Optimal. Leaf size=35 \[ \frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]
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Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4619, 4677, 8} \[ \frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 4619
Rule 4677
Rubi steps
\begin {align*} \int \sin ^{-1}(a x)^2 \, dx &=x \sin ^{-1}(a x)^2-(2 a) \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 \int 1 \, dx\\ &=-2 x+\frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ \frac {2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 36, normalized size = 1.03 \[ \frac {a x \arcsin \left (a x\right )^{2} - 2 \, a x + 2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 33, normalized size = 0.94 \[ x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 1.06 \[ \frac {a x \arcsin \left (a x \right )^{2}-2 a x +2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 33, normalized size = 0.94 \[ x \arcsin \left (a x\right )^{2} - 2 \, x + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 32, normalized size = 0.91 \[ x\,\left ({\mathrm {asin}\left (a\,x\right )}^2-2\right )+\frac {2\,\mathrm {asin}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 32, normalized size = 0.91 \[ \begin {cases} x \operatorname {asin}^{2}{\left (a x \right )} - 2 x + \frac {2 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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