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 = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 43
Rule 266
Rule 5034
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*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 1.00 \[ -\frac {\log \left (a^2 x^4+1\right )}{12 a^3}+\frac {x^4}{12 a}+\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 39, normalized size = 0.95 \[ \frac {2 \, a^{3} x^{6} \operatorname {arccot}\left (a x^{2}\right ) + a^{2} x^{4} - \log \left (a^{2} x^{4} + 1\right )}{12 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 0.98 \[ \frac {1}{6} \, x^{6} \arctan \left (\frac {1}{a x^{2}}\right ) + \frac {1}{12} \, {\left (\frac {x^{4}}{a^{2}} - \frac {\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 0.88 \[ \frac {x^{4}}{12 a}+\frac {x^{6} \mathrm {arccot}\left (a \,x^{2}\right )}{6}-\frac {\ln \left (a^{2} x^{4}+1\right )}{12 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 38, normalized size = 0.93 \[ \frac {1}{6} \, x^{6} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{12} \, {\left (\frac {x^{4}}{a^{2}} - \frac {\log \left (a^{2} x^{4} + 1\right )}{a^{4}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 35, normalized size = 0.85 \[ \frac {x^6\,\mathrm {acot}\left (a\,x^2\right )}{6}-\frac {\ln \left (a^2\,x^4+1\right )}{12\,a^3}+\frac {x^4}{12\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.78, size = 39, normalized size = 0.95 \[ \begin {cases} \frac {x^{6} \operatorname {acot}{\left (a x^{2} \right )}}{6} + \frac {x^{4}}{12 a} - \frac {\log {\left (a^{2} x^{4} + 1 \right )}}{12 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi x^{6}}{12} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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