Optimal. Leaf size=41 \[ -\frac{\log \left (a^2 x^4+1\right )}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.0253619, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5034, 266, 43} \[ -\frac{\log \left (a^2 x^4+1\right )}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Rule 5034
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \cot ^{-1}\left (a x^2\right ) \, dx &=\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac{1}{3} a \int \frac{x^7}{1+a^2 x^4} \, dx\\ &=\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac{1}{12} a \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^4\right )\\ &=\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )+\frac{1}{12} a \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^4\right )\\ &=\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right )-\frac{\log \left (1+a^2 x^4\right )}{12 a^3}\\ \end{align*}
Mathematica [A] time = 0.0144449, size = 41, normalized size = 1. \[ -\frac{\log \left (a^2 x^4+1\right )}{12 a^3}+\frac{x^4}{12 a}+\frac{1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 36, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{12\,a}}+{\frac{{x}^{6}{\rm arccot} \left (a{x}^{2}\right )}{6}}-{\frac{\ln \left ({a}^{2}{x}^{4}+1 \right ) }{12\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974629, size = 51, normalized size = 1.24 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{arccot}\left (a x^{2}\right ) + \frac{1}{12} \,{\left (\frac{x^{4}}{a^{2}} - \frac{\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1604, size = 88, normalized size = 2.15 \begin{align*} \frac{2 \, a^{3} x^{6} \operatorname{arccot}\left (a x^{2}\right ) + a^{2} x^{4} - \log \left (a^{2} x^{4} + 1\right )}{12 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.93952, size = 39, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{acot}{\left (a x^{2} \right )}}{6} + \frac{x^{4}}{12 a} - \frac{\log{\left (a^{2} x^{4} + 1 \right )}}{12 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi x^{6}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10586, size = 54, normalized size = 1.32 \begin{align*} \frac{1}{6} \, x^{6} \arctan \left (\frac{1}{a x^{2}}\right ) + \frac{1}{12} \,{\left (\frac{x^{4}}{a^{2}} - \frac{\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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