Integrand size = 14, antiderivative size = 38 \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=-\frac {\sqrt {1+a^2 x^2}}{x}-i a \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5168, 821, 272, 65, 214} \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=-\frac {\sqrt {a^2 x^2+1}}{x}-i a \text {arctanh}\left (\sqrt {a^2 x^2+1}\right ) \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 5168
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a x}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}+(i a) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}+\frac {1}{2} (i a) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}+\frac {i \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} \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}-i a \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=-\frac {\sqrt {1+a^2 x^2}}{x}+i a \log (x)-i a \log \left (1+\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\sqrt {a^{2} x^{2}+1}}{x}-i a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(34\) |
risch | \(-\frac {\sqrt {a^{2} x^{2}+1}}{x}-i a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(34\) |
meijerg | \(-\frac {\sqrt {a^{2} x^{2}+1}}{x}+\frac {i a \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }\right )}{2 \sqrt {\pi }}\) | \(64\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.74 \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=\frac {-i \, a x \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + i \, a x \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - a x - \sqrt {a^{2} x^{2} + 1}}{x} \]
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Time = 1.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=- a \sqrt {1 + \frac {1}{a^{2} x^{2}}} - i a \operatorname {asinh}{\left (\frac {1}{a x} \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=-i \, a \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{x} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.97 \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=-i \, a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} + 1 \right |}\right ) + i \, a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1} - 1 \right |}\right ) + \frac {2 \, {\left | a \right |}}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1} \]
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Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {e^{i \arctan (a x)}}{x^2} \, dx=-\frac {\sqrt {a^2\,x^2+1}}{x}-a\,\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )\,1{}\mathrm {i} \]
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