Integrand size = 14, antiderivative size = 25 \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5168, 858, 221, 272, 65, 214} \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=-\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+i \text {arcsinh}(a x) \]
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Rule 65
Rule 214
Rule 221
Rule 272
Rule 858
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a x}{x \sqrt {1+a^2 x^2}} \, dx \\ & = (i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = i \text {arcsinh}(a x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = i \text {arcsinh}(a x)+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{a^2} \\ & = i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=i \text {arcsinh}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22 ) = 44\).
Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92
method | result | size |
default | \(-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}\) | \(48\) |
meijerg | \(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{2 \sqrt {\pi }}+i \operatorname {arcsinh}\left (a x \right )\) | \(53\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=-\log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) \]
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Time = 1.63 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=i a \left (\begin {cases} \frac {\log {\left (2 a^{2} x + 2 \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2}} \right )}}{\sqrt {a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) - \operatorname {asinh}{\left (\frac {1}{a x} \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=i \, \operatorname {arsinh}\left (a x\right ) - \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72 \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=-\frac {i \, a \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{{\left | a \right |}} - \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{i \arctan (a x)}}{x} \, dx=-\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )+\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}} \]
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