Optimal. Leaf size=32 \[ \frac{1}{4} \sqrt{1-x^2} x+\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{4} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0101239, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {4627, 321, 216} \[ \frac{1}{4} \sqrt{1-x^2} x+\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{4} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4627
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x \sin ^{-1}(x) \, dx &=\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{2} \int \frac{x^2}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{4} x \sqrt{1-x^2}+\frac{1}{2} x^2 \sin ^{-1}(x)-\frac{1}{4} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{4} x \sqrt{1-x^2}-\frac{1}{4} \sin ^{-1}(x)+\frac{1}{2} x^2 \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0091866, size = 28, normalized size = 0.88 \[ \frac{1}{4} \left (\sqrt{1-x^2} x+\left (2 x^2-1\right ) \sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.8 \begin{align*} -{\frac{\arcsin \left ( x \right ) }{4}}+{\frac{{x}^{2}\arcsin \left ( x \right ) }{2}}+{\frac{x}{4}\sqrt{-{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4477, size = 32, normalized size = 1. \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x - \frac{1}{4} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98438, size = 68, normalized size = 2.12 \begin{align*} \frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.181605, size = 24, normalized size = 0.75 \begin{align*} \frac{x^{2} \operatorname{asin}{\left (x \right )}}{2} + \frac{x \sqrt{1 - x^{2}}}{4} - \frac{\operatorname{asin}{\left (x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07066, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} \arcsin \left (x\right ) + \frac{1}{4} \, \sqrt{-x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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