Integrand size = 10, antiderivative size = 37 \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )-\frac {\arctan \left (a x^2\right )}{4 a^2} \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4947, 281, 327, 209} \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=-\frac {\arctan \left (a x^2\right )}{4 a^2}+\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right ) \]
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Rule 209
Rule 281
Rule 327
Rule 4947
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )+\frac {1}{2} a \int \frac {x^5}{1+a^2 x^4} \, dx \\ & = \frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )+\frac {1}{4} a \text {Subst}\left (\int \frac {x^2}{1+a^2 x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )-\frac {\text {Subst}\left (\int \frac {1}{1+a^2 x^2} \, dx,x,x^2\right )}{4 a} \\ & = \frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )-\frac {\arctan \left (a x^2\right )}{4 a^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {x^2}{4 a}+\frac {1}{4} x^4 \cot ^{-1}\left (a x^2\right )-\frac {\arctan \left (a x^2\right )}{4 a^2} \]
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Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84
method | result | size |
parallelrisch | \(\frac {\operatorname {arccot}\left (a \,x^{2}\right ) a^{2} x^{4}+a \,x^{2}+\operatorname {arccot}\left (a \,x^{2}\right )}{4 a^{2}}\) | \(31\) |
default | \(\frac {x^{4} \operatorname {arccot}\left (a \,x^{2}\right )}{4}+\frac {a \left (\frac {x^{2}}{2 a^{2}}-\frac {\arctan \left (a \,x^{2}\right )}{2 a^{3}}\right )}{2}\) | \(36\) |
parts | \(\frac {x^{4} \operatorname {arccot}\left (a \,x^{2}\right )}{4}+\frac {a \left (\frac {x^{2}}{2 a^{2}}-\frac {\arctan \left (a \,x^{2}\right )}{2 a^{3}}\right )}{2}\) | \(36\) |
risch | \(\frac {i x^{4} \ln \left (i a \,x^{2}+1\right )}{8}-\frac {i x^{4} \ln \left (-i a \,x^{2}+1\right )}{8}+\frac {\pi \,x^{4}}{8}+\frac {x^{2}}{4 a}-\frac {\arctan \left (a \,x^{2}\right )}{4 a^{2}}+\frac {1}{8 \pi \,a^{2}}\) | \(67\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {a x^{2} + {\left (a^{2} x^{4} + 1\right )} \operatorname {arccot}\left (a x^{2}\right )}{4 \, a^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=\begin {cases} \frac {x^{4} \operatorname {acot}{\left (a x^{2} \right )}}{4} + \frac {x^{2}}{4 a} + \frac {\operatorname {acot}{\left (a x^{2} \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{4} \, x^{4} \operatorname {arccot}\left (a x^{2}\right ) + \frac {1}{4} \, a {\left (\frac {x^{2}}{a^{2}} - \frac {\arctan \left (a x^{2}\right )}{a^{3}}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{4} \, {\left (\frac {x^{4} \arctan \left (\frac {1}{a x^{2}}\right )}{a} + \frac {x^{2}}{a^{2}} + \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{a^{3}}\right )} a \]
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Time = 0.71 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int x^3 \cot ^{-1}\left (a x^2\right ) \, dx=\frac {x^4\,\mathrm {acot}\left (a\,x^2\right )}{4}-\frac {\mathrm {atan}\left (a\,x^2\right )}{4\,a^2}+\frac {x^2}{4\,a} \]
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