Integrand size = 22, antiderivative size = 374 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac {3 a^2 \arctan (a x)}{8 c^2}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {i a^2 \arctan (a x)^4}{2 c^2}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {2 a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{2 c^2} \]
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Time = 0.75 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5086, 5038, 4946, 5044, 4988, 2497, 5004, 5112, 5116, 6745, 5050, 5012, 205, 211} \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c^2}-\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{c^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (a^2 x^2+1\right )}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (a^2 x^2+1\right )}+\frac {i a^2 \arctan (a x)^4}{2 c^2}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a^2 \arctan (a x)}{8 c^2}-\frac {2 a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{2 c^2}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (a^2 x^2+1\right )}-\frac {3 a^3 x}{8 c^2 \left (a^2 x^2+1\right )}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x} \]
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Rule 205
Rule 211
Rule 2497
Rule 4946
Rule 4988
Rule 5004
Rule 5012
Rule 5038
Rule 5044
Rule 5050
Rule 5086
Rule 5112
Rule 5116
Rule 6745
Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = a^4 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\arctan (a x)^3}{x^3} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = -\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {1}{2} \left (3 a^3\right ) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(3 a) \int \frac {\arctan (a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {\left (i a^2\right ) \int \frac {\arctan (a x)^3}{x (i+a x)} \, dx}{c^2}\right ) \\ & = \frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}+\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {1}{2} \left (3 a^4\right ) \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {(3 a) \int \frac {\arctan (a x)^2}{x^2} \, dx}{2 c^2}-\frac {\left (3 a^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {\left (3 a^3\right ) \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right ) \\ & = \frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}-\frac {1}{4} \left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\left (3 a^2\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx}{c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {\left (3 i a^3\right ) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2}\right ) \\ & = -\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\left (3 i a^2\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx}{c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}-\frac {\left (3 a^3\right ) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^2}\right )-\frac {\left (3 a^3\right ) \int \frac {1}{c+a^2 c x^2} \, dx}{8 c} \\ & = -\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac {3 a^2 \arctan (a x)}{8 c^2}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^2}\right )-\frac {\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^2} \\ & = -\frac {3 a^3 x}{8 c^2 \left (1+a^2 x^2\right )}-\frac {3 a^2 \arctan (a x)}{8 c^2}+\frac {3 a^2 \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 i a^2 \arctan (a x)^2}{2 c^2}-\frac {3 a \arctan (a x)^2}{2 c^2 x}+\frac {3 a^3 x \arctan (a x)^2}{4 c^2 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)^3}{4 c^2}-\frac {\arctan (a x)^3}{2 c^2 x^2}-\frac {a^2 \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {3 a^2 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}-2 \left (-\frac {i a^2 \arctan (a x)^4}{4 c^2}+\frac {a^2 \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {3 i a^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 a^2 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^2}+\frac {3 i a^2 \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^2}\right ) \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.65 \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 \left (i \pi ^4-48 i \arctan (a x)^2-\frac {48 \arctan (a x)^2}{a x}-\frac {16 \left (1+a^2 x^2\right ) \arctan (a x)^3}{a^2 x^2}-16 i \arctan (a x)^4+12 \arctan (a x) \cos (2 \arctan (a x))-8 \arctan (a x)^3 \cos (2 \arctan (a x))-64 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+96 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-96 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )-96 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+48 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-6 \sin (2 \arctan (a x))+12 \arctan (a x)^2 \sin (2 \arctan (a x))\right )}{32 c^2} \]
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Time = 96.56 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(a^{2} \left (\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {\left (6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}-3 i-6 \arctan \left (a x \right )\right ) \left (a x -i\right )}{32 c^{2} \left (a x +i\right )}+\frac {\left (-6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}+3 i-6 \arctan \left (a x \right )\right ) \left (a x +i\right )}{32 c^{2} \left (a x -i\right )}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c^{2} a^{2} x^{2}}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {3 i \arctan \left (a x \right )^{2}}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}+\frac {i \arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(531\) |
default | \(a^{2} \left (\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {\left (6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}-3 i-6 \arctan \left (a x \right )\right ) \left (a x -i\right )}{32 c^{2} \left (a x +i\right )}+\frac {\left (-6 i \arctan \left (a x \right )^{2}+4 \arctan \left (a x \right )^{3}+3 i-6 \arctan \left (a x \right )\right ) \left (a x +i\right )}{32 c^{2} \left (a x -i\right )}-\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 c^{2} a^{2} x^{2}}-\frac {3 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {3 i \arctan \left (a x \right )^{2}}{c^{2}}+\frac {3 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}-\frac {3 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )}{c^{2}}+\frac {i \arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}+\frac {6 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {2 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}-\frac {12 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(531\) |
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\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{7} + 2 a^{2} x^{5} + x^{3}}\, dx}{c^{2}} \]
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\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^3}{x^3 \left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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