Integrand size = 8, antiderivative size = 53 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {x \cot ^{-1}(a x)}{a}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {\log \left (1+a^2 x^2\right )}{2 a^2} \]
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Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4947, 5037, 4931, 266, 5005} \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {\log \left (a^2 x^2+1\right )}{2 a^2}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {x \cot ^{-1}(a x)}{a} \]
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Rule 266
Rule 4931
Rule 4947
Rule 5005
Rule 5037
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \cot ^{-1}(a x)^2+a \int \frac {x^2 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {\int \cot ^{-1}(a x) \, dx}{a}-\frac {\int \frac {\cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{a} \\ & = \frac {x \cot ^{-1}(a x)}{a}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\int \frac {x}{1+a^2 x^2} \, dx \\ & = \frac {x \cot ^{-1}(a x)}{a}+\frac {\cot ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)^2+\frac {\log \left (1+a^2 x^2\right )}{2 a^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {2 a x \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2+\log \left (1+a^2 x^2\right )}{2 a^2} \]
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Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}+2 \,\operatorname {arccot}\left (a x \right ) a x +\operatorname {arccot}\left (a x \right )^{2}+\ln \left (a^{2} x^{2}+1\right )}{2 a^{2}}\) | \(44\) |
parts | \(\frac {x^{2} \operatorname {arccot}\left (a x \right )^{2}}{2}+\frac {-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{a^{2}}\) | \(55\) |
derivativedivides | \(\frac {\frac {a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}}{2}-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{a^{2}}\) | \(57\) |
default | \(\frac {\frac {a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}}{2}-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\operatorname {arccot}\left (a x \right ) a x +\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}}{a^{2}}\) | \(57\) |
risch | \(-\frac {\left (a^{2} x^{2}+1\right ) \ln \left (i a x +1\right )^{2}}{8 a^{2}}+\frac {\left (i \pi \,a^{2} x^{2}+x^{2} \ln \left (-i a x +1\right ) a^{2}+2 i a x +\ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{4 a^{2}}-\frac {i \pi \,x^{2} \ln \left (-i a x +1\right )}{4}+\frac {\pi ^{2} x^{2}}{8}-\frac {x^{2} \ln \left (-i a x +1\right )^{2}}{8}-\frac {i x \ln \left (-i a x +1\right )}{2 a}+\frac {\pi x}{2 a}-\frac {\pi \arctan \left (a x \right )}{2 a^{2}}-\frac {\ln \left (-i a x +1\right )^{2}}{8 a^{2}}+\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a^{2}}\) | \(178\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {2 \, a x \operatorname {arccot}\left (a x\right ) + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )^{2} + \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int x \cot ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{2} \operatorname {acot}^{2}{\left (a x \right )}}{2} + \frac {x \operatorname {acot}{\left (a x \right )}}{a} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a^{2}} + \frac {\operatorname {acot}^{2}{\left (a x \right )}}{2 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{2}}{8} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {1}{2} \, x^{2} \operatorname {arccot}\left (a x\right )^{2} + a {\left (\frac {x}{a^{2}} - \frac {\arctan \left (a x\right )}{a^{3}}\right )} \operatorname {arccot}\left (a x\right ) - \frac {\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right )}{2 \, a^{2}} \]
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\[ \int x \cot ^{-1}(a x)^2 \, dx=\int { x \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int x \cot ^{-1}(a x)^2 \, dx=\frac {x^2\,{\mathrm {acot}\left (a\,x\right )}^2}{2}+\frac {\frac {{\mathrm {acot}\left (a\,x\right )}^2}{2}+a\,x\,\mathrm {acot}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2+1\right )}{2}}{a^2} \]
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