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 \\ & = -\left ((i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\right )+\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. 120 vs. \(2 (22 ) = 44\).
Time = 0.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.84
method | result | size |
default | \(\sqrt {a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}-\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\) | \(121\) |
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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 = 0.26 (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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\[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=- i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x^{2} - i x}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-i \arctan (a x)}}{x} \, dx=-i \, a {\left (\frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {i \, \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right )}{a}\right )} \]
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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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