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 = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 8
Rule 4619
Rule 4677
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ 2 \sqrt {1-x^2} \sin ^{-1}(x)-2 x+x \sin ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 23, normalized size = 0.92 \[ x \arcsin \relax (x)^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \relax (x) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.02, size = 23, normalized size = 0.92 \[ x \arcsin \relax (x)^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \relax (x) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 24, normalized size = 0.96 \[ x \arcsin \relax (x )^{2}-2 x +2 \sqrt {-x^{2}+1}\, \arcsin \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 23, normalized size = 0.92 \[ x \arcsin \relax (x)^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \relax (x) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 22, normalized size = 0.88 \[ 2\,\mathrm {asin}\relax (x)\,\sqrt {1-x^2}+x\,\left ({\mathrm {asin}\relax (x)}^2-2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 22, normalized size = 0.88 \[ x \operatorname {asin}^{2}{\relax (x )} - 2 x + 2 \sqrt {1 - x^{2}} \operatorname {asin}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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