Integrand size = 23, antiderivative size = 167 \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {a x}{c^2 d^2}-\frac {i b}{2 c^3 d^2 (i-c x)}+\frac {i b \arctan (c x)}{2 c^3 d^2}-\frac {b x \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^3 d^2 (i-c x)}+\frac {2 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^2}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d^2} \]
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Time = 0.14 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4996, 4930, 266, 4972, 641, 46, 209, 4964, 2449, 2352} \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\frac {a+b \arctan (c x)}{c^3 d^2 (-c x+i)}+\frac {2 i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3 d^2}-\frac {a x}{c^2 d^2}+\frac {i b \arctan (c x)}{2 c^3 d^2}-\frac {b x \arctan (c x)}{c^2 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^3 d^2}-\frac {i b}{2 c^3 d^2 (-c x+i)}+\frac {b \log \left (c^2 x^2+1\right )}{2 c^3 d^2} \]
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Rule 46
Rule 209
Rule 266
Rule 641
Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 4972
Rule 4996
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a+b \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^2 d^2 (-i+c x)^2}-\frac {2 i (a+b \arctan (c x))}{c^2 d^2 (-i+c x)}\right ) \, dx \\ & = -\frac {(2 i) \int \frac {a+b \arctan (c x)}{-i+c x} \, dx}{c^2 d^2}-\frac {\int (a+b \arctan (c x)) \, dx}{c^2 d^2}+\frac {\int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{c^2 d^2} \\ & = -\frac {a x}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^3 d^2 (i-c x)}+\frac {2 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^2}-\frac {(2 i b) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d^2}+\frac {b \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^2 d^2}-\frac {b \int \arctan (c x) \, dx}{c^2 d^2} \\ & = -\frac {a x}{c^2 d^2}-\frac {b x \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^3 d^2 (i-c x)}+\frac {2 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^2}-\frac {(2 b) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^3 d^2}+\frac {b \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^2 d^2}+\frac {b \int \frac {x}{1+c^2 x^2} \, dx}{c d^2} \\ & = -\frac {a x}{c^2 d^2}-\frac {b x \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^3 d^2 (i-c x)}+\frac {2 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^2}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d^2}+\frac {b \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^2 d^2} \\ & = -\frac {a x}{c^2 d^2}-\frac {i b}{2 c^3 d^2 (i-c x)}-\frac {b x \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^3 d^2 (i-c x)}+\frac {2 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^2}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d^2}+\frac {(i b) \int \frac {1}{1+c^2 x^2} \, dx}{2 c^2 d^2} \\ & = -\frac {a x}{c^2 d^2}-\frac {i b}{2 c^3 d^2 (i-c x)}+\frac {i b \arctan (c x)}{2 c^3 d^2}-\frac {b x \arctan (c x)}{c^2 d^2}+\frac {a+b \arctan (c x)}{c^3 d^2 (i-c x)}+\frac {2 i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d^2}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 d^2}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d^2} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=-\frac {4 a c x+\frac {4 a}{-i+c x}-8 a \arctan (c x)+4 i a \log \left (1+c^2 x^2\right )+b \left (-8 \arctan (c x)^2+\cos (2 \arctan (c x))-2 \log \left (1+c^2 x^2\right )-4 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))+2 \arctan (c x) \left (2 c x+i \cos (2 \arctan (c x))-4 i \log \left (1+e^{2 i \arctan (c x)}\right )+\sin (2 \arctan (c x))\right )\right )}{4 c^3 d^2} \]
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Time = 0.73 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.62
method | result | size |
derivativedivides | \(\frac {-\frac {a c x}{d^{2}}-\frac {a}{d^{2} \left (c x -i\right )}+\frac {2 a \arctan \left (c x \right )}{d^{2}}+\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 d^{2}}-\frac {b \arctan \left (c x \right ) c x}{d^{2}}-\frac {b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}-\frac {2 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}+\frac {b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d^{2}}+\frac {3 i b \arctan \left (c x \right )}{4 d^{2}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}+\frac {i b}{2 d^{2} \left (c x -i\right )}-\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d^{2}}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}-\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d^{2}}}{c^{3}}\) | \(271\) |
