Optimal. Leaf size=49 \[ -\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{10 a^5} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 272, 45}
\begin {gather*} -\frac {x^2}{10 a^3}+\frac {\log \left (a^2 x^2+1\right )}{10 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {x^4}{20 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4947
Rubi steps
\begin {align*} \int x^4 \cot ^{-1}(a x) \, dx &=\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {1}{5} a \int \frac {x^5}{1+a^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {1}{10} a \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {1}{10} a \text {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{10 a^5}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.00 \begin {gather*} -\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{10 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 46, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \mathrm {arccot}\left (a x \right )}{5}+\frac {a^{4} x^{4}}{20}-\frac {a^{2} x^{2}}{10}+\frac {\ln \left (a^{2} x^{2}+1\right )}{10}}{a^{5}}\) | \(46\) |
default | \(\frac {\frac {a^{5} x^{5} \mathrm {arccot}\left (a x \right )}{5}+\frac {a^{4} x^{4}}{20}-\frac {a^{2} x^{2}}{10}+\frac {\ln \left (a^{2} x^{2}+1\right )}{10}}{a^{5}}\) | \(46\) |
risch | \(\frac {i x^{5} \ln \left (i a x +1\right )}{10}-\frac {i x^{5} \ln \left (-i a x +1\right )}{10}+\frac {x^{5} \pi }{10}+\frac {x^{4}}{20 a}-\frac {x^{2}}{10 a^{3}}+\frac {\ln \left (-a^{2} x^{2}-1\right )}{10 a^{5}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 46, normalized size = 0.94 \begin {gather*} \frac {1}{5} \, x^{5} \operatorname {arccot}\left (a x\right ) + \frac {1}{20} \, a {\left (\frac {a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.33, size = 45, normalized size = 0.92 \begin {gather*} \frac {4 \, a^{5} x^{5} \operatorname {arccot}\left (a x\right ) + a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} + 1\right )}{20 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 46, normalized size = 0.94 \begin {gather*} \begin {cases} \frac {x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {x^{4}}{20 a} - \frac {x^{2}}{10 a^{3}} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi x^{5}}{10} & \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 = 74, normalized size = 1.51 \begin {gather*} \frac {1}{20} \, {\left (\frac {4 \, x^{5} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{4} {\left (\frac {2}{a^{2} x^{2}} - \frac {3}{a^{4} x^{4}} - 1\right )}}{a^{2}} + \frac {2 \, \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{6}} - \frac {2 \, \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{6}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.70, size = 56, normalized size = 1.14 \begin {gather*} \left \{\begin {array}{cl} \frac {\pi \,x^5}{10} & \text {\ if\ \ }a=0\\ \frac {2\,\ln \left (a^2\,x^2+1\right )-2\,a^2\,x^2+a^4\,x^4}{20\,a^5}+\frac {x^5\,\mathrm {acot}\left (a\,x\right )}{5} & \text {\ if\ \ }a\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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