Optimal. Leaf size=35 \[ \frac {1}{2} x^2 \cos ^{-1}\left (a x^2\right )-\frac {\sqrt {1-a^2 x^4}}{2 a} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6715, 4620, 261} \[ \frac {1}{2} x^2 \cos ^{-1}\left (a x^2\right )-\frac {\sqrt {1-a^2 x^4}}{2 a} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4620
Rule 6715
Rubi steps
\begin {align*} \int x \cos ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \cos ^{-1}(a x) \, dx,x,x^2\right )\\ &=\frac {1}{2} x^2 \cos ^{-1}\left (a x^2\right )+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^4}}{2 a}+\frac {1}{2} x^2 \cos ^{-1}\left (a x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 1.00 \[ \frac {1}{2} x^2 \cos ^{-1}\left (a x^2\right )-\frac {\sqrt {1-a^2 x^4}}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 31, normalized size = 0.89 \[ \frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 31, normalized size = 0.89 \[ \frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.91 \[ \frac {x^{2} a \arccos \left (a \,x^{2}\right )-\sqrt {-a^{2} x^{4}+1}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 31, normalized size = 0.89 \[ \frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 29, normalized size = 0.83 \[ \frac {x^2\,\mathrm {acos}\left (a\,x^2\right )}{2}-\frac {\sqrt {1-a^2\,x^4}}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 32, normalized size = 0.91 \[ \begin {cases} \frac {x^{2} \operatorname {acos}{\left (a x^{2} \right )}}{2} - \frac {\sqrt {- a^{2} x^{4} + 1}}{2 a} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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