default | \(\frac {-\frac {a c x}{d^{2}}-\frac {a}{d^{2} \left (c x -i\right )}+\frac {2 a \arctan \left (c x \right )}{d^{2}}+\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 d^{2}}-\frac {b \arctan \left (c x \right ) c x}{d^{2}}-\frac {b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}-\frac {2 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}+\frac {b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d^{2}}+\frac {3 i b \arctan \left (c x \right )}{4 d^{2}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}+\frac {i b}{2 d^{2} \left (c x -i\right )}-\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d^{2}}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}-\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d^{2}}}{c^{3}}\) | \(271\) |
parts | \(-\frac {a x}{c^{2} d^{2}}+\frac {a}{d^{2} c^{3} \left (-c x +i\right )}-\frac {2 i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{3} d^{2}}+\frac {2 a \arctan \left (c x \right )}{d^{2} c^{3}}-\frac {b x \arctan \left (c x \right )}{c^{2} d^{2}}-\frac {b \arctan \left (c x \right )}{c^{3} d^{2} \left (c x -i\right )}+\frac {3 i b \arctan \left (c x \right )}{4 c^{3} d^{2}}-\frac {b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{3} d^{2}}-\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{c^{3} d^{2}}+\frac {b \ln \left (c x -i\right )^{2}}{2 c^{3} d^{2}}+\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 c^{3} d^{2}}-\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 c^{3} d^{2}}+\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 c^{3} d^{2}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2} c^{3}}-\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 c^{3} d^{2}}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 c^{3} d^{2}}+\frac {i b}{2 c^{3} d^{2} \left (c x -i\right )}\) | \(316\) |
risch | \(-\frac {b \ln \left (i c x +1\right )^{2}}{2 c^{3} d^{2}}+\left (\frac {i b x}{2 c^{2} d^{2}}+\frac {i b}{2 c^{3} d^{2} \left (c x -i\right )}\right ) \ln \left (i c x +1\right )+\frac {3 i b \arctan \left (c x \right )}{4 c^{3} d^{2}}-\frac {b}{2 c^{3} d^{2}}-\frac {b \ln \left (-i c x +1\right )}{4 d^{2} c^{3} \left (-i c x -1\right )}-\frac {i b}{2 c^{3} d^{2} \left (-c x +i\right )}+\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 c^{3} d^{2}}+\frac {b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{d^{2} c^{3}}-\frac {b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{2} c^{3}}-\frac {b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{2} c^{3}}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2} c^{3}}+\frac {i b \ln \left (-i c x +1\right ) x}{4 d^{2} c^{2} \left (-i c x -1\right )}+\frac {2 a \arctan \left (c x \right )}{d^{2} c^{3}}-\frac {i a}{d^{2} c^{3}}-\frac {a x}{c^{2} d^{2}}-\frac {i \ln \left (-i c x +1\right ) b x}{2 d^{2} c^{2}}+\frac {\ln \left (-i c x +1\right ) b}{2 d^{2} c^{3}}+\frac {i a}{d^{2} c^{3} \left (-i c x -1\right )}\) | \(349\) |
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\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\frac {\left (- i b c^{2} x^{2} + 2 b c x \log {\left (i c x + 1 \right )} - b c x - 2 i b \log {\left (i c x + 1 \right )} - i b\right ) \log {\left (- i c x + 1 \right )}}{2 c^{4} d^{2} x - 2 i c^{3} d^{2}} - \frac {\int \left (- \frac {b}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \left (- \frac {2 b \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {2 a c^{3} x^{3}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {2 b c^{2} x^{2}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {2 i a c^{2} x^{2}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {i b c^{3} x^{3}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {3 b c^{2} x^{2} \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {4 i b c x \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \left (- \frac {i b c^{3} x^{3} \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx}{2 c^{2} d^{2}} \]
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\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))}{(d+i c d x)^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
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