Integrand size = 10, antiderivative size = 135 \[ \int x^4 \cot ^{-1}(a x)^2 \, dx=-\frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}-\frac {x^2 \cot ^{-1}(a x)}{5 a^3}+\frac {x^4 \cot ^{-1}(a x)}{10 a}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {3 \arctan (a x)}{10 a^5}-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{5 a^5}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^5} \]
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Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {4947, 5037, 308, 209, 327, 5041, 4965, 2449, 2352} \[ \int x^4 \cot ^{-1}(a x)^2 \, dx=\frac {3 \arctan (a x)}{10 a^5}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{5 a^5}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}-\frac {2 \log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{5 a^5}-\frac {3 x}{10 a^4}-\frac {x^2 \cot ^{-1}(a x)}{5 a^3}+\frac {x^3}{30 a^2}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {x^4 \cot ^{-1}(a x)}{10 a} \]
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Rule 209
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 4947
Rule 4965
Rule 5037
Rule 5041
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {1}{5} (2 a) \int \frac {x^5 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {2 \int x^3 \cot ^{-1}(a x) \, dx}{5 a}-\frac {2 \int \frac {x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a} \\ & = \frac {x^4 \cot ^{-1}(a x)}{10 a}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {1}{10} \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {2 \int x \cot ^{-1}(a x) \, dx}{5 a^3}+\frac {2 \int \frac {x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^3} \\ & = -\frac {x^2 \cot ^{-1}(a x)}{5 a^3}+\frac {x^4 \cot ^{-1}(a x)}{10 a}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {1}{10} \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {2 \int \frac {\cot ^{-1}(a x)}{i-a x} \, dx}{5 a^4}-\frac {\int \frac {x^2}{1+a^2 x^2} \, dx}{5 a^2} \\ & = -\frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}-\frac {x^2 \cot ^{-1}(a x)}{5 a^3}+\frac {x^4 \cot ^{-1}(a x)}{10 a}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{5 a^5}+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{10 a^4}+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{5 a^4}-\frac {2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^4} \\ & = -\frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}-\frac {x^2 \cot ^{-1}(a x)}{5 a^3}+\frac {x^4 \cot ^{-1}(a x)}{10 a}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {3 \arctan (a x)}{10 a^5}-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{5 a^5}+\frac {(2 i) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^5} \\ & = -\frac {3 x}{10 a^4}+\frac {x^3}{30 a^2}-\frac {x^2 \cot ^{-1}(a x)}{5 a^3}+\frac {x^4 \cot ^{-1}(a x)}{10 a}+\frac {i \cot ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \cot ^{-1}(a x)^2+\frac {3 \arctan (a x)}{10 a^5}-\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{5 a^5}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^5} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70 \[ \int x^4 \cot ^{-1}(a x)^2 \, dx=\frac {a x \left (-9+a^2 x^2\right )+6 \left (i+a^5 x^5\right ) \cot ^{-1}(a x)^2+3 \cot ^{-1}(a x) \left (-3-2 a^2 x^2+a^4 x^4-4 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )+6 i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{30 a^5} \]
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Time = 0.49 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.52
method | result | size |
parts | \(\frac {x^{5} \operatorname {arccot}\left (a x \right )^{2}}{5}+\frac {\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )}{10}-\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{5}+\frac {a^{3} x^{3}}{30}-\frac {3 a x}{10}+\frac {3 \arctan \left (a x \right )}{10}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{10}}{a^{5}}\) | \(205\) |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )}{10}-\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{5}+\frac {a^{3} x^{3}}{30}-\frac {3 a x}{10}+\frac {3 \arctan \left (a x \right )}{10}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{10}}{a^{5}}\) | \(206\) |
default | \(\frac {\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )^{2}}{5}+\frac {a^{4} x^{4} \operatorname {arccot}\left (a x \right )}{10}-\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{5}+\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{5}+\frac {a^{3} x^{3}}{30}-\frac {3 a x}{10}+\frac {3 \arctan \left (a x \right )}{10}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{10}}{a^{5}}\) | \(206\) |
risch | \(\frac {i \pi \ln \left (i a x +1\right ) x^{5}}{10}-\frac {i \ln \left (i a x +1\right ) x^{2}}{10 a^{3}}+\frac {i \ln \left (i a x +1\right ) x^{4}}{20 a}-\frac {\pi \,x^{2}}{10 a^{3}}+\frac {\pi \,x^{4}}{20 a}+\frac {23 i \ln \left (a^{2} x^{2}+1\right )}{150 a^{5}}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{5 a^{5}}-\frac {47 i \ln \left (-i a x +1\right )}{600 a^{5}}+\frac {\ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {i \pi ^{2}}{20 a^{5}}-\frac {137 i \ln \left (i a x +1\right )}{600 a^{5}}+\frac {i \ln \left (i a x +1\right )^{2}}{20 a^{5}}-\frac {i \ln \left (-i a x +1\right )^{2}}{20 a^{5}}+\frac {\pi \ln \left (a^{2} x^{2}+1\right )}{10 a^{5}}-\frac {137 \pi }{300 a^{5}}+\frac {\pi ^{2} x^{5}}{20}+\frac {413 i}{2250 a^{5}}-\frac {\ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {\ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {i \ln \left (-i a x +1\right ) x^{4}}{20 a}+\frac {i \ln \left (-i a x +1\right ) x^{2}}{10 a^{3}}+\frac {i \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{10 a^{5}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{5 a^{5}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{5 a^{5}}-\frac {i \pi \ln \left (-i a x +1\right ) x^{5}}{10}-\frac {3 x}{10 a^{4}}+\frac {x^{3}}{30 a^{2}}+\frac {3 \arctan \left (a x \right )}{20 a^{5}}\) | \(393\) |
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\[ \int x^4 \cot ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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\[ \int x^4 \cot ^{-1}(a x)^2 \, dx=\int x^{4} \operatorname {acot}^{2}{\left (a x \right )}\, dx \]
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\[ \int x^4 \cot ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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\[ \int x^4 \cot ^{-1}(a x)^2 \, dx=\int { x^{4} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int x^4 \cot ^{-1}(a x)^2 \, dx=\int x^4\,{\mathrm {acot}\left (a\,x\right )}^2 \,d x \]
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