Optimal. Leaf size=25 \[ -2 x+2 \sqrt {1-x^2} \sin ^{-1}(x)+x \sin ^{-1}(x)^2 \]
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Rubi [A]
time = 0.02, 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 = {4715, 4767, 8}
\begin {gather*} 2 \sqrt {1-x^2} \text {ArcSin}(x)+x \text {ArcSin}(x)^2-2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4715
Rule 4767
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 \begin {gather*} -2 x+2 \sqrt {1-x^2} \sin ^{-1}(x)+x \sin ^{-1}(x)^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 24, normalized size = 0.96
method | result | size |
default | \(-2 x +x \arcsin \left (x \right )^{2}+2 \arcsin \left (x \right ) \sqrt {-x^{2}+1}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.87, size = 23, normalized size = 0.92 \begin {gather*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.94, size = 23, normalized size = 0.92 \begin {gather*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 22, normalized size = 0.88 \begin {gather*} x \operatorname {asin}^{2}{\left (x \right )} - 2 x + 2 \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 23, normalized size = 0.92 \begin {gather*} x \arcsin \left (x\right )^{2} + 2 \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - 2 \, x \end {gather*}
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 \begin {gather*} 2\,\mathrm {asin}\left (x\right )\,\sqrt {1-x^2}+x\,\left ({\mathrm {asin}\left (x\right )}^2-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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