Integrand size = 10, antiderivative size = 29 \[ \int e^{-i \arctan (a x)} \, dx=-\frac {i \sqrt {1+a^2 x^2}}{a}+\frac {\text {arcsinh}(a x)}{a} \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5167, 655, 221} \[ \int e^{-i \arctan (a x)} \, dx=\frac {\text {arcsinh}(a x)}{a}-\frac {i \sqrt {a^2 x^2+1}}{a} \]
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Rule 221
Rule 655
Rule 5167
Rubi steps \begin{align*} \text {integral}& = \int \frac {1-i a x}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {i \sqrt {1+a^2 x^2}}{a}+\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {i \sqrt {1+a^2 x^2}}{a}+\frac {\text {arcsinh}(a x)}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int e^{-i \arctan (a x)} \, dx=\frac {-i \sqrt {1+a^2 x^2}+\text {arcsinh}(a x)}{a} \]
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66
method | result | size |
risch | \(-\frac {i \sqrt {a^{2} x^{2}+1}}{a}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}\) | \(48\) |
default | \(-\frac {i \left (\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\right )}{a}\) | \(100\) |
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int e^{-i \arctan (a x)} \, dx=\frac {-i \, \sqrt {a^{2} x^{2} + 1} - \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{a} \]
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\[ \int e^{-i \arctan (a x)} \, dx=- i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int e^{-i \arctan (a x)} \, dx=\frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {i \, \sqrt {a^{2} x^{2} + 1}}{a} \]
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Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int e^{-i \arctan (a x)} \, dx=-\frac {\log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \frac {i \, \sqrt {a^{2} x^{2} + 1}}{a} \]
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Time = 0.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int e^{-i \arctan (a x)} \, dx=\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{a} \]
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