Integrand size = 23, antiderivative size = 307 \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {5 a b d^3 x}{2 c}+\frac {13 i b^2 d^3 x}{10 c}-\frac {1}{4} b^2 d^3 x^2-\frac {1}{30} i b^2 c d^3 x^3-\frac {13 i b^2 d^3 \arctan (c x)}{10 c^2}-\frac {5 b^2 d^3 x \arctan (c x)}{2 c}-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))+\frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}-\frac {12 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {6 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{5 c^2} \]
[Out]
Time = 0.44 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {4996, 4974, 4930, 266, 4946, 327, 209, 272, 45, 1600, 4964, 2449, 2352, 308} \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {12 i b d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{5 c^2}+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))-\frac {5 a b d^3 x}{2 c}-\frac {13 i b^2 d^3 \arctan (c x)}{10 c^2}-\frac {5 b^2 d^3 x \arctan (c x)}{2 c}-\frac {6 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (c^2 x^2+1\right )}{2 c^2}-\frac {1}{30} i b^2 c d^3 x^3+\frac {13 i b^2 d^3 x}{10 c}-\frac {1}{4} b^2 d^3 x^2 \]
[In]
[Out]
Rule 45
Rule 209
Rule 266
Rule 272
Rule 308
Rule 327
Rule 1600
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 4974
Rule 4996
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {i (d+i c d x)^3 (a+b \arctan (c x))^2}{c}-\frac {i (d+i c d x)^4 (a+b \arctan (c x))^2}{c d}\right ) \, dx \\ & = \frac {i \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx}{c}-\frac {i \int (d+i c d x)^4 (a+b \arctan (c x))^2 \, dx}{c d} \\ & = \frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}+\frac {(2 b) \int \left (-15 d^5 (a+b \arctan (c x))-11 i c d^5 x (a+b \arctan (c x))+5 c^2 d^5 x^2 (a+b \arctan (c x))+i c^3 d^5 x^3 (a+b \arctan (c x))-\frac {16 i \left (i d^5-c d^5 x\right ) (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{5 c d^2}-\frac {b \int \left (-7 d^4 (a+b \arctan (c x))-4 i c d^4 x (a+b \arctan (c x))+c^2 d^4 x^2 (a+b \arctan (c x))-\frac {8 i \left (i d^4-c d^4 x\right ) (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{2 c d} \\ & = \frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}-\frac {(32 i b) \int \frac {\left (i d^5-c d^5 x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c d^2}+\frac {(4 i b) \int \frac {\left (i d^4-c d^4 x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c d}+\left (2 i b d^3\right ) \int x (a+b \arctan (c x)) \, dx-\frac {1}{5} \left (22 i b d^3\right ) \int x (a+b \arctan (c x)) \, dx+\frac {\left (7 b d^3\right ) \int (a+b \arctan (c x)) \, dx}{2 c}-\frac {\left (6 b d^3\right ) \int (a+b \arctan (c x)) \, dx}{c}-\frac {1}{2} \left (b c d^3\right ) \int x^2 (a+b \arctan (c x)) \, dx+\left (2 b c d^3\right ) \int x^2 (a+b \arctan (c x)) \, dx+\frac {1}{5} \left (2 i b c^2 d^3\right ) \int x^3 (a+b \arctan (c x)) \, dx \\ & = -\frac {5 a b d^3 x}{2 c}-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))+\frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}-\frac {(32 i b) \int \frac {a+b \arctan (c x)}{-\frac {i}{d^5}-\frac {c x}{d^5}} \, dx}{5 c d^2}+\frac {(4 i b) \int \frac {a+b \arctan (c x)}{-\frac {i}{d^4}-\frac {c x}{d^4}} \, dx}{c d}+\frac {\left (7 b^2 d^3\right ) \int \arctan (c x) \, dx}{2 c}-\frac {\left (6 b^2 d^3\right ) \int \arctan (c x) \, dx}{c}-\left (i b^2 c d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {1}{5} \left (11 i b^2 c d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {1}{6} \left (b^2 c^2 d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{3} \left (2 b^2 c^2 d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{10} \left (i b^2 c^3 d^3\right ) \int \frac {x^4}{1+c^2 x^2} \, dx \\ & = -\frac {5 a b d^3 x}{2 c}+\frac {6 i b^2 d^3 x}{5 c}-\frac {5 b^2 d^3 x \arctan (c x)}{2 c}-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))+\frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}-\frac {12 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{5 c^2}-\frac {1}{2} \left (7 b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx+\left (6 b^2 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx+\frac {\left (i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c}-\frac {\left (11 i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c}-\frac {\left (4 i b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{c}+\frac {\left (32 i b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{5 c}+\frac {1}{12} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{3} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (i b^2 c^3 d^3\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = -\frac {5 a b d^3 x}{2 c}+\frac {13 i b^2 d^3 x}{10 c}-\frac {1}{30} i b^2 c d^3 x^3-\frac {6 i b^2 d^3 \arctan (c x)}{5 c^2}-\frac {5 b^2 d^3 x \arctan (c x)}{2 c}-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))+\frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}-\frac {12 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{5 c^2}+\frac {5 b^2 d^3 \log \left (1+c^2 x^2\right )}{4 c^2}+\frac {\left (4 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{c^2}-\frac {\left (32 b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{5 c^2}-\frac {\left (i b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{10 c}+\frac {1}{12} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{3} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {5 a b d^3 x}{2 c}+\frac {13 i b^2 d^3 x}{10 c}-\frac {1}{4} b^2 d^3 x^2-\frac {1}{30} i b^2 c d^3 x^3-\frac {13 i b^2 d^3 \arctan (c x)}{10 c^2}-\frac {5 b^2 d^3 x \arctan (c x)}{2 c}-\frac {6}{5} i b d^3 x^2 (a+b \arctan (c x))+\frac {1}{2} b c d^3 x^3 (a+b \arctan (c x))+\frac {1}{10} i b c^2 d^3 x^4 (a+b \arctan (c x))+\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c^2}-\frac {d^3 (1+i c x)^5 (a+b \arctan (c x))^2}{5 c^2}-\frac {12 i b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac {6 b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{5 c^2} \\ \end{align*}
Time = 2.09 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.06 \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {d^3 \left (-18 i a b-15 b^2-150 a b c x+78 i b^2 c x+30 a^2 c^2 x^2-72 i a b c^2 x^2-15 b^2 c^2 x^2+60 i a^2 c^3 x^3+30 a b c^3 x^3-2 i b^2 c^3 x^3-45 a^2 c^4 x^4+6 i a b c^4 x^4-12 i a^2 c^5 x^5+3 b^2 (1-4 i c x) (-i+c x)^4 \arctan (c x)^2+6 b \arctan (c x) \left (b \left (-13 i-25 c x-12 i c^2 x^2+5 c^3 x^3+i c^4 x^4\right )+a \left (25+10 c^2 x^2+20 i c^3 x^3-15 c^4 x^4-4 i c^5 x^5\right )-24 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+72 i a b \log \left (1+c^2 x^2\right )+90 b^2 \log \left (1+c^2 x^2\right )-72 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{60 c^2} \]
[In]
[Out]
Time = 1.74 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.47
method | result | size |
parts | \(d^{3} a^{2} \left (-\frac {1}{5} i c^{3} x^{5}-\frac {3}{4} c^{2} x^{4}+i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {b^{2} d^{3} \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {6 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i c^{3} x^{3}}{30}-\frac {6 i \arctan \left (c x \right ) c^{2} x^{2}}{5}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+i \arctan \left (c x \right )^{2} c^{3} x^{3}+\frac {5 \arctan \left (c x \right )^{2}}{4}-\frac {5 c x \arctan \left (c x \right )}{2}-\frac {3 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {3 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{5}-\frac {3 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{5}-\frac {13 i \arctan \left (c x \right )}{10}+\frac {13 i c x}{10}-\frac {c^{2} x^{2}}{4}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}\right )}{c^{2}}+\frac {2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{4}+i \arctan \left (c x \right ) c^{3} x^{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {5 c x}{4}+\frac {i c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}-\frac {3 i c^{2} x^{2}}{5}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(452\) |
derivativedivides | \(\frac {d^{3} a^{2} \left (-\frac {1}{5} i c^{5} x^{5}-\frac {3}{4} c^{4} x^{4}+i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+b^{2} d^{3} \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {6 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i c^{3} x^{3}}{30}-\frac {6 i \arctan \left (c x \right ) c^{2} x^{2}}{5}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+i \arctan \left (c x \right )^{2} c^{3} x^{3}+\frac {5 \arctan \left (c x \right )^{2}}{4}-\frac {5 c x \arctan \left (c x \right )}{2}-\frac {3 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {3 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{5}-\frac {3 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{5}-\frac {13 i \arctan \left (c x \right )}{10}+\frac {13 i c x}{10}-\frac {c^{2} x^{2}}{4}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{4}+i \arctan \left (c x \right ) c^{3} x^{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {5 c x}{4}+\frac {i c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}-\frac {3 i c^{2} x^{2}}{5}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(455\) |
