Integrand size = 22, antiderivative size = 227 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}+\frac {a^3 \log (x)}{c}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c} \]
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Time = 0.53 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5038, 4946, 272, 36, 29, 31, 5004, 5044, 4988, 5112, 6745} \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c}+\frac {a^3 \arctan (a x)^4}{4 c}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{c}+\frac {a^3 \log (x)}{c}+\frac {a^2 \arctan (a x)^3}{c x}-\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \log \left (a^2 x^2+1\right )}{2 c}-\frac {\arctan (a x)^3}{3 c x^3}-\frac {a \arctan (a x)^2}{2 c x^2} \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 4946
Rule 4988
Rule 5004
Rule 5038
Rule 5044
Rule 5112
Rule 6745
Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\arctan (a x)^3}{x^4} \, dx}{c} \\ & = -\frac {\arctan (a x)^3}{3 c x^3}+a^4 \int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx+\frac {a \int \frac {\arctan (a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx}{c}-\frac {a^2 \int \frac {\arctan (a x)^3}{x^2} \, dx}{c} \\ & = -\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}+\frac {a \int \frac {\arctan (a x)^2}{x^3} \, dx}{c}-\frac {a^3 \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c}-\frac {\left (3 a^3\right ) \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c} \\ & = -\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}+\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (1+a^2 x^2\right )} \, dx}{c}-\frac {\left (i a^3\right ) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx}{c}-\frac {\left (3 i a^3\right ) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx}{c} \\ & = -\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {a^2 \int \frac {\arctan (a x)}{x^2} \, dx}{c}-\frac {a^4 \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{c}+\frac {\left (2 a^4\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}+\frac {\left (6 a^4\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c}-\frac {\left (i a^4\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}-\frac {\left (3 i a^4\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = -\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c} \\ & = -\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}-\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c} \\ & = -\frac {a^2 \arctan (a x)}{c x}-\frac {a^3 \arctan (a x)^2}{2 c}-\frac {a \arctan (a x)^2}{2 c x^2}+\frac {4 i a^3 \arctan (a x)^3}{3 c}-\frac {\arctan (a x)^3}{3 c x^3}+\frac {a^2 \arctan (a x)^3}{c x}+\frac {a^3 \arctan (a x)^4}{4 c}+\frac {a^3 \log (x)}{c}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c}-\frac {4 a^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {a^3 \left (\frac {1}{12} \left (2 i \pi ^3-\frac {12 \arctan (a x)}{a x}+\left (-16 i-\frac {4}{a^3 x^3}+\frac {12}{a x}\right ) \arctan (a x)^3+3 \arctan (a x)^4+\arctan (a x)^2 \left (-6-\frac {6}{a^2 x^2}-48 \log \left (1-e^{-2 i \arctan (a x)}\right )\right )+12 \log (a x)-6 \log \left (1+a^2 x^2\right )\right )-4 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )}{c} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 77.33 (sec) , antiderivative size = 1831, normalized size of antiderivative = 8.07
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1831\) |
default | \(\text {Expression too large to display}\) | \(1831\) |
parts | \(\text {Expression too large to display}\) | \(1930\) |
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\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]
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\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \]
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Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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