Integrand size = 23, antiderivative size = 205 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\frac {3 b d^3 x}{4 c^3}+\frac {13 i b d^3 x^2}{35 c^2}-\frac {b d^3 x^3}{4 c}-\frac {13}{70} i b d^3 x^4+\frac {1}{10} b c d^3 x^5+\frac {1}{42} i b c^2 d^3 x^6-\frac {3 b d^3 \arctan (c x)}{4 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {13 i b d^3 \log \left (1+c^2 x^2\right )}{35 c^4} \]
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Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {45, 4992, 12, 1816, 649, 209, 266} \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))-\frac {3 b d^3 \arctan (c x)}{4 c^4}+\frac {3 b d^3 x}{4 c^3}+\frac {1}{42} i b c^2 d^3 x^6+\frac {13 i b d^3 x^2}{35 c^2}-\frac {13 i b d^3 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac {1}{10} b c d^3 x^5-\frac {b d^3 x^3}{4 c}-\frac {13}{70} i b d^3 x^4 \]
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Rule 12
Rule 45
Rule 209
Rule 266
Rule 649
Rule 1816
Rule 4992
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-(b c) \int \frac {d^3 x^4 \left (35+84 i c x-70 c^2 x^2-20 i c^3 x^3\right )}{140 \left (1+c^2 x^2\right )} \, dx \\ & = \frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {1}{140} \left (b c d^3\right ) \int \frac {x^4 \left (35+84 i c x-70 c^2 x^2-20 i c^3 x^3\right )}{1+c^2 x^2} \, dx \\ & = \frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {1}{140} \left (b c d^3\right ) \int \left (-\frac {105}{c^4}-\frac {104 i x}{c^3}+\frac {105 x^2}{c^2}+\frac {104 i x^3}{c}-70 x^4-20 i c x^5+\frac {105+104 i c x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = \frac {3 b d^3 x}{4 c^3}+\frac {13 i b d^3 x^2}{35 c^2}-\frac {b d^3 x^3}{4 c}-\frac {13}{70} i b d^3 x^4+\frac {1}{10} b c d^3 x^5+\frac {1}{42} i b c^2 d^3 x^6+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {\left (b d^3\right ) \int \frac {105+104 i c x}{1+c^2 x^2} \, dx}{140 c^3} \\ & = \frac {3 b d^3 x}{4 c^3}+\frac {13 i b d^3 x^2}{35 c^2}-\frac {b d^3 x^3}{4 c}-\frac {13}{70} i b d^3 x^4+\frac {1}{10} b c d^3 x^5+\frac {1}{42} i b c^2 d^3 x^6+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {\left (3 b d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 c^3}-\frac {\left (26 i b d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{35 c^2} \\ & = \frac {3 b d^3 x}{4 c^3}+\frac {13 i b d^3 x^2}{35 c^2}-\frac {b d^3 x^3}{4 c}-\frac {13}{70} i b d^3 x^4+\frac {1}{10} b c d^3 x^5+\frac {1}{42} i b c^2 d^3 x^6-\frac {3 b d^3 \arctan (c x)}{4 c^4}+\frac {1}{4} d^3 x^4 (a+b \arctan (c x))+\frac {3}{5} i c d^3 x^5 (a+b \arctan (c x))-\frac {1}{2} c^2 d^3 x^6 (a+b \arctan (c x))-\frac {1}{7} i c^3 d^3 x^7 (a+b \arctan (c x))-\frac {13 i b d^3 \log \left (1+c^2 x^2\right )}{35 c^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.75 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\frac {d^3 \left (3 a c^4 x^4 \left (35+84 i c x-70 c^2 x^2-20 i c^3 x^3\right )+b c x \left (315+156 i c x-105 c^2 x^2-78 i c^3 x^3+42 c^4 x^4+10 i c^5 x^5\right )+3 b \left (-105+35 c^4 x^4+84 i c^5 x^5-70 c^6 x^6-20 i c^7 x^7\right ) \arctan (c x)-156 i b \log \left (1+c^2 x^2\right )\right )}{420 c^4} \]