default | \(\frac {d^{3} a^{2} \left (-\frac {1}{5} i c^{5} x^{5}-\frac {3}{4} c^{4} x^{4}+i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+b^{2} d^{3} \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{10}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )^{2}}{4}+\frac {6 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i c^{3} x^{3}}{30}-\frac {6 i \arctan \left (c x \right ) c^{2} x^{2}}{5}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{2}+i \arctan \left (c x \right )^{2} c^{3} x^{3}+\frac {5 \arctan \left (c x \right )^{2}}{4}-\frac {5 c x \arctan \left (c x \right )}{2}-\frac {3 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {3 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {3 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{5}+\frac {3 \ln \left (c x -i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right )^{2}}{10}-\frac {3 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{5}+\frac {3 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{5}-\frac {3 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{5}-\frac {13 i \arctan \left (c x \right )}{10}+\frac {13 i c x}{10}-\frac {c^{2} x^{2}}{4}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i \arctan \left (c x \right )^{2} c^{5} x^{5}}{5}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{4}+i \arctan \left (c x \right ) c^{3} x^{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {5 c x}{4}+\frac {i c^{4} x^{4}}{20}+\frac {c^{3} x^{3}}{4}-\frac {3 i c^{2} x^{2}}{5}+\frac {3 i \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 \arctan \left (c x \right )}{4}\right )}{c^{2}}\) | \(455\) |
risch | \(-\frac {x^{2} d^{3} b^{2}}{4}+\frac {3 b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{2 c^{2}}-\frac {5 a b \,d^{3} x}{2 c}-\frac {19 b^{2} d^{3}}{12 c^{2}}+\frac {3 d^{3} b^{2} \ln \left (-i c x +1\right ) x^{2}}{5}+\frac {a b c \,d^{3} x^{3}}{2}-\frac {6 i b \,d^{3} x^{2} a}{5}+\frac {a^{2} d^{3} x^{2}}{2}-\frac {3 a^{2} c^{2} d^{3} x^{4}}{4}+\frac {6 i b \,d^{3} a \ln \left (c^{2} x^{2}+1\right )}{5 c^{2}}-d^{3} c a b \ln \left (-i c x +1\right ) x^{3}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{5}}{5}-\frac {5 i b^{2} d^{3} \ln \left (-i c x +1\right ) x}{4 c}+\frac {i b^{2} d^{3} c \ln \left (-i c x +1\right ) x^{3}}{4}+\frac {i b \,d^{3} c^{2} x^{4} a}{10}-\frac {i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{4}+\frac {i b^{2} d^{3} \left (4 c^{5} x^{5}-15 i c^{4} x^{4}-20 c^{3} x^{3}+10 i c^{2} x^{2}+i\right ) \ln \left (i c x +1\right )^{2}}{80 c^{2}}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}+\frac {i d^{3} a b \ln \left (-i c x +1\right ) x^{2}}{2}+\frac {5 b \,d^{3} a \arctan \left (c x \right )}{2 c^{2}}+\frac {49 d^{3} a^{2}}{20 c^{2}}-\frac {d^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}+\frac {6 b^{2} d^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{2}}-\frac {49 d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{80 c^{2}}+\left (-\frac {i b^{2} d^{3} \left (4 c^{3} x^{5}-15 i c^{2} x^{4}-20 c \,x^{3}+10 i x^{2}\right ) \ln \left (-i c x +1\right )}{40}+\frac {b \,d^{3} \left (-8 a \,c^{5} x^{5}+30 i a \,c^{4} x^{4}+2 b \,c^{4} x^{4}-10 i b \,c^{3} x^{3}+40 a \,c^{3} x^{3}-20 i a \,c^{2} x^{2}-24 b \,c^{2} x^{2}+50 i b c x +49 b \ln \left (-i c x +1\right )\right )}{40 c^{2}}\right ) \ln \left (i c x +1\right )-\frac {3 i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{4}}{4}-\frac {i a^{2} c^{3} d^{3} x^{5}}{5}-\frac {i b^{2} c \,d^{3} x^{3}}{30}-\frac {13 i b^{2} d^{3} \arctan \left (c x \right )}{10 c^{2}}+\frac {13 i b^{2} d^{3} x}{10 c}+i a^{2} c \,d^{3} x^{3}+\frac {3 d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}-\frac {6 b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{5 c^{2}}+\frac {6 b^{2} d^{3} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{2}}-\frac {43 i b \,d^{3} a}{10 c^{2}}-\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right ) x^{4}}{20}\) | \(804\) |
[In]
[Out]
\[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \]
[In]
[Out]
Timed out. \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \]
[In]
[Out]
\[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x \,d x } \]
[In]
[Out]
Timed out. \[ \int x (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]
[In]
[Out]