Optimal. Leaf size=29 \[ \frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]
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Rubi [A] time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4677, 8} \[ \frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 4677
Rubi steps
\begin {align*} \int \frac {x \sin ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac {\int 1 \, dx}{a}\\ &=\frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ \frac {x}{a}-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 26, normalized size = 0.90 \[ \frac {a x - \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 27, normalized size = 0.93 \[ \frac {x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 62, normalized size = 2.14 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2} \arcsin \left (a x \right )-\arcsin \left (a x \right )+a x \sqrt {-a^{2} x^{2}+1}\right )}{a^{2} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 27, normalized size = 0.93 \[ \frac {x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\mathrm {asin}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 24, normalized size = 0.83 \[ \begin {cases} \frac {x}{a} - \frac {\sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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