Integrand size = 14, antiderivative size = 93 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=-\frac {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {3 i a \sqrt {1+a^2 x^2}}{x}-\frac {4 i a^2 \sqrt {1+a^2 x^2}}{i-a x}+\frac {9}{2} a^2 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5168, 6874, 272, 44, 65, 214, 270, 665} \[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=\frac {9}{2} a^2 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-\frac {4 i a^2 \sqrt {a^2 x^2+1}}{-a x+i}+\frac {3 i a \sqrt {a^2 x^2+1}}{x}-\frac {\sqrt {a^2 x^2+1}}{2 x^2} \]
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Rule 44
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^3 (1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = \int \left (\frac {1}{x^3 \sqrt {1+a^2 x^2}}-\frac {3 i a}{x^2 \sqrt {1+a^2 x^2}}-\frac {4 a^2}{x \sqrt {1+a^2 x^2}}+\frac {4 a^3}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx \\ & = -\left ((3 i a) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx\right )-\left (4 a^2\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx+\left (4 a^3\right ) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx \\ & = \frac {3 i a \sqrt {1+a^2 x^2}}{x}-\frac {4 i a^2 \sqrt {1+a^2 x^2}}{i-a x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )-\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {3 i a \sqrt {1+a^2 x^2}}{x}-\frac {4 i a^2 \sqrt {1+a^2 x^2}}{i-a x}-4 \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )-\frac {1}{4} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {3 i a \sqrt {1+a^2 x^2}}{x}-\frac {4 i a^2 \sqrt {1+a^2 x^2}}{i-a x}+4 a^2 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right ) \\ & = -\frac {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {3 i a \sqrt {1+a^2 x^2}}{x}-\frac {4 i a^2 \sqrt {1+a^2 x^2}}{i-a x}+\frac {9}{2} a^2 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=\sqrt {1+a^2 x^2} \left (-\frac {1}{2 x^2}+\frac {3 i a}{x}+\frac {4 i a^2}{-i+a x}\right )-\frac {9}{2} a^2 \log (x)+\frac {9}{2} a^2 \log \left (1+\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {i \left (6 a^{3} x^{3}+i a^{2} x^{2}+6 a x +i\right )}{2 x^{2} \sqrt {a^{2} x^{2}+1}}-\frac {a^{2} \left (-9 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {8 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a \left (x -\frac {i}{a}\right )}\right )}{2}\) | \(108\) |
default | \(-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {9 a^{2} \left (\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2}-3 i a \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+4 a^{2} \left (\frac {x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}+\frac {3 \sqrt {a^{2} x^{2}+1}\, x}{8}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 \sqrt {a^{2}}}\right )\right )+6 a^{2} \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\) | \(551\) |
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Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.40 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=\frac {14 i \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 9 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 9 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + \sqrt {a^{2} x^{2} + 1} {\left (14 i \, a^{2} x^{2} + 5 \, a x + i\right )}}{2 \, {\left (a x^{3} - i \, x^{2}\right )}} \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{6} - 3 i a^{2} x^{5} - 3 a x^{4} + i x^{3}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{6} - 3 i a^{2} x^{5} - 3 a x^{4} + i x^{3}}\, dx\right ) \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{3}} \,d x } \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{3}} \,d x } \]
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Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^3} \, dx=-\frac {a^2\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{2}-\frac {\sqrt {a^2\,x^2+1}}{2\,x^2}+\frac {a\,\sqrt {a^2\,x^2+1}\,3{}\mathrm {i}}{x}-\frac {a^3\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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