Integrand size = 8, antiderivative size = 31 \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+\frac {\log \left (1+a^2 x^4\right )}{4 a} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4947, 266} \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\frac {\log \left (a^2 x^4+1\right )}{4 a}+\frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right ) \]
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Rule 266
Rule 4947
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+a \int \frac {x^3}{1+a^2 x^4} \, dx \\ & = \frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+\frac {\log \left (1+a^2 x^4\right )}{4 a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{2} x^2 \cot ^{-1}\left (a x^2\right )+\frac {\log \left (1+a^2 x^4\right )}{4 a} \]
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Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90
method | result | size |
parts | \(\frac {x^{2} \operatorname {arccot}\left (a \,x^{2}\right )}{2}+\frac {\ln \left (a^{2} x^{4}+1\right )}{4 a}\) | \(28\) |
parallelrisch | \(\frac {2 \,\operatorname {arccot}\left (a \,x^{2}\right ) a \,x^{2}+\ln \left (a^{2} x^{4}+1\right )}{4 a}\) | \(29\) |
derivativedivides | \(\frac {\operatorname {arccot}\left (a \,x^{2}\right ) a \,x^{2}+\frac {\ln \left (a^{2} x^{4}+1\right )}{2}}{2 a}\) | \(30\) |
default | \(\frac {\operatorname {arccot}\left (a \,x^{2}\right ) a \,x^{2}+\frac {\ln \left (a^{2} x^{4}+1\right )}{2}}{2 a}\) | \(30\) |
risch | \(\frac {i x^{2} \ln \left (i a \,x^{2}+1\right )}{4}-\frac {i x^{2} \ln \left (-i a \,x^{2}+1\right )}{4}+\frac {\pi \,x^{2}}{4}+\frac {\ln \left (-a^{2} x^{4}-1\right )}{4 a}\) | \(56\) |
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Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\frac {2 \, a x^{2} \operatorname {arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \]
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Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\begin {cases} \frac {x^{2} \operatorname {acot}{\left (a x^{2} \right )}}{2} + \frac {\log {\left (a^{2} x^{4} + 1 \right )}}{4 a} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\frac {2 \, a x^{2} \operatorname {arccot}\left (a x^{2}\right ) + \log \left (a^{2} x^{4} + 1\right )}{4 \, a} \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\frac {1}{4} \, {\left (\frac {2 \, x^{2} \arctan \left (\frac {1}{a x^{2}}\right )}{a} + \frac {\log \left (\frac {1}{a^{2} x^{4}} + 1\right )}{a^{2}} - \frac {\log \left (\frac {1}{a^{2} x^{4}}\right )}{a^{2}}\right )} a \]
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Time = 0.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int x \cot ^{-1}\left (a x^2\right ) \, dx=\frac {x^2\,\mathrm {acot}\left (a\,x^2\right )}{2}+\frac {\ln \left (a^2\,x^4+1\right )}{4\,a} \]
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