Integrand size = 21, antiderivative size = 178 \[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=-\frac {16 b d^4 x}{15 c}-\frac {4 i b d^4 (i-c x)^2}{15 c^2}-\frac {4 b d^4 (i-c x)^3}{45 c^2}+\frac {i b d^4 (i-c x)^4}{30 c^2}+\frac {b d^4 (i-c x)^5}{30 c^2}+\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 c^2}-\frac {d^4 (1+i c x)^6 (a+b \arctan (c x))}{6 c^2}+\frac {32 i b d^4 \log (i+c x)}{15 c^2} \]
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Time = 0.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 4992, 12, 78} \[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=-\frac {d^4 (1+i c x)^6 (a+b \arctan (c x))}{6 c^2}+\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 c^2}+\frac {b d^4 (-c x+i)^5}{30 c^2}+\frac {i b d^4 (-c x+i)^4}{30 c^2}-\frac {4 b d^4 (-c x+i)^3}{45 c^2}-\frac {4 i b d^4 (-c x+i)^2}{15 c^2}+\frac {32 i b d^4 \log (c x+i)}{15 c^2}-\frac {16 b d^4 x}{15 c} \]
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Rule 12
Rule 45
Rule 78
Rule 4992
Rubi steps \begin{align*} \text {integral}& = \frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 c^2}-\frac {d^4 (1+i c x)^6 (a+b \arctan (c x))}{6 c^2}-(b c) \int \frac {d^4 (i-c x)^4 (i+5 c x)}{30 c^2 (i+c x)} \, dx \\ & = \frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 c^2}-\frac {d^4 (1+i c x)^6 (a+b \arctan (c x))}{6 c^2}-\frac {\left (b d^4\right ) \int \frac {(i-c x)^4 (i+5 c x)}{i+c x} \, dx}{30 c} \\ & = \frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 c^2}-\frac {d^4 (1+i c x)^6 (a+b \arctan (c x))}{6 c^2}-\frac {\left (b d^4\right ) \int \left (32+5 (i-c x)^4+16 i (-i+c x)-8 (-i+c x)^2-4 i (-i+c x)^3-\frac {64 i}{i+c x}\right ) \, dx}{30 c} \\ & = -\frac {16 b d^4 x}{15 c}-\frac {4 i b d^4 (i-c x)^2}{15 c^2}-\frac {4 b d^4 (i-c x)^3}{45 c^2}+\frac {i b d^4 (i-c x)^4}{30 c^2}+\frac {b d^4 (i-c x)^5}{30 c^2}+\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{5 c^2}-\frac {d^4 (1+i c x)^6 (a+b \arctan (c x))}{6 c^2}+\frac {32 i b d^4 \log (i+c x)}{15 c^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.48 \[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=-\frac {13 b d^4 x}{6 c}+\frac {1}{2} a d^4 x^2-\frac {16}{15} i b d^4 x^2+\frac {4}{3} i a c d^4 x^3+\frac {5}{9} b c d^4 x^3-\frac {3}{2} a c^2 d^4 x^4+\frac {1}{5} i b c^2 d^4 x^4-\frac {4}{5} i a c^3 d^4 x^5-\frac {1}{30} b c^3 d^4 x^5+\frac {1}{6} a c^4 d^4 x^6+\frac {13 b d^4 \arctan (c x)}{6 c^2}+\frac {1}{2} b d^4 x^2 \arctan (c x)+\frac {4}{3} i b c d^4 x^3 \arctan (c x)-\frac {3}{2} b c^2 d^4 x^4 \arctan (c x)-\frac {4}{5} i b c^3 d^4 x^5 \arctan (c x)+\frac {1}{6} b c^4 d^4 x^6 \arctan (c x)+\frac {16 i b d^4 \log \left (1+c^2 x^2\right )}{15 c^2} \]
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Time = 1.64 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.97
method | result | size |
parts | \(d^{4} a \left (\frac {1}{6} c^{4} x^{6}-\frac {4}{5} i c^{3} x^{5}-\frac {3}{2} c^{2} x^{4}+\frac {4}{3} i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d^{4} b \left (\frac {\arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {4 i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{2}+\frac {4 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {13 c x}{6}-\frac {c^{5} x^{5}}{30}+\frac {i c^{4} x^{4}}{5}+\frac {5 c^{3} x^{3}}{9}-\frac {16 i c^{2} x^{2}}{15}+\frac {16 i \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {13 \arctan \left (c x \right )}{6}\right )}{c^{2}}\) | \(173\) |