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Time = 1.97 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.79
method | result | size |
parts | \(a \,d^{3} \left (-\frac {1}{7} i c^{3} x^{7}-\frac {1}{2} c^{2} x^{6}+\frac {3}{5} i c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {b \,d^{3} \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(162\) |
derivativedivides | \(\frac {a \,d^{3} \left (-\frac {1}{7} i c^{7} x^{7}-\frac {1}{2} c^{6} x^{6}+\frac {3}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+b \,d^{3} \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(168\) |
default | \(\frac {a \,d^{3} \left (-\frac {1}{7} i c^{7} x^{7}-\frac {1}{2} c^{6} x^{6}+\frac {3}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+b \,d^{3} \left (-\frac {i \arctan \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {3 c x}{4}+\frac {i c^{6} x^{6}}{42}+\frac {c^{5} x^{5}}{10}-\frac {13 i c^{4} x^{4}}{70}-\frac {c^{3} x^{3}}{4}+\frac {13 i c^{2} x^{2}}{35}-\frac {13 i \ln \left (c^{2} x^{2}+1\right )}{35}-\frac {3 \arctan \left (c x \right )}{4}\right )}{c^{4}}\) | \(168\) |
parallelrisch | \(-\frac {-252 i x^{5} a \,c^{5} d^{3}-156 i x^{2} b \,c^{2} d^{3}-252 i c^{5} b \,d^{3} \arctan \left (c x \right ) x^{5}+210 b \,c^{6} d^{3} \arctan \left (c x \right ) x^{6}+60 i c^{7} b \,d^{3} \arctan \left (c x \right ) x^{7}+210 a \,c^{6} d^{3} x^{6}+78 i x^{4} b \,c^{4} d^{3}-42 b \,c^{5} d^{3} x^{5}+156 i b \,d^{3} \ln \left (c^{2} x^{2}+1\right )-105 x^{4} \arctan \left (c x \right ) b \,c^{4} d^{3}-105 a \,c^{4} d^{3} x^{4}+105 b \,c^{3} d^{3} x^{3}+60 i x^{7} a \,c^{7} d^{3}-10 i x^{6} b \,c^{6} d^{3}-315 b c \,d^{3} x +315 b \,d^{3} \arctan \left (c x \right )}{420 c^{4}}\) | \(221\) |
risch | \(-\frac {d^{3} b \left (20 c^{3} x^{7}-70 i c^{2} x^{6}-84 x^{5} c +35 i x^{4}\right ) \ln \left (i c x +1\right )}{280}+\frac {d^{3} c^{3} b \,x^{7} \ln \left (-i c x +1\right )}{14}-\frac {13 i b \,d^{3} x^{4}}{70}-\frac {d^{3} c^{2} a \,x^{6}}{2}+\frac {i b \,c^{2} d^{3} x^{6}}{42}-\frac {3 d^{3} c b \,x^{5} \ln \left (-i c x +1\right )}{10}+\frac {i d^{3} x^{4} b \ln \left (-i c x +1\right )}{8}+\frac {b c \,d^{3} x^{5}}{10}+\frac {13 i b \,d^{3} x^{2}}{35 c^{2}}+\frac {d^{3} a \,x^{4}}{4}-\frac {13 i d^{3} b \ln \left (11025 c^{2} x^{2}+11025\right )}{35 c^{4}}-\frac {i d^{3} c^{2} x^{6} b \ln \left (-i c x +1\right )}{4}-\frac {b \,d^{3} x^{3}}{4 c}-\frac {i d^{3} c^{3} a \,x^{7}}{7}+\frac {3 b \,d^{3} x}{4 c^{3}}-\frac {3 b \,d^{3} \arctan \left (c x \right )}{4 c^{4}}+\frac {3 i d^{3} c a \,x^{5}}{5}\) | \(270\) |
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Time = 0.24 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.99 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\frac {-120 i \, a c^{7} d^{3} x^{7} - 20 \, {\left (21 \, a - i \, b\right )} c^{6} d^{3} x^{6} - 84 \, {\left (-6 i \, a - b\right )} c^{5} d^{3} x^{5} + 6 \, {\left (35 \, a - 26 i \, b\right )} c^{4} d^{3} x^{4} - 210 \, b c^{3} d^{3} x^{3} + 312 i \, b c^{2} d^{3} x^{2} + 630 \, b c d^{3} x - 627 i \, b d^{3} \log \left (\frac {c x + i}{c}\right ) + 3 i \, b d^{3} \log \left (\frac {c x - i}{c}\right ) + 3 \, {\left (20 \, b c^{7} d^{3} x^{7} - 70 i \, b c^{6} d^{3} x^{6} - 84 \, b c^{5} d^{3} x^{5} + 35 i \, b c^{4} d^{3} x^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{840 \, c^{4}} \]
