sqrt{1 - x^{4}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05492, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x^{2}\right ) + \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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