Optimal. Leaf size=31 \[ \frac{\log \left (a^2 x^4+1\right )}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.0091751, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5034, 260} \[ \frac{\log \left (a^2 x^4+1\right )}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Rule 5034
Rule 260
Rubi steps
\begin{align*} \int x \cot ^{-1}\left (a x^2\right ) \, dx &=\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right )+a \int \frac{x^3}{1+a^2 x^4} \, dx\\ &=\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right )+\frac{\log \left (1+a^2 x^4\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.006245, size = 31, normalized size = 1. \[ \frac{\log \left (a^2 x^4+1\right )}{4 a}+\frac{1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 28, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}{\rm arccot} \left (a{x}^{2}\right )}{2}}+{\frac{\ln \left ({a}^{2}{x}^{4}+1 \right ) }{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993607, size = 38, normalized size = 1.23 \begin{align*} \frac{2 \, a x^{2} \operatorname{arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21044, size = 68, normalized size = 2.19 \begin{align*} \frac{2 \, a x^{2} \operatorname{arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.839381, size = 31, normalized size = 1. \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acot}{\left (a x^{2} \right )}}{2} + \frac{\log{\left (a^{2} x^{4} + 1 \right )}}{4 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 = 1.09056, size = 39, normalized size = 1.26 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\frac{1}{a x^{2}}\right ) + \frac{\log \left (a^{2} x^{4} + 1\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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