Integrand size = 12, antiderivative size = 42 \[ \int e^{-i \arctan (a x)} x \, dx=\frac {(2-i a x) \sqrt {1+a^2 x^2}}{2 a^2}+\frac {i \text {arcsinh}(a x)}{2 a^2} \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5168, 794, 221} \[ \int e^{-i \arctan (a x)} x \, dx=\frac {i \text {arcsinh}(a x)}{2 a^2}+\frac {\sqrt {a^2 x^2+1} (2-i a x)}{2 a^2} \]
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Rule 221
Rule 794
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {x (1-i a x)}{\sqrt {1+a^2 x^2}} \, dx \\ & = \frac {(2-i a x) \sqrt {1+a^2 x^2}}{2 a^2}+\frac {i \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a} \\ & = \frac {(2-i a x) \sqrt {1+a^2 x^2}}{2 a^2}+\frac {i \text {arcsinh}(a x)}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int e^{-i \arctan (a x)} x \, dx=\frac {(2-i a x) \sqrt {1+a^2 x^2}+i \text {arcsinh}(a x)}{2 a^2} \]
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Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.40
method | result | size |
risch | \(-\frac {i \left (a x +2 i\right ) \sqrt {a^{2} x^{2}+1}}{2 a^{2}}+\frac {i \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a \sqrt {a^{2}}}\) | \(59\) |
default | \(-\frac {i \left (\frac {\sqrt {a^{2} x^{2}+1}\, x}{2}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}\right )}{a}+\frac {\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}}{a^{2}}\) | \(150\) |
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Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02 \[ \int e^{-i \arctan (a x)} x \, dx=\frac {\sqrt {a^{2} x^{2} + 1} {\left (-i \, a x + 2\right )} - i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a^{2}} \]
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\[ \int e^{-i \arctan (a x)} x \, dx=- i \int \frac {x \sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int e^{-i \arctan (a x)} x \, dx=-\frac {i \, \sqrt {a^{2} x^{2} + 1} x}{2 \, a} + \frac {i \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{2}} + \frac {\sqrt {a^{2} x^{2} + 1}}{a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.26 \[ \int e^{-i \arctan (a x)} x \, dx=-\frac {1}{2} \, \sqrt {a^{2} x^{2} + 1} {\left (\frac {i \, x}{a} - \frac {2}{a^{2}}\right )} - \frac {i \, \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a {\left | a \right |}} \]
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Time = 0.46 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int e^{-i \arctan (a x)} x \, dx=\frac {\left (\frac {1}{\sqrt {a^2}}-\frac {x\,\sqrt {a^2}\,1{}\mathrm {i}}{2\,a}\right )\,\sqrt {a^2\,x^2+1}+\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{2\,a}}{\sqrt {a^2}} \]
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