Integrand size = 23, antiderivative size = 216 \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {i b^2}{2 c^2 d^2 (i-c x)}-\frac {i b^2 \arctan (c x)}{2 c^2 d^2}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 216, 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, 4974, 4972, 641, 46, 209, 5004, 4964, 5114, 6745} \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^2 d^2}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (-c x+i)}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (-c x+i)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^2 d^2}-\frac {i b^2 \arctan (c x)}{2 c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^2 d^2}+\frac {i b^2}{2 c^2 d^2 (-c x+i)} \]
[In]
[Out]
Rule 46
Rule 209
Rule 641
Rule 4964
Rule 4972
Rule 4974
Rule 4996
Rule 5004
Rule 5114
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {i (a+b \arctan (c x))^2}{c d^2 (-i+c x)^2}-\frac {(a+b \arctan (c x))^2}{c d^2 (-i+c x)}\right ) \, dx \\ & = -\frac {i \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{c d^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{-i+c x} \, dx}{c d^2} \\ & = -\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}-\frac {(2 i b) \int \left (-\frac {i (a+b \arctan (c x))}{2 (-i+c x)^2}+\frac {i (a+b \arctan (c x))}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c d^2}-\frac {(2 b) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c d^2} \\ & = -\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}-\frac {b \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{c d^2}+\frac {b \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c d^2}-\frac {\left (i b^2\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c d^2} \\ & = -\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c d^2} \\ & = -\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c d^2} \\ & = -\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c d^2} \\ & = \frac {i b^2}{2 c^2 d^2 (i-c x)}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {\left (i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 c d^2} \\ & = \frac {i b^2}{2 c^2 d^2 (i-c x)}-\frac {i b^2 \arctan (c x)}{2 c^2 d^2}-\frac {b (a+b \arctan (c x))}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2}{2 c^2 d^2}-\frac {i (a+b \arctan (c x))^2}{c^2 d^2 (i-c x)}+\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2} \\ \end{align*}
Time = 0.96 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.39 \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {\frac {12 i a^2}{-i+c x}-12 i a^2 \arctan (c x)-6 a^2 \log \left (1+c^2 x^2\right )-6 i a b \left (4 \arctan (c x)^2-\cos (2 \arctan (c x))+2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )-2 i \arctan (c x) \left (\cos (2 \arctan (c x))-2 \log \left (1+e^{2 i \arctan (c x)}\right )-i \sin (2 \arctan (c x))\right )+i \sin (2 \arctan (c x))\right )+b^2 \left (-8 i \arctan (c x)^3+3 \cos (2 \arctan (c x))+6 i \arctan (c x) \cos (2 \arctan (c x))-6 \arctan (c x)^2 \cos (2 \arctan (c x))+12 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )-12 i \arctan (c x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )-3 i \sin (2 \arctan (c x))+6 \arctan (c x) \sin (2 \arctan (c x))+6 i \arctan (c x)^2 \sin (2 \arctan (c x))\right )}{12 c^2 d^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.03 (sec) , antiderivative size = 857, normalized size of antiderivative = 3.97
method | result | size |
derivativedivides | \(\frac {-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}+\frac {a b}{d^{2} \left (c x -i\right )}+\frac {i a b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2}}+\frac {i a^{2}}{d^{2} \left (c x -i\right )}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}}{c^{2}}\) | \(857\) |
default | \(\frac {-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2}}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}+\frac {a b}{d^{2} \left (c x -i\right )}+\frac {i a b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{d^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2}}+\frac {i a^{2}}{d^{2} \left (c x -i\right )}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}}{c^{2}}\) | \(857\) |
parts | \(-\frac {i a^{2}}{d^{2} c^{2} \left (-c x +i\right )}-\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{2}}-\frac {i a^{2} \arctan \left (c x \right )}{d^{2} c^{2}}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{c x -i}-\arctan \left (c x \right )^{2} \ln \left (c x -i\right )+\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )-\frac {2 i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3} \arctan \left (c x \right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2} \arctan \left (c x \right )^{2}+\frac {\arctan \left (c x \right )^{2}}{2}+i \pi \arctan \left (c x \right )^{2}-\frac {2 i \arctan \left (c x \right ) \left (c x +i\right )}{4 c x -4 i}-\frac {c x +i}{4 \left (c x -i\right )}-i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}\right )}{d^{2} c^{2}}+\frac {i a b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2} c^{2}}-\frac {2 a b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2} c^{2}}-\frac {i a b \ln \left (c x -i\right )^{2}}{2 d^{2} c^{2}}-\frac {a b \arctan \left (\frac {c x}{2}\right )}{4 d^{2} c^{2}}+\frac {a b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2} c^{2}}+\frac {a b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2} c^{2}}+\frac {a b}{d^{2} c^{2} \left (c x -i\right )}+\frac {i a b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{d^{2} c^{2}}+\frac {a b \arctan \left (c x \right )}{2 d^{2} c^{2}}-\frac {i a b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2} c^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} c^{2} \left (c x -i\right )}+\frac {i a b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2} c^{2}}\) | \(902\) |
[In]
[Out]
\[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (i \, c d x + d\right )}^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x (a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
[In]
[Out]