Integrand size = 10, antiderivative size = 111 \[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2-\frac {\arctan (a x)}{3 a^3}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^3} \]
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Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4947, 5037, 327, 209, 5041, 4965, 2449, 2352} \[ \int x^2 \cot ^{-1}(a x)^2 \, dx=-\frac {\arctan (a x)}{3 a^3}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{3 a^3}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{3 a^3}+\frac {x}{3 a^2}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {x^2 \cot ^{-1}(a x)}{3 a} \]
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Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4947
Rule 4965
Rule 5037
Rule 5041
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {1}{3} (2 a) \int \frac {x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2 \int x \cot ^{-1}(a x) \, dx}{3 a}-\frac {2 \int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a} \\ & = \frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {1}{3} \int \frac {x^2}{1+a^2 x^2} \, dx+\frac {2 \int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{3 a^2} \\ & = \frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {\int \frac {1}{1+a^2 x^2} \, dx}{3 a^2}+\frac {2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2} \\ & = \frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2-\frac {\arctan (a x)}{3 a^3}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {(2 i) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^3} \\ & = \frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2-\frac {\arctan (a x)}{3 a^3}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^3} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68 \[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\frac {a x+\left (-i+a^3 x^3\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (1+a^2 x^2+2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{3 a^3} \]
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Time = 0.46 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.67
method | result | size |
parts | \(\frac {x^{3} \operatorname {arccot}\left (a x \right )^{2}}{3}+\frac {\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}}{a^{3}}\) | \(185\) |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )^{2}}{3}+\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}}{a^{3}}\) | \(186\) |
default | \(\frac {\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )^{2}}{3}+\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}}{a^{3}}\) | \(186\) |
risch | \(\frac {\pi ^{2} x^{3}}{12}+\frac {\pi \,x^{2}}{6 a}+\frac {\ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a^{3}}-\frac {2 i \ln \left (a^{2} x^{2}+1\right )}{9 a^{3}}+\frac {5 i \ln \left (-i a x +1\right )}{36 a^{3}}+\frac {11 \pi }{18 a^{3}}-\frac {\ln \left (-i a x +1\right )^{2} x^{3}}{12}-\frac {\ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {17 i}{54 a^{3}}+\frac {x}{3 a^{2}}-\frac {i \ln \left (-i a x +1\right ) x^{2}}{6 a}+\frac {i \ln \left (i a x +1\right ) x^{2}}{6 a}-\frac {\pi \ln \left (a^{2} x^{2}+1\right )}{6 a^{3}}+\frac {11 i \ln \left (i a x +1\right )}{36 a^{3}}-\frac {i \pi \ln \left (-i a x +1\right ) x^{3}}{6}+\frac {i \ln \left (-i a x +1\right )^{2}}{12 a^{3}}-\frac {i \pi ^{2}}{12 a^{3}}-\frac {i \ln \left (i a x +1\right )^{2}}{12 a^{3}}-\frac {i \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{6 a^{3}}-\frac {\arctan \left (a x \right )}{6 a^{3}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{3 a^{3}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a^{3}}+\frac {i \pi \ln \left (i a x +1\right ) x^{3}}{6}\) | \(342\) |
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\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int x^{2} \operatorname {acot}^{2}{\left (a x \right )}\, dx \]
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\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int x^2\,{\mathrm {acot}\left (a\,x\right )}^2 \,d x \]
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