Integrand size = 20, antiderivative size = 156 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{2 c^2 x}+\frac {a^3 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{4 c^2}-\frac {\arctan (a x)}{2 c^2 x^2}-\frac {a^2 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^2 \arctan (a x)^2}{c^2}-\frac {2 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2} \]
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Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5086, 5038, 4946, 331, 209, 5044, 4988, 2497, 5050, 205, 211} \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a^2 \arctan (a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {i a^2 \arctan (a x)^2}{c^2}-\frac {a^2 \arctan (a x)}{4 c^2}-\frac {2 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {i a^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^2}+\frac {a^3 x}{4 c^2 \left (a^2 x^2+1\right )}-\frac {\arctan (a x)}{2 c^2 x^2}-\frac {a}{2 c^2 x} \]
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Rule 205
Rule 209
Rule 211
Rule 331
Rule 2497
Rule 4946
Rule 4988
Rule 5038
Rule 5044
Rule 5050
Rule 5086
Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = a^4 \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\arctan (a x)}{x^3} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = -\frac {\arctan (a x)}{2 c^2 x^2}-\frac {a^2 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {1}{2} a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {a \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^2}{2 c^2}+\frac {\left (i a^2\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^2}\right ) \\ & = -\frac {a}{2 c^2 x}+\frac {a^3 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{2 c^2 x^2}-\frac {a^2 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a^3 \int \frac {1}{1+a^2 x^2} \, dx}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^2}{2 c^2}+\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {a^3 \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right )+\frac {a^3 \int \frac {1}{c+a^2 c x^2} \, dx}{4 c} \\ & = -\frac {a}{2 c^2 x}+\frac {a^3 x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{4 c^2}-\frac {\arctan (a x)}{2 c^2 x^2}-\frac {a^2 \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-2 \left (-\frac {i a^2 \arctan (a x)^2}{2 c^2}+\frac {a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}\right ) \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60 \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 \left (-\frac {4}{a x}+8 i \arctan (a x)^2+\arctan (a x) \left (-4-\frac {4}{a^2 x^2}-2 \cos (2 \arctan (a x))-16 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+8 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+\sin (2 \arctan (a x))\right )}{8 c^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.65 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.62
method | result | size |
parts | \(\frac {\arctan \left (a x \right ) a^{2} \ln \left (a^{2} x^{2}+1\right )}{c^{2}}-\frac {a^{2} \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{2 c^{2} x^{2}}-\frac {2 \arctan \left (a x \right ) a^{2} \ln \left (x \right )}{c^{2}}-\frac {a \left (-4 a^{2} \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{2 a}-\frac {i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{2 a}\right )-\frac {a^{2} x}{2 \left (a^{2} x^{2}+1\right )}+\frac {a \arctan \left (a x \right )}{2}+\frac {1}{x}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{2}\right )}{2 c^{2}}\) | \(253\) |
derivativedivides | \(a^{2} \left (-\frac {\arctan \left (a x \right )}{2 c^{2} a^{2} x^{2}}-\frac {2 \arctan \left (a x \right ) \ln \left (a x \right )}{c^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{c^{2}}-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {1}{a x}-\frac {a x}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}\right )\) | \(286\) |
default | \(a^{2} \left (-\frac {\arctan \left (a x \right )}{2 c^{2} a^{2} x^{2}}-\frac {2 \arctan \left (a x \right ) \ln \left (a x \right )}{c^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{c^{2}}-\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\frac {1}{a x}-\frac {a x}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2}}{2 c^{2}}\right )\) | \(286\) |
risch | \(-\frac {a}{2 c^{2} x}+\frac {a^{2} \arctan \left (a x \right )}{8 c^{2}}+\frac {i a^{2}}{8 c^{2} \left (i a x +1\right )}-\frac {i a^{2} \operatorname {dilog}\left (i a x +1\right )}{c^{2}}-\frac {i a^{2} \ln \left (i a x +1\right )^{2}}{4 c^{2}}+\frac {i a^{2} \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{2 c^{2}}-\frac {i a^{2} \ln \left (i a x \right )}{4 c^{2}}+\frac {i a^{2} \ln \left (i a x +1\right )}{4 c^{2}}+\frac {i \ln \left (i a x +1\right )}{4 c^{2} x^{2}}+\frac {i a^{2} \operatorname {dilog}\left (-i a x +1\right )}{c^{2}}-\frac {i a^{2}}{8 c^{2} \left (-i a x +1\right )}+\frac {i a^{2} \ln \left (-i a x +1\right )^{2}}{4 c^{2}}-\frac {i a^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{2 c^{2}}+\frac {i a^{2} \ln \left (-i a x \right )}{4 c^{2}}-\frac {i a^{2} \ln \left (-i a x +1\right )}{4 c^{2}}-\frac {i \ln \left (-i a x +1\right )}{4 c^{2} x^{2}}+\frac {a^{3} \ln \left (-i a x +1\right ) x}{16 c^{2} \left (-i a x -1\right )}-\frac {i a^{2} \ln \left (-i a x +1\right )}{8 c^{2} \left (-i a x +1\right )}+\frac {i a^{2} \ln \left (-i a x +1\right )}{16 c^{2} \left (-i a x -1\right )}+\frac {i a^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{2 c^{2}}+\frac {a^{3} \ln \left (i a x +1\right ) x}{16 c^{2} \left (i a x -1\right )}+\frac {i a^{2} \ln \left (i a x +1\right )}{8 c^{2} \left (i a x +1\right )}-\frac {i a^{2} \ln \left (i a x +1\right )}{16 c^{2} \left (i a x -1\right )}-\frac {i a^{2} \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{2 c^{2}}\) | \(469\) |
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\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}{\left (a x \right )}}{a^{4} x^{7} + 2 a^{2} x^{5} + x^{3}}\, dx}{c^{2}} \]
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\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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