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 = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\sqrt {a^{2} x^{2}+1}}{x}+i a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(34\) |
default | \(-\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}+2 a^{2} \left (\frac {\sqrt {a^{2} x^{2}+1}\, x}{2}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}\right )-i a \left (\sqrt {a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )+i a \left (\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\right )\) | \(195\) |
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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 = 0.26 (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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\[ \int \frac {e^{-i \arctan (a x)}}{x^2} \, dx=- i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x^{3} - i x^{2}}\, dx \]
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\[ \int \frac {e^{-i \arctan (a x)}}{x^2} \, dx=\int { \frac {\sqrt {a^{2} x^{2} + 1}}{{\left (i \, a x + 1\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {e^{-i \arctan (a x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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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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