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 = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41
method | result | size |
meijerg | \(\frac {\operatorname {arcsinh}\left (a x \right )}{a}+\frac {i \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}\) | \(41\) |
default | \(\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {i \sqrt {a^{2} x^{2}+1}}{a}\) | \(48\) |
risch | \(\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {i \sqrt {a^{2} x^{2}+1}}{a}\) | \(48\) |
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none
Time = 0.26 (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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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.45 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int e^{i \arctan (a x)} \, dx=\begin {cases} \frac {\log {\left (2 a^{2} x + 2 \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2}} \right )}}{\sqrt {a^{2}}} + \frac {i \sqrt {a^{2} x^{2} + 1}}{a} & \text {for}\: a^{2} \neq 0 \\\frac {i a x^{2}}{2} + x & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (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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none
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 = 0.04 (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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