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 = 0.02 (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} \[ \int e^{i \arctan (a x)} x \, dx=\frac {(2+i a x) \sqrt {a^2 x^2+1}}{2 a^2}-\frac {i \text {arcsinh}(a x)}{2 a^2} \]
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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 = 0.20 (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 | \(\frac {\sqrt {a^{2} x^{2}+1}}{a^{2}}+i a \left (\frac {x \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\right )\) | \(72\) |
meijerg | \(\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{2 a^{2} \sqrt {\pi }}+\frac {i \left (\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}} \sqrt {a^{2} x^{2}+1}}{a^{2}}-\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \operatorname {arcsinh}\left (a x \right )}{a^{3}}\right )}{2 a \sqrt {\pi }\, \sqrt {a^{2}}}\) | \(88\) |
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none
Time = 0.28 (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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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (34) = 68\).
Time = 0.54 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.86 \[ \int e^{i \arctan (a x)} x \, dx=\begin {cases} \sqrt {a^{2} x^{2} + 1} \left (\frac {i x}{2 a} + \frac {1}{a^{2}}\right ) - \frac {i \log {\left (2 a^{2} x + 2 \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2}} \right )}}{2 a \sqrt {a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {i a x^{3}}{3} + \frac {x^{2}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.21 (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 = 0.04 (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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