Integrand size = 12, antiderivative size = 92 \[ \int e^{-3 i \arctan (a x)} x \, dx=-\frac {9 \sqrt {1+a^2 x^2}}{2 a^2}-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {9 i \text {arcsinh}(a x)}{2 a^2} \]
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Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5168, 1647, 1607, 12, 807, 679, 221} \[ \int e^{-3 i \arctan (a x)} x \, dx=-\frac {9 i \text {arcsinh}(a x)}{2 a^2}-\frac {\left (a^2 x^2+1\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {3 \left (a^2 x^2+1\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {9 \sqrt {a^2 x^2+1}}{2 a^2} \]
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Rule 12
Rule 221
Rule 679
Rule 807
Rule 1607
Rule 1647
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {x (1-i a x)^2}{(1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = (i a) \int \frac {\left (-\frac {i x}{a}-x^2\right ) \sqrt {1+a^2 x^2}}{(1+i a x)^2} \, dx \\ & = (i a) \int \frac {\left (-\frac {i}{a}-x\right ) x \sqrt {1+a^2 x^2}}{(1+i a x)^2} \, dx \\ & = a^2 \int \frac {x \left (1+a^2 x^2\right )^{3/2}}{a^2 (1+i a x)^3} \, dx \\ & = \int \frac {x \left (1+a^2 x^2\right )^{3/2}}{(1+i a x)^3} \, dx \\ & = -\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {(3 i) \int \frac {\left (1+a^2 x^2\right )^{3/2}}{(1+i a x)^2} \, dx}{a} \\ & = -\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {(9 i) \int \frac {\sqrt {1+a^2 x^2}}{1+i a x} \, dx}{2 a} \\ & = -\frac {9 \sqrt {1+a^2 x^2}}{2 a^2}-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {(9 i) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a} \\ & = -\frac {9 \sqrt {1+a^2 x^2}}{2 a^2}-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {9 i \text {arcsinh}(a x)}{2 a^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65 \[ \int e^{-3 i \arctan (a x)} x \, dx=\sqrt {1+a^2 x^2} \left (-\frac {3}{a^2}+\frac {i x}{2 a}+\frac {4 i}{a^2 (-i+a x)}\right )-\frac {9 i \text {arcsinh}(a x)}{2 a^2} \]
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Time = 0.33 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15
method | result | size |
risch | \(\frac {i \left (a x +6 i\right ) \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {i \left (\frac {9 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}-\frac {8 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{2} \left (x -\frac {i}{a}\right )}\right )}{2 a}\) | \(106\) |
default | \(-\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{4}}+\frac {i \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{3}}\) | \(463\) |
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Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.78 \[ \int e^{-3 i \arctan (a x)} x \, dx=\frac {8 i \, a x - 9 \, {\left (-i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \sqrt {a^{2} x^{2} + 1} {\left (i \, a^{2} x^{2} - 5 \, a x + 14 i\right )} + 8}{2 \, {\left (a^{3} x - i \, a^{2}\right )}} \]
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\[ \int e^{-3 i \arctan (a x)} x \, dx=i \left (\int \frac {x \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{3} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
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Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.22 \[ \int e^{-3 i \arctan (a x)} x \, dx=-\frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4} x^{2} - 2 i \, a^{3} x - a^{2}} - \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, a^{3} x + 2 \, a^{2}} - \frac {6 \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{3} x + a^{2}} - \frac {9 i \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{2}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1}}{2 \, a^{2}} \]
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\[ \int e^{-3 i \arctan (a x)} x \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{{\left (i \, a x + 1\right )}^{3}} \,d x } \]
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Time = 0.48 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.14 \[ \int e^{-3 i \arctan (a x)} x \, dx=-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,\sqrt {a^2}}{a^2}-\frac {x\,\sqrt {a^2}\,1{}\mathrm {i}}{2\,a}\right )}{\sqrt {a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,9{}\mathrm {i}}{2\,a\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{a\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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