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Time = 2.57 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.60 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=- \frac {i a c^{3} d^{3} x^{7}}{7} - \frac {b d^{3} x^{3}}{4 c} + \frac {13 i b d^{3} x^{2}}{35 c^{2}} + \frac {3 b d^{3} x}{4 c^{3}} - \frac {b d^{3} \left (- \frac {i \log {\left (353 b c d^{3} x - 353 i b d^{3} \right )}}{280} + \frac {351 i \log {\left (353 b c d^{3} x + 353 i b d^{3} \right )}}{560}\right )}{c^{4}} - x^{6} \left (\frac {a c^{2} d^{3}}{2} - \frac {i b c^{2} d^{3}}{42}\right ) - x^{5} \left (- \frac {3 i a c d^{3}}{5} - \frac {b c d^{3}}{10}\right ) - x^{4} \left (- \frac {a d^{3}}{4} + \frac {13 i b d^{3}}{70}\right ) + \left (- \frac {b c^{3} d^{3} x^{7}}{14} + \frac {i b c^{2} d^{3} x^{6}}{4} + \frac {3 b c d^{3} x^{5}}{10} - \frac {i b d^{3} x^{4}}{8}\right ) \log {\left (i c x + 1 \right )} + \frac {\left (40 b c^{7} d^{3} x^{7} - 140 i b c^{6} d^{3} x^{6} - 168 b c^{5} d^{3} x^{5} + 70 i b c^{4} d^{3} x^{4} - 67 i b d^{3}\right ) \log {\left (- i c x + 1 \right )}}{560 c^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.27 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=-\frac {1}{7} i \, a c^{3} d^{3} x^{7} - \frac {1}{2} \, a c^{2} d^{3} x^{6} + \frac {3}{5} i \, a c d^{3} x^{5} - \frac {1}{84} i \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{3} + \frac {1}{4} \, a d^{3} x^{4} - \frac {1}{30} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{2} d^{3} + \frac {3}{20} i \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d^{3} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{3} \]
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\[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )} x^{3} \,d x } \]
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Time = 0.93 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.91 \[ \int x^3 (d+i c d x)^3 (a+b \arctan (c x)) \, dx=-\frac {\frac {d^3\,\left (315\,b\,\mathrm {atan}\left (c\,x\right )+b\,\ln \left (c^2\,x^2+1\right )\,156{}\mathrm {i}\right )}{420}+\frac {b\,c^3\,d^3\,x^3}{4}-\frac {3\,b\,c\,d^3\,x}{4}-\frac {b\,c^2\,d^3\,x^2\,13{}\mathrm {i}}{35}}{c^4}+\frac {d^3\,\left (105\,a\,x^4+105\,b\,x^4\,\mathrm {atan}\left (c\,x\right )-b\,x^4\,78{}\mathrm {i}\right )}{420}-\frac {c^3\,d^3\,\left (a\,x^7\,60{}\mathrm {i}+b\,x^7\,\mathrm {atan}\left (c\,x\right )\,60{}\mathrm {i}\right )}{420}+\frac {c\,d^3\,\left (a\,x^5\,252{}\mathrm {i}+42\,b\,x^5+b\,x^5\,\mathrm {atan}\left (c\,x\right )\,252{}\mathrm {i}\right )}{420}-\frac {c^2\,d^3\,\left (210\,a\,x^6+210\,b\,x^6\,\mathrm {atan}\left (c\,x\right )-b\,x^6\,10{}\mathrm {i}\right )}{420} \]
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