Integrand size = 16, antiderivative size = 72 \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=\frac {2 i (i+a)^2 x}{b^3}+\frac {(1-i a) x^2}{b^2}+\frac {2 i x^3}{3 b}-\frac {x^4}{4}-\frac {2 (1-i a)^3 \log (i+a+b x)}{b^4} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 72, 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} \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=-\frac {2 (1-i a)^3 \log (a+b x+i)}{b^4}+\frac {2 i (a+i)^2 x}{b^3}+\frac {(1-i a) x^2}{b^2}+\frac {2 i x^3}{3 b}-\frac {x^4}{4} \]
[In]
[Out]
Rule 78
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 (1+i a+i b x)}{1-i a-i b x} \, dx \\ & = \int \left (\frac {2 i (i+a)^2}{b^3}+\frac {2 (1-i a) x}{b^2}+\frac {2 i x^2}{b}-x^3+\frac {2 (-1+i a)^3}{b^3 (i+a+b x)}\right ) \, dx \\ & = \frac {2 i (i+a)^2 x}{b^3}+\frac {(1-i a) x^2}{b^2}+\frac {2 i x^3}{3 b}-\frac {x^4}{4}-\frac {2 (1-i a)^3 \log (i+a+b x)}{b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=\frac {2 i (i+a)^2 x}{b^3}+\frac {(1-i a) x^2}{b^2}+\frac {2 i x^3}{3 b}-\frac {x^4}{4}-\frac {2 (1-i a)^3 \log (i+a+b x)}{b^4} \]
[In]
[Out]
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.51
method | result | size |
parallelrisch | \(-\frac {3 x^{4} b^{4}-8 i b^{3} x^{3}+12 i a \,b^{2} x^{2}-72 \ln \left (b x +a +i\right ) a^{2}+24 i \ln \left (b x +a +i\right ) a^{3}-24 i a^{2} b x -12 b^{2} x^{2}+24 \ln \left (b x +a +i\right )-72 i \ln \left (b x +a +i\right ) a +24 b x i+48 a b x}{12 b^{4}}\) | \(109\) |
default | \(\frac {i \left (\frac {1}{4} i b^{3} x^{4}+\frac {2}{3} b^{2} x^{3}-i b \,x^{2}-a b \,x^{2}+4 i a x +2 a^{2} x -2 x \right )}{b^{3}}+\frac {\frac {\left (-2 i a^{3} b +6 a^{2} b +6 i a b -2 b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (-2 i a^{4}+4 a^{3}+2 i+4 a -\frac {\left (-2 i a^{3} b +6 a^{2} b +6 i a b -2 b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}}{b^{3}}\) | \(170\) |
risch | \(-\frac {x^{4}}{4}+\frac {2 i x^{3}}{3 b}+\frac {x^{2}}{b^{2}}-\frac {i a \,x^{2}}{b^{2}}-\frac {4 a x}{b^{3}}+\frac {2 i a^{2} x}{b^{3}}-\frac {2 i x}{b^{3}}+\frac {3 \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{2}}{b^{4}}-\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{3}}{b^{4}}-\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b^{4}}+\frac {3 i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a}{b^{4}}-\frac {6 i \arctan \left (b x +a \right ) a^{2}}{b^{4}}-\frac {2 \arctan \left (b x +a \right ) a^{3}}{b^{4}}+\frac {2 i \arctan \left (b x +a \right )}{b^{4}}+\frac {6 \arctan \left (b x +a \right ) a}{b^{4}}\) | \(211\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=-\frac {3 \, b^{4} x^{4} - 8 i \, b^{3} x^{3} + 12 \, {\left (i \, a - 1\right )} b^{2} x^{2} + 24 \, {\left (-i \, a^{2} + 2 \, a + i\right )} b x + 24 \, {\left (i \, a^{3} - 3 \, a^{2} - 3 i \, a + 1\right )} \log \left (\frac {b x + a + i}{b}\right )}{12 \, b^{4}} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04 \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=- \frac {x^{4}}{4} - x^{2} \left (\frac {i a}{b^{2}} - \frac {1}{b^{2}}\right ) - x \left (- \frac {2 i a^{2}}{b^{3}} + \frac {4 a}{b^{3}} + \frac {2 i}{b^{3}}\right ) + \frac {2 i x^{3}}{3 b} - \frac {2 i \left (a + i\right )^{3} \log {\left (a + b x + i \right )}}{b^{4}} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (56) = 112\).
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.61 \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=-\frac {3 \, b^{3} x^{4} - 8 i \, b^{2} x^{3} + 12 \, {\left (i \, a - 1\right )} b x^{2} + 24 \, {\left (-i \, a^{2} + 2 \, a + i\right )} x}{12 \, b^{3}} - \frac {2 \, {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b^{4}} + \frac {{\left (-i \, a^{3} + 3 \, a^{2} + 3 i \, a - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{4}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15 \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=-\frac {2 \, {\left (i \, a^{3} - 3 \, a^{2} - 3 i \, a + 1\right )} \log \left (b x + a + i\right )}{b^{4}} - \frac {3 \, b^{4} x^{4} - 8 i \, b^{3} x^{3} + 12 i \, a b^{2} x^{2} - 24 i \, a^{2} b x - 12 \, b^{2} x^{2} + 48 \, a b x + 24 i \, b x}{12 \, b^{4}} \]
[In]
[Out]
Time = 0.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.12 \[ \int e^{2 i \arctan (a+b x)} x^3 \, dx=-x^3\,\left (\frac {\left (-1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3\,b}-\frac {\left (1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3\,b}\right )-\frac {x^4}{4}+\ln \left (x+\frac {a+1{}\mathrm {i}}{b}\right )\,\left (\frac {6\,a^2-2}{b^4}+\frac {\left (6\,a-2\,a^3\right )\,1{}\mathrm {i}}{b^4}\right )-\frac {x^2\,\left (-1+a\,1{}\mathrm {i}\right )\,\left (\frac {\left (-1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}-\frac {\left (1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}\right )\,1{}\mathrm {i}}{2\,b}+\frac {x\,{\left (-1+a\,1{}\mathrm {i}\right )}^2\,\left (\frac {\left (-1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}-\frac {\left (1+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b}\right )}{b^2} \]
[In]
[Out]