Integrand size = 16, antiderivative size = 99 \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=-\frac {2 (1+i a)^3 x}{b^4}-\frac {i (i-a)^2 x^2}{b^3}+\frac {2 (1+i a) x^3}{3 b^2}-\frac {i x^4}{2 b}-\frac {x^5}{5}-\frac {2 i (i-a)^4 \log (i-a-b x)}{b^5} \]
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Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5203, 78} \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=-\frac {2 i (-a+i)^4 \log (-a-b x+i)}{b^5}-\frac {2 (1+i a)^3 x}{b^4}-\frac {i (-a+i)^2 x^2}{b^3}+\frac {2 (1+i a) x^3}{3 b^2}-\frac {i x^4}{2 b}-\frac {x^5}{5} \]
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Rule 78
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 (1-i a-i b x)}{1+i a+i b x} \, dx \\ & = \int \left (\frac {2 (-1-i a)^3}{b^4}-\frac {2 i (-i+a)^2 x}{b^3}+\frac {2 (1+i a) x^2}{b^2}-\frac {2 i x^3}{b}-x^4-\frac {2 i (-i+a)^4}{b^4 (-i+a+b x)}\right ) \, dx \\ & = -\frac {2 (1+i a)^3 x}{b^4}-\frac {i (i-a)^2 x^2}{b^3}+\frac {2 (1+i a) x^3}{3 b^2}-\frac {i x^4}{2 b}-\frac {x^5}{5}-\frac {2 i (i-a)^4 \log (i-a-b x)}{b^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.96 \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=-\frac {2 (1+i a)^3 x}{b^4}-\frac {i (-i+a)^2 x^2}{b^3}+\frac {2 (1+i a) x^3}{3 b^2}-\frac {i x^4}{2 b}-\frac {x^5}{5}-\frac {2 i (-i+a)^4 \log (i-a-b x)}{b^5} \]
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Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {i \left (-\frac {1}{5} i b^{4} x^{5}+\frac {1}{2} b^{3} x^{4}+\frac {2}{3} i b^{2} x^{3}-\frac {2}{3} a \,b^{2} x^{3}-2 i a b \,x^{2}+a^{2} b \,x^{2}+6 i a^{2} x -2 a^{3} x -x^{2} b -2 i x +6 a x \right )}{b^{4}}+\frac {\left (-2 i a^{4}-8 a^{3}+12 i a^{2}+8 a -2 i\right ) \ln \left (-b x -a +i\right )}{b^{5}}\) | \(125\) |
risch | \(-\frac {x^{5}}{5}+\frac {2 i a \,x^{3}}{3 b^{2}}+\frac {2 x^{3}}{3 b^{2}}-\frac {i x^{4}}{2 b}-\frac {2 a \,x^{2}}{b^{3}}-\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{4}}{b^{5}}+\frac {6 a^{2} x}{b^{4}}-\frac {8 i \arctan \left (b x +a \right ) a^{3}}{b^{5}}+\frac {2 i a^{3} x}{b^{4}}-\frac {2 x}{b^{4}}+\frac {8 i \arctan \left (b x +a \right ) a}{b^{5}}-\frac {4 \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{3}}{b^{5}}-\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b^{5}}+\frac {4 \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a}{b^{5}}+\frac {i x^{2}}{b^{3}}+\frac {6 i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{2}}{b^{5}}-\frac {6 i a x}{b^{4}}+\frac {2 \arctan \left (b x +a \right ) a^{4}}{b^{5}}-\frac {i a^{2} x^{2}}{b^{3}}-\frac {12 \arctan \left (b x +a \right ) a^{2}}{b^{5}}+\frac {2 \arctan \left (b x +a \right )}{b^{5}}\) | \(292\) |
parallelrisch | \(\frac {60-90 a^{2} b^{2} x^{2}-600 a^{2}+300 a^{4}+300 \ln \left (b x +a -i\right ) a^{4}-600 \ln \left (b x +a -i\right ) a^{2}-600 i \ln \left (b x +a -i\right ) a^{3}+300 i \ln \left (b x +a -i\right ) a +9 i x^{5} b^{5}+60 i \ln \left (b x +a -i\right ) a^{5}+240 \ln \left (b x +a -i\right ) x \,a^{3} b -240 \ln \left (b x +a -i\right ) x a b +60 i \ln \left (b x +a -i\right ) x b +60 i \ln \left (b x +a -i\right ) x \,a^{4} b -360 i \ln \left (b x +a -i\right ) x \,a^{2} b +300 i a +60 \ln \left (b x +a -i\right )+6 b^{6} x^{6}+30 b^{2} x^{2}-5 x^{4} b^{4}+20 a \,b^{3} x^{3}+60 i a^{5}+90 i a \,b^{2} x^{2}-600 i a^{3}-5 i a \,b^{4} x^{4}+10 i a^{2} b^{3} x^{3}+6 a \,b^{5} x^{5}-30 i x^{2} a^{3} b^{2}-10 i b^{3} x^{3}}{30 b^{5} \left (-b x -a +i\right )}\) | \(310\) |
