Integrand size = 22, antiderivative size = 76 \[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c} \]
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Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1607, 4988, 5004, 5112, 6745} \[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{c}+\frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 c} \]
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Rule 1607
Rule 4988
Rule 5004
Rule 5112
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \int \frac {\arctan (a x)^2}{x (c-i a c x)} \, dx \\ & = \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {(2 a) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {(i a) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c} \\ & = \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08 \[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\frac {-i \pi ^3+16 i \arctan (a x)^3+24 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )}{24 c} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (70 ) = 140\).
Time = 6.00 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.54
method | result | size |
derivativedivides | \(\frac {\frac {a \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {a \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}}{a}\) | \(193\) |
default | \(\frac {\frac {a \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {a \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}-\frac {2 i a \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}+\frac {2 a \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{c}}{a}\) | \(193\) |
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\[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{-i \, a c x^{2} + c x} \,d x } \]
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\[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\frac {i \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a x^{2} + i x}\, dx}{c} \]
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\[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{-i \, a c x^{2} + c x} \,d x } \]
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\[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{-i \, a c x^{2} + c x} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^2}{c x-i a c x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{c\,x-a\,c\,x^2\,1{}\mathrm {i}} \,d x \]
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