Integrand size = 8, antiderivative size = 49 \[ \int x^4 \cot ^{-1}(a x) \, dx=-\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{10 a^5} \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4947, 272, 45} \[ \int x^4 \cot ^{-1}(a x) \, dx=-\frac {x^2}{10 a^3}+\frac {\log \left (a^2 x^2+1\right )}{10 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {x^4}{20 a} \]
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Rule 45
Rule 272
Rule 4947
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int x^4 \cot ^{-1}(a x) \, dx=-\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{10 a^5} \]
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Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{5}+\frac {a^{4} x^{4}}{20}-\frac {a^{2} x^{2}}{10}+\frac {\ln \left (a^{2} x^{2}+1\right )}{10}}{a^{5}}\) | \(46\) |
default | \(\frac {\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{5}+\frac {a^{4} x^{4}}{20}-\frac {a^{2} x^{2}}{10}+\frac {\ln \left (a^{2} x^{2}+1\right )}{10}}{a^{5}}\) | \(46\) |
parallelrisch | \(\frac {4 a^{5} x^{5} \operatorname {arccot}\left (a x \right )+a^{4} x^{4}-2 a^{2} x^{2}+2+2 \ln \left (a^{2} x^{2}+1\right )}{20 a^{5}}\) | \(47\) |
parts | \(\frac {x^{5} \operatorname {arccot}\left (a x \right )}{5}+\frac {a \left (\frac {\frac {1}{2} a^{2} x^{4}-x^{2}}{2 a^{4}}+\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a^{6}}\right )}{5}\) | \(49\) |
risch | \(\frac {i x^{5} \ln \left (i a x +1\right )}{10}-\frac {i x^{5} \ln \left (-i a x +1\right )}{10}+\frac {x^{5} \pi }{10}+\frac {x^{4}}{20 a}-\frac {x^{2}}{10 a^{3}}+\frac {\ln \left (-a^{2} x^{2}-1\right )}{10 a^{5}}\) | \(68\) |
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Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int x^4 \cot ^{-1}(a x) \, dx=\frac {4 \, a^{5} x^{5} \operatorname {arccot}\left (a x\right ) + a^{4} x^{4} - 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} + 1\right )}{20 \, a^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int x^4 \cot ^{-1}(a x) \, dx=\begin {cases} \frac {x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {x^{4}}{20 a} - \frac {x^{2}}{10 a^{3}} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{10 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi x^{5}}{10} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int x^4 \cot ^{-1}(a x) \, dx=\frac {1}{5} \, x^{5} \operatorname {arccot}\left (a x\right ) + \frac {1}{20} \, a {\left (\frac {a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.51 \[ \int x^4 \cot ^{-1}(a x) \, dx=\frac {1}{20} \, {\left (\frac {4 \, x^{5} \arctan \left (\frac {1}{a x}\right )}{a} - \frac {x^{4} {\left (\frac {2}{a^{2} x^{2}} - \frac {3}{a^{4} x^{4}} - 1\right )}}{a^{2}} + \frac {2 \, \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{6}} - \frac {2 \, \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{6}}\right )} a \]
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Time = 0.87 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.14 \[ \int x^4 \cot ^{-1}(a x) \, dx=\left \{\begin {array}{cl} \frac {\pi \,x^5}{10} & \text {\ if\ \ }a=0\\ \frac {2\,\ln \left (a^2\,x^2+1\right )-2\,a^2\,x^2+a^4\,x^4}{20\,a^5}+\frac {x^5\,\mathrm {acot}\left (a\,x\right )}{5} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
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