Optimal. Leaf size=41 \[ \frac {x^4}{12 a}+\frac {1}{6} x^6 \cot ^{-1}\left (a x^2\right )-\frac {\log \left (1+a^2 x^4\right )}{12 a^3} \]
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Rubi [A]
time = 0.02, 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 = {4947, 272, 45}
\begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4947
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 \text {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 \text {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 = 0.01, size = 41, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 40, normalized size = 0.98
method | result | size |
default | \(\frac {x^{6} \mathrm {arccot}\left (a \,x^{2}\right )}{6}+\frac {a \left (\frac {x^{4}}{4 a^{2}}-\frac {\ln \left (a^{2} x^{4}+1\right )}{4 a^{4}}\right )}{3}\) | \(40\) |
risch | \(\frac {i x^{6} \ln \left (i a \,x^{2}+1\right )}{12}-\frac {i x^{6} \ln \left (-i a \,x^{2}+1\right )}{12}+\frac {x^{6} \pi }{12}+\frac {x^{4}}{12 a}-\frac {\ln \left (-a^{2} x^{4}-1\right )}{12 a^{3}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 38, normalized size = 0.93 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 39, normalized size = 0.95 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 39, normalized size = 0.95 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 40, normalized size = 0.98 \begin {gather*} \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 {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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