Integrand size = 14, antiderivative size = 64 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=-\frac {\sqrt {1+a^2 x^2}}{x}+\frac {4 a \sqrt {1+a^2 x^2}}{i-a x}+3 i a \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.41 (sec) , antiderivative size = 64, 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, 270, 272, 65, 214, 665} \[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=3 i a \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {4 a \sqrt {a^2 x^2+1}}{-a x+i}-\frac {\sqrt {a^2 x^2+1}}{x} \]
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Rule 65
Rule 214
Rule 270
Rule 272
Rule 665
Rule 5168
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^2}{x^2 (1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = \int \left (\frac {1}{x^2 \sqrt {1+a^2 x^2}}-\frac {3 i a}{x \sqrt {1+a^2 x^2}}+\frac {4 i a^2}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx \\ & = -\left ((3 i a) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\right )+\left (4 i a^2\right ) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}+\frac {4 a \sqrt {1+a^2 x^2}}{i-a x}-\frac {1}{2} (3 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 {4 a \sqrt {1+a^2 x^2}}{i-a x}-\frac {(3 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}+\frac {4 a \sqrt {1+a^2 x^2}}{i-a x}+3 i a \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=\sqrt {1+a^2 x^2} \left (-\frac {1}{x}-\frac {4 a}{-i+a x}\right )-3 i a \log (x)+3 i a \log \left (1+\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-\frac {\sqrt {a^{2} x^{2}+1}}{x}+i a \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {4 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a \left (x -\frac {i}{a}\right )}\right )\) | \(82\) |
default | \(-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+4 a^{2} \left (\frac {x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 \sqrt {a^{2} x^{2}+1}\, x}{8}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 \sqrt {a^{2}}}\right )-3 i a \left (\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 )\right )-\frac {2 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}}+9 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 )-\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 )^{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 )\right )}{a}\) | \(580\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (53) = 106\).
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=-\frac {5 \, a^{2} x^{2} - 5 i \, a x + 3 \, {\left (-i \, a^{2} x^{2} - a x\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (i \, a^{2} x^{2} + a x\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + \sqrt {a^{2} x^{2} + 1} {\left (5 \, a x - i\right )}}{a x^{2} - i \, x} \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{5} - 3 i a^{2} x^{4} - 3 a x^{3} + i x^{2}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{5} - 3 i a^{2} x^{4} - 3 a x^{3} + i x^{2}}\, dx\right ) \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{2}} \,d x } \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{2}} \,d x } \]
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Time = 0.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^2} \, dx=a\,\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )\,3{}\mathrm {i}-\frac {\sqrt {a^2\,x^2+1}}{x}+\frac {4\,a^2\,\sqrt {a^2\,x^2+1}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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