derivativedivides | \(\frac {d^{4} a \left (\frac {1}{6} c^{6} x^{6}-\frac {4}{5} i c^{5} x^{5}-\frac {3}{2} c^{4} x^{4}+\frac {4}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{4} b \left (\frac {\arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {4 i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{2}+\frac {4 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {13 c x}{6}-\frac {c^{5} x^{5}}{30}+\frac {i c^{4} x^{4}}{5}+\frac {5 c^{3} x^{3}}{9}-\frac {16 i c^{2} x^{2}}{15}+\frac {16 i \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {13 \arctan \left (c x \right )}{6}\right )}{c^{2}}\) | \(179\) |
default | \(\frac {d^{4} a \left (\frac {1}{6} c^{6} x^{6}-\frac {4}{5} i c^{5} x^{5}-\frac {3}{2} c^{4} x^{4}+\frac {4}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d^{4} b \left (\frac {\arctan \left (c x \right ) c^{6} x^{6}}{6}-\frac {4 i \arctan \left (c x \right ) c^{5} x^{5}}{5}-\frac {3 c^{4} x^{4} \arctan \left (c x \right )}{2}+\frac {4 i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {13 c x}{6}-\frac {c^{5} x^{5}}{30}+\frac {i c^{4} x^{4}}{5}+\frac {5 c^{3} x^{3}}{9}-\frac {16 i c^{2} x^{2}}{15}+\frac {16 i \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {13 \arctan \left (c x \right )}{6}\right )}{c^{2}}\) | \(179\) |
parallelrisch | \(\frac {15 b \,c^{6} d^{4} \arctan \left (c x \right ) x^{6}+18 i x^{4} b \,c^{4} d^{4}+15 a \,c^{6} d^{4} x^{6}+96 i b \,d^{4} \ln \left (c^{2} x^{2}+1\right )-3 b \,c^{5} d^{4} x^{5}+120 i x^{3} \arctan \left (c x \right ) b \,c^{3} d^{4}-135 x^{4} \arctan \left (c x \right ) b \,c^{4} d^{4}-72 i c^{5} b \,d^{4} \arctan \left (c x \right ) x^{5}-135 a \,c^{4} d^{4} x^{4}-72 i x^{5} a \,c^{5} d^{4}+50 b \,c^{3} d^{4} x^{3}+120 i x^{3} a \,c^{3} d^{4}+45 x^{2} \arctan \left (c x \right ) b \,c^{2} d^{4}+45 x^{2} d^{4} c^{2} a -96 i x^{2} b \,c^{2} d^{4}-195 b c \,d^{4} x +195 b \,d^{4} \arctan \left (c x \right )}{90 c^{2}}\) | \(236\) |
risch | \(-\frac {i d^{4} b \left (5 c^{4} x^{6}-24 i c^{3} x^{5}-45 c^{2} x^{4}+40 i c \,x^{3}+15 x^{2}\right ) \ln \left (i c x +1\right )}{60}+\frac {a \,c^{4} d^{4} x^{6}}{6}+\frac {i b \,c^{2} d^{4} x^{4}}{5}+\frac {2 d^{4} c^{3} b \,x^{5} \ln \left (-i c x +1\right )}{5}-\frac {b \,c^{3} d^{4} x^{5}}{30}-\frac {16 i b \,d^{4} x^{2}}{15}-\frac {3 a \,c^{2} d^{4} x^{4}}{2}+\frac {16 i d^{4} b \ln \left (4225 c^{2} x^{2}+4225\right )}{15 c^{2}}-\frac {2 d^{4} c b \,x^{3} \ln \left (-i c x +1\right )}{3}+\frac {i d^{4} x^{2} b \ln \left (-i c x +1\right )}{4}+\frac {5 b c \,d^{4} x^{3}}{9}-\frac {4 i a \,c^{3} d^{4} x^{5}}{5}+\frac {a \,d^{4} x^{2}}{2}+\frac {i d^{4} c^{4} x^{6} b \ln \left (-i c x +1\right )}{12}+\frac {4 i a c \,d^{4} x^{3}}{3}-\frac {13 b \,d^{4} x}{6 c}+\frac {13 d^{4} b \arctan \left (c x \right )}{6 c^{2}}-\frac {3 i d^{4} c^{2} x^{4} b \ln \left (-i c x +1\right )}{4}\) | \(299\) |
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Time = 0.26 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.16 \[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {30 \, a c^{6} d^{4} x^{6} - 6 \, {\left (24 i \, a + b\right )} c^{5} d^{4} x^{5} - 18 \, {\left (15 \, a - 2 i \, b\right )} c^{4} d^{4} x^{4} - 20 \, {\left (-12 i \, a - 5 \, b\right )} c^{3} d^{4} x^{3} + 6 \, {\left (15 \, a - 32 i \, b\right )} c^{2} d^{4} x^{2} - 390 \, b c d^{4} x + 387 i \, b d^{4} \log \left (\frac {c x + i}{c}\right ) - 3 i \, b d^{4} \log \left (\frac {c x - i}{c}\right ) - 3 \, {\left (-5 i \, b c^{6} d^{4} x^{6} - 24 \, b c^{5} d^{4} x^{5} + 45 i \, b c^{4} d^{4} x^{4} + 40 \, b c^{3} d^{4} x^{3} - 15 i \, b c^{2} d^{4} x^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{180 \, c^{2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (156) = 312\).
