Integrand size = 23, antiderivative size = 166 \[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=\frac {5 b d^2 x}{12 c^3}+\frac {i b d^2 x^2}{5 c^2}-\frac {5 b d^2 x^3}{36 c}-\frac {1}{10} i b d^2 x^4+\frac {1}{30} b c d^2 x^5-\frac {5 b d^2 \arctan (c x)}{12 c^4}+\frac {1}{4} d^2 x^4 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))-\frac {i b d^2 \log \left (1+c^2 x^2\right )}{5 c^4} \]
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Time = 0.11 (sec) , antiderivative size = 166, 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} \[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))+\frac {1}{4} d^2 x^4 (a+b \arctan (c x))-\frac {5 b d^2 \arctan (c x)}{12 c^4}+\frac {5 b d^2 x}{12 c^3}+\frac {i b d^2 x^2}{5 c^2}-\frac {i b d^2 \log \left (c^2 x^2+1\right )}{5 c^4}+\frac {1}{30} b c d^2 x^5-\frac {5 b d^2 x^3}{36 c}-\frac {1}{10} i b d^2 x^4 \]
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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^2 x^4 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))-(b c) \int \frac {d^2 x^4 \left (15+24 i c x-10 c^2 x^2\right )}{60 \left (1+c^2 x^2\right )} \, dx \\ & = \frac {1}{4} d^2 x^4 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))-\frac {1}{60} \left (b c d^2\right ) \int \frac {x^4 \left (15+24 i c x-10 c^2 x^2\right )}{1+c^2 x^2} \, dx \\ & = \frac {1}{4} d^2 x^4 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))-\frac {1}{60} \left (b c d^2\right ) \int \left (-\frac {25}{c^4}-\frac {24 i x}{c^3}+\frac {25 x^2}{c^2}+\frac {24 i x^3}{c}-10 x^4+\frac {25+24 i c x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = \frac {5 b d^2 x}{12 c^3}+\frac {i b d^2 x^2}{5 c^2}-\frac {5 b d^2 x^3}{36 c}-\frac {1}{10} i b d^2 x^4+\frac {1}{30} b c d^2 x^5+\frac {1}{4} d^2 x^4 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))-\frac {\left (b d^2\right ) \int \frac {25+24 i c x}{1+c^2 x^2} \, dx}{60 c^3} \\ & = \frac {5 b d^2 x}{12 c^3}+\frac {i b d^2 x^2}{5 c^2}-\frac {5 b d^2 x^3}{36 c}-\frac {1}{10} i b d^2 x^4+\frac {1}{30} b c d^2 x^5+\frac {1}{4} d^2 x^4 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))-\frac {\left (5 b d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{12 c^3}-\frac {\left (2 i b d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{5 c^2} \\ & = \frac {5 b d^2 x}{12 c^3}+\frac {i b d^2 x^2}{5 c^2}-\frac {5 b d^2 x^3}{36 c}-\frac {1}{10} i b d^2 x^4+\frac {1}{30} b c d^2 x^5-\frac {5 b d^2 \arctan (c x)}{12 c^4}+\frac {1}{4} d^2 x^4 (a+b \arctan (c x))+\frac {2}{5} i c d^2 x^5 (a+b \arctan (c x))-\frac {1}{6} c^2 d^2 x^6 (a+b \arctan (c x))-\frac {i b d^2 \log \left (1+c^2 x^2\right )}{5 c^4} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.75 \[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=\frac {d^2 \left (3 a c^4 x^4 \left (15+24 i c x-10 c^2 x^2\right )+b c x \left (75+36 i c x-25 c^2 x^2-18 i c^3 x^3+6 c^4 x^4\right )+3 b \left (-25+15 c^4 x^4+24 i c^5 x^5-10 c^6 x^6\right ) \arctan (c x)-36 i b \log \left (1+c^2 x^2\right )\right )}{180 c^4} \]
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Time = 1.51 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79
method | result | size |
parts | \(a \,d^{2} \left (-\frac {1}{6} c^{2} x^{6}+\frac {2}{5} i c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {b \,d^{2} \left (-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{6}+\frac {2 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {5 c x}{12}+\frac {c^{5} x^{5}}{30}-\frac {i c^{4} x^{4}}{10}-\frac {5 c^{3} x^{3}}{36}+\frac {i c^{2} x^{2}}{5}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {5 \arctan \left (c x \right )}{12}\right )}{c^{4}}\) | \(131\) |
derivativedivides | \(\frac {a \,d^{2} \left (-\frac {1}{6} c^{6} x^{6}+\frac {2}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+b \,d^{2} \left (-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{6}+\frac {2 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {5 c x}{12}+\frac {c^{5} x^{5}}{30}-\frac {i c^{4} x^{4}}{10}-\frac {5 c^{3} x^{3}}{36}+\frac {i c^{2} x^{2}}{5}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {5 \arctan \left (c x \right )}{12}\right )}{c^{4}}\) | \(137\) |
