Integrand size = 14, antiderivative size = 52 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.42 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5168, 6874, 221, 272, 65, 214, 665} \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {4 i \sqrt {a^2 x^2+1}}{-a x+i}+i \text {arcsinh}(a x) \]
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Rule 65
Rule 214
Rule 221
Rule 272
Rule 665
Rule 5168
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^2}{x (1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = \int \left (\frac {i a}{\sqrt {1+a^2 x^2}}+\frac {1}{x \sqrt {1+a^2 x^2}}-\frac {4 a}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx \\ & = (i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx-(4 a) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx \\ & = \frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+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 ) \\ & = \frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+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} \\ & = \frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\frac {4 i \sqrt {1+a^2 x^2}}{-i+a x}+i \text {arcsinh}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (45 ) = 90\).
Time = 0.29 (sec) , antiderivative size = 649, normalized size of antiderivative = 12.48
method | result | size |
default | \(\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}-\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}-i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )+\frac {i \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a}\) | \(649\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (41) = 82\).
Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.92 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\frac {-4 i \, a x - {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + {\left (-i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - 4 i \, \sqrt {a^{2} x^{2} + 1} - 4}{a x - i} \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{4} - 3 i a^{2} x^{3} - 3 a x^{2} + i x}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{4} - 3 i a^{2} x^{3} - 3 a x^{2} + i x}\, dx\right ) \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x} \,d x } \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x} \,d x } \]
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Time = 0.49 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.42 \[ \int \frac {e^{-3 i \arctan (a x)}}{x} \, dx=-\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )+\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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