Optimal. Leaf size=25 \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]
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Rubi [A] time = 0.0323284, antiderivative size = 25, 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 = {4619, 4677, 8} \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 4619
Rule 4677
Rule 8
Rubi steps
\begin{align*} \int \sin ^{-1}(x)^2 \, dx &=x \sin ^{-1}(x)^2-2 \int \frac{x \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx\\ &=2 \sqrt{1-x^2} \sin ^{-1}(x)+x \sin ^{-1}(x)^2-2 \int 1 \, dx\\ &=-2 x+2 \sqrt{1-x^2} \sin ^{-1}(x)+x \sin ^{-1}(x)^2\\ \end{align*}
Mathematica [A] time = 0.0058889, size = 25, normalized size = 1. \[ 2 \sqrt{1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 24, normalized size = 1. \begin{align*} -2\,x+x \left ( \arcsin \left ( x \right ) \right ) ^{2}+2\,\arcsin \left ( x \right ) \sqrt{-{x}^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42581, size = 31, normalized size = 1.24 \begin{align*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8521, size = 68, normalized size = 2.72 \begin{align*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.177275, size = 22, normalized size = 0.88 \begin{align*} x \operatorname{asin}^{2}{\left (x \right )} - 2 x + 2 \sqrt{1 - x^{2}} \operatorname{asin}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05263, size = 31, normalized size = 1.24 \begin{align*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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