default | \(\frac {a \,d^{2} \left (-\frac {1}{6} c^{6} x^{6}+\frac {2}{5} i c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+b \,d^{2} \left (-\frac {\arctan \left (c x \right ) c^{6} x^{6}}{6}+\frac {2 i \arctan \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}+\frac {5 c x}{12}+\frac {c^{5} x^{5}}{30}-\frac {i c^{4} x^{4}}{10}-\frac {5 c^{3} x^{3}}{36}+\frac {i c^{2} x^{2}}{5}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{5}-\frac {5 \arctan \left (c x \right )}{12}\right )}{c^{4}}\) | \(137\) |
parallelrisch | \(-\frac {30 x^{6} \arctan \left (c x \right ) b \,c^{6} d^{2}-72 i c^{5} b \,d^{2} \arctan \left (c x \right ) x^{5}+30 a \,c^{6} d^{2} x^{6}-72 i x^{5} a \,c^{5} d^{2}-6 b \,c^{5} d^{2} x^{5}+18 i x^{4} b \,c^{4} d^{2}-45 x^{4} \arctan \left (c x \right ) b \,c^{4} d^{2}-45 a \,c^{4} d^{2} x^{4}+25 b \,c^{3} d^{2} x^{3}-36 i x^{2} b \,c^{2} d^{2}+36 i b \,d^{2} \ln \left (c^{2} x^{2}+1\right )-75 b c \,d^{2} x +75 b \arctan \left (c x \right ) d^{2}}{180 c^{4}}\) | \(178\) |
risch | \(\frac {i d^{2} b \left (10 c^{2} x^{6}-24 i c \,x^{5}-15 x^{4}\right ) \ln \left (i c x +1\right )}{120}-\frac {a \,c^{2} d^{2} x^{6}}{6}-\frac {i d^{2} c^{2} x^{6} b \ln \left (-i c x +1\right )}{12}-\frac {d^{2} c b \,x^{5} \ln \left (-i c x +1\right )}{5}+\frac {b c \,d^{2} x^{5}}{30}+\frac {2 i d^{2} c a \,x^{5}}{5}+\frac {d^{2} a \,x^{4}}{4}+\frac {i d^{2} x^{4} b \ln \left (-i c x +1\right )}{8}-\frac {i b \,d^{2} x^{4}}{10}-\frac {5 b \,d^{2} x^{3}}{36 c}+\frac {i b \,d^{2} x^{2}}{5 c^{2}}+\frac {5 b \,d^{2} x}{12 c^{3}}-\frac {5 b \,d^{2} \arctan \left (c x \right )}{12 c^{4}}-\frac {i d^{2} b \ln \left (625 c^{2} x^{2}+625\right )}{5 c^{4}}\) | \(216\) |
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Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.04 \[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {60 \, a c^{6} d^{2} x^{6} + 12 \, {\left (-12 i \, a - b\right )} c^{5} d^{2} x^{5} - 18 \, {\left (5 \, a - 2 i \, b\right )} c^{4} d^{2} x^{4} + 50 \, b c^{3} d^{2} x^{3} - 72 i \, b c^{2} d^{2} x^{2} - 150 \, b c d^{2} x + 147 i \, b d^{2} \log \left (\frac {c x + i}{c}\right ) - 3 i \, b d^{2} \log \left (\frac {c x - i}{c}\right ) + 3 \, {\left (10 i \, b c^{6} d^{2} x^{6} + 24 \, b c^{5} d^{2} x^{5} - 15 i \, b c^{4} d^{2} x^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{360 \, c^{4}} \]
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Time = 2.14 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.63 \[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=- \frac {a c^{2} d^{2} x^{6}}{6} - \frac {5 b d^{2} x^{3}}{36 c} + \frac {i b d^{2} x^{2}}{5 c^{2}} + \frac {5 b d^{2} x}{12 c^{3}} - \frac {b d^{2} \left (- \frac {i \log {\left (291 b c d^{2} x - 291 i b d^{2} \right )}}{120} + \frac {71 i \log {\left (291 b c d^{2} x + 291 i b d^{2} \right )}}{210}\right )}{c^{4}} - x^{5} \left (- \frac {2 i a c d^{2}}{5} - \frac {b c d^{2}}{30}\right ) - x^{4} \left (- \frac {a d^{2}}{4} + \frac {i b d^{2}}{10}\right ) + \left (\frac {i b c^{2} d^{2} x^{6}}{12} + \frac {b c d^{2} x^{5}}{5} - \frac {i b d^{2} x^{4}}{8}\right ) \log {\left (i c x + 1 \right )} + \frac {\left (- 70 i b c^{6} d^{2} x^{6} - 168 b c^{5} d^{2} x^{5} + 105 i b c^{4} d^{2} x^{4} - 59 i b d^{2}\right ) \log {\left (- i c x + 1 \right )}}{840 c^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.11 \[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {1}{6} \, a c^{2} d^{2} x^{6} + \frac {2}{5} i \, a c d^{2} x^{5} + \frac {1}{4} \, a d^{2} x^{4} - \frac {1}{90} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{2} d^{2} + \frac {1}{10} i \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{2} \]
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\[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )} x^{3} \,d x } \]
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Time = 0.84 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.92 \[ \int x^3 (d+i c d x)^2 (a+b \arctan (c x)) \, dx=-\frac {\frac {d^2\,\left (75\,b\,\mathrm {atan}\left (c\,x\right )+b\,\ln \left (c^2\,x^2+1\right )\,36{}\mathrm {i}\right )}{180}+\frac {5\,b\,c^3\,d^2\,x^3}{36}-\frac {5\,b\,c\,d^2\,x}{12}-\frac {b\,c^2\,d^2\,x^2\,1{}\mathrm {i}}{5}}{c^4}+\frac {d^2\,\left (45\,a\,x^4+45\,b\,x^4\,\mathrm {atan}\left (c\,x\right )-b\,x^4\,18{}\mathrm {i}\right )}{180}-\frac {c^2\,d^2\,\left (30\,a\,x^6+30\,b\,x^6\,\mathrm {atan}\left (c\,x\right )\right )}{180}+\frac {c\,d^2\,\left (a\,x^5\,72{}\mathrm {i}+6\,b\,x^5+b\,x^5\,\mathrm {atan}\left (c\,x\right )\,72{}\mathrm {i}\right )}{180} \]
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