Integrand size = 22, antiderivative size = 332 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {141 a x}{256 c^3 \left (1+a^2 x^2\right )}+\frac {141 \arctan (a x)}{256 c^3}-\frac {3 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 a x \arctan (a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)^3}{32 c^3}+\frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^3}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 i \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^3} \]
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Time = 0.50 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5086, 5044, 4988, 5004, 5112, 5116, 6745, 5050, 5012, 205, 211, 5020} \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {\arctan (a x)^3}{2 c^3 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {33 a x \arctan (a x)^2}{32 c^3 \left (a^2 x^2+1\right )}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac {33 \arctan (a x)}{32 c^3 \left (a^2 x^2+1\right )}-\frac {3 \arctan (a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {141 a x}{256 c^3 \left (a^2 x^2+1\right )}+\frac {3 a x}{128 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 c^3}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c^3}-\frac {i \arctan (a x)^4}{4 c^3}-\frac {11 \arctan (a x)^3}{32 c^3}+\frac {141 \arctan (a x)}{256 c^3}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}+\frac {3 i \operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{4 c^3} \]
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Rule 205
Rule 211
Rule 4988
Rule 5004
Rule 5012
Rule 5020
Rule 5044
Rule 5050
Rule 5086
Rule 5112
Rule 5116
Rule 6745
Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = \frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{4} (3 a) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^2 \int \frac {x \arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = -\frac {3 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^3}+\frac {1}{32} (3 a) \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {i \int \frac {\arctan (a x)^3}{x (i+a x)} \, dx}{c^3}-\frac {(9 a) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-\frac {(3 a) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c} \\ & = \frac {3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 a x \arctan (a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)^3}{32 c^3}+\frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^3}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {(3 a) \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}+\frac {(9 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{128 c}+\frac {\left (9 a^2\right ) \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}+\frac {\left (3 a^2\right ) \int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c} \\ & = \frac {3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 a x}{256 c^3 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 a x \arctan (a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)^3}{32 c^3}+\frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^3}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {(3 i a) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}+\frac {(9 a) \int \frac {1}{c+a^2 c x^2} \, dx}{256 c^2}+\frac {(9 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}+\frac {(3 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c} \\ & = \frac {3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {141 a x}{256 c^3 \left (1+a^2 x^2\right )}+\frac {9 \arctan (a x)}{256 c^3}-\frac {3 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 a x \arctan (a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)^3}{32 c^3}+\frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^3}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3}-\frac {(3 a) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{2 c^3}+\frac {(9 a) \int \frac {1}{c+a^2 c x^2} \, dx}{64 c^2}+\frac {(3 a) \int \frac {1}{c+a^2 c x^2} \, dx}{8 c^2} \\ & = \frac {3 a x}{128 c^3 \left (1+a^2 x^2\right )^2}+\frac {141 a x}{256 c^3 \left (1+a^2 x^2\right )}+\frac {141 \arctan (a x)}{256 c^3}-\frac {3 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )}-\frac {3 a x \arctan (a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {33 a x \arctan (a x)^2}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \arctan (a x)^3}{32 c^3}+\frac {\arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^4}{4 c^3}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c^3}+\frac {3 i \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c^3} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.63 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {-16 i \pi ^4+256 i \arctan (a x)^4-576 \arctan (a x) \cos (2 \arctan (a x))+384 \arctan (a x)^3 \cos (2 \arctan (a x))-12 \arctan (a x) \cos (4 \arctan (a x))+32 \arctan (a x)^3 \cos (4 \arctan (a x))+1024 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+1536 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+1536 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-768 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )+288 \sin (2 \arctan (a x))-576 \arctan (a x)^2 \sin (2 \arctan (a x))+3 \sin (4 \arctan (a x))-24 \arctan (a x)^2 \sin (4 \arctan (a x))}{1024 c^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 93.26 (sec) , antiderivative size = 1841, normalized size of antiderivative = 5.55
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1841\) |
default | \(\text {Expression too large to display}\) | \(1841\) |
parts | \(\text {Expression too large to display}\) | \(2278\) |
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\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]
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\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{7} + 3 a^{4} x^{5} + 3 a^{2} x^{3} + x}\, dx}{c^{3}} \]
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\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]
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\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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