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 = 0.0236618, antiderivative size = 35, normalized size of antiderivative = 1., 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 6715
Rule 4620
Rule 261
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*}
Mathematica [A] time = 0.0161599, size = 35, normalized size = 1. \[ \frac{1}{2} x^2 \cos ^{-1}\left (a x^2\right )-\frac{\sqrt{1-a^2 x^4}}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{2\,a} \left ({x}^{2}a\arccos \left ( a{x}^{2} \right ) -\sqrt{-{a}^{2}{x}^{4}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44838, size = 42, normalized size = 1.2 \begin{align*} \frac{a x^{2} \arccos \left (a x^{2}\right ) - \sqrt{-a^{2} x^{4} + 1}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37848, size = 68, normalized size = 1.94 \begin{align*} \frac{a x^{2} \arccos \left (a x^{2}\right ) - \sqrt{-a^{2} x^{4} + 1}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.229752, size = 32, normalized size = 0.91 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32775, size = 42, normalized size = 1.2 \begin{align*} \frac{a x^{2} \arccos \left (a x^{2}\right ) - \sqrt{-a^{2} x^{4} + 1}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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