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Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06 \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=-\frac {6 \, b^{5} x^{5} + 15 i \, b^{4} x^{4} + 20 \, {\left (-i \, a - 1\right )} b^{3} x^{3} + 30 \, {\left (i \, a^{2} + 2 \, a - i\right )} b^{2} x^{2} + 60 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} b x + 60 \, {\left (i \, a^{4} + 4 \, a^{3} - 6 i \, a^{2} - 4 \, a + i\right )} \log \left (\frac {b x + a - i}{b}\right )}{30 \, b^{5}} \]
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Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.15 \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=- \frac {x^{5}}{5} - x^{3} \left (- \frac {2 i a}{3 b^{2}} - \frac {2}{3 b^{2}}\right ) - x^{2} \left (\frac {i a^{2}}{b^{3}} + \frac {2 a}{b^{3}} - \frac {i}{b^{3}}\right ) - x \left (- \frac {2 i a^{3}}{b^{4}} - \frac {6 a^{2}}{b^{4}} + \frac {6 i a}{b^{4}} + \frac {2}{b^{4}}\right ) - \frac {i x^{4}}{2 b} - \frac {2 i \left (a - i\right )^{4} \log {\left (a + b x - i \right )}}{b^{5}} \]
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Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06 \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=-\frac {6 \, b^{4} x^{5} + 15 i \, b^{3} x^{4} - 20 \, {\left (i \, a + 1\right )} b^{2} x^{3} - 30 \, {\left (-i \, a^{2} - 2 \, a + i\right )} b x^{2} - 60 \, {\left (i \, a^{3} + 3 \, a^{2} - 3 i \, a - 1\right )} x}{30 \, b^{4}} - \frac {2 \, {\left (i \, a^{4} + 4 \, a^{3} - 6 i \, a^{2} - 4 \, a + i\right )} \log \left (i \, b x + i \, a + 1\right )}{b^{5}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (73) = 146\).
Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.17 \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=\frac {i \, {\left (i \, b x + i \, a + 1\right )}^{5} {\left (-\frac {15 i \, {\left (2 \, a b - 3 i \, b\right )}}{{\left (i \, b x + i \, a + 1\right )} b} - \frac {20 \, {\left (3 \, a^{2} b^{2} - 10 i \, a b^{2} - 7 \, b^{2}\right )}}{{\left (i \, b x + i \, a + 1\right )}^{2} b^{2}} + \frac {60 i \, {\left (a^{3} b^{3} - 6 i \, a^{2} b^{3} - 9 \, a b^{3} + 4 i \, b^{3}\right )}}{{\left (i \, b x + i \, a + 1\right )}^{3} b^{3}} + \frac {30 \, {\left (a^{4} b^{4} - 12 i \, a^{3} b^{4} - 30 \, a^{2} b^{4} + 28 i \, a b^{4} + 9 \, b^{4}\right )}}{{\left (i \, b x + i \, a + 1\right )}^{4} b^{4}} + 6\right )}}{30 \, b^{5}} - \frac {2 \, {\left (-i \, a^{4} - 4 \, a^{3} + 6 i \, a^{2} + 4 \, a - i\right )} \log \left (\frac {1}{\sqrt {{\left (b x + a\right )}^{2} + 1} {\left | b \right |}}\right )}{b^{5}} \]
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Time = 0.19 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.67 \[ \int e^{-2 i \arctan (a+b x)} x^4 \, dx=\ln \left (x+\frac {a-\mathrm {i}}{b}\right )\,\left (\frac {8\,a-8\,a^3}{b^5}-\frac {\left (2\,a^4-12\,a^2+2\right )\,1{}\mathrm {i}}{b^5}\right )+x^4\,\left (\frac {a-\mathrm {i}}{4\,b}-\frac {a+1{}\mathrm {i}}{4\,b}\right )-\frac {x^5}{5}+\frac {x^2\,\left (\frac {a-\mathrm {i}}{b}-\frac {a+1{}\mathrm {i}}{b}\right )\,{\left (a-\mathrm {i}\right )}^2}{2\,b^2}-\frac {x^3\,\left (\frac {a-\mathrm {i}}{b}-\frac {a+1{}\mathrm {i}}{b}\right )\,\left (a-\mathrm {i}\right )}{3\,b}-\frac {x\,\left (\frac {a-\mathrm {i}}{b}-\frac {a+1{}\mathrm {i}}{b}\right )\,{\left (a-\mathrm {i}\right )}^3}{b^3} \]
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