Time = 2.75 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.02 \[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {a c^{4} d^{4} x^{6}}{6} - \frac {13 b d^{4} x}{6 c} + \frac {b d^{4} \left (- \frac {i \log {\left (709 b c d^{4} x - 709 i b d^{4} \right )}}{60} + \frac {117 i \log {\left (709 b c d^{4} x + 709 i b d^{4} \right )}}{70}\right )}{c^{2}} + x^{5} \left (- \frac {4 i a c^{3} d^{4}}{5} - \frac {b c^{3} d^{4}}{30}\right ) + x^{4} \left (- \frac {3 a c^{2} d^{4}}{2} + \frac {i b c^{2} d^{4}}{5}\right ) + x^{3} \cdot \left (\frac {4 i a c d^{4}}{3} + \frac {5 b c d^{4}}{9}\right ) + x^{2} \left (\frac {a d^{4}}{2} - \frac {16 i b d^{4}}{15}\right ) + \left (- \frac {i b c^{4} d^{4} x^{6}}{12} - \frac {2 b c^{3} d^{4} x^{5}}{5} + \frac {3 i b c^{2} d^{4} x^{4}}{4} + \frac {2 b c d^{4} x^{3}}{3} - \frac {i b d^{4} x^{2}}{4}\right ) \log {\left (i c x + 1 \right )} + \frac {\left (35 i b c^{6} d^{4} x^{6} + 168 b c^{5} d^{4} x^{5} - 315 i b c^{4} d^{4} x^{4} - 280 b c^{3} d^{4} x^{3} + 105 i b c^{2} d^{4} x^{2} + 201 i b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{420 c^{2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (138) = 276\).
Time = 0.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.63 \[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {1}{6} \, a c^{4} d^{4} x^{6} - \frac {4}{5} i \, a c^{3} d^{4} x^{5} - \frac {3}{2} \, a c^{2} d^{4} x^{4} + \frac {1}{90} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{4} d^{4} - \frac {1}{5} i \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c^{3} d^{4} + \frac {4}{3} i \, a c d^{4} x^{3} - \frac {1}{2} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b c^{2} d^{4} + \frac {2}{3} i \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b c d^{4} + \frac {1}{2} \, a d^{4} x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d^{4} \]
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\[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\int { {\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )} x \,d x } \]
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Time = 0.85 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.07 \[ \int x (d+i c d x)^4 (a+b \arctan (c x)) \, dx=\frac {\frac {d^4\,\left (195\,b\,\mathrm {atan}\left (c\,x\right )+b\,\ln \left (c^2\,x^2+1\right )\,96{}\mathrm {i}\right )}{90}-\frac {13\,b\,c\,d^4\,x}{6}}{c^2}+\frac {d^4\,\left (45\,a\,x^2+45\,b\,x^2\,\mathrm {atan}\left (c\,x\right )-b\,x^2\,96{}\mathrm {i}\right )}{90}+\frac {c^4\,d^4\,\left (15\,a\,x^6+15\,b\,x^6\,\mathrm {atan}\left (c\,x\right )\right )}{90}+\frac {c\,d^4\,\left (a\,x^3\,120{}\mathrm {i}+50\,b\,x^3+b\,x^3\,\mathrm {atan}\left (c\,x\right )\,120{}\mathrm {i}\right )}{90}-\frac {c^3\,d^4\,\left (a\,x^5\,72{}\mathrm {i}+3\,b\,x^5+b\,x^5\,\mathrm {atan}\left (c\,x\right )\,72{}\mathrm {i}\right )}{90}-\frac {c^2\,d^4\,\left (135\,a\,x^4+135\,b\,x^4\,\mathrm {atan}\left (c\,x\right )-b\,x^4\,18{}\mathrm {i}\right )}{90} \]
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