Integrand size = 14, antiderivative size = 139 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {51}{8} a^4 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.48 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5168, 6874, 272, 44, 65, 214, 277, 270, 665} \[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=\frac {19 a^2 \sqrt {a^2 x^2+1}}{8 x^2}-\frac {\sqrt {a^2 x^2+1}}{4 x^4}+\frac {i a \sqrt {a^2 x^2+1}}{x^3}-\frac {51}{8} a^4 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {4 i a^4 \sqrt {a^2 x^2+1}}{-a x+i}-\frac {6 i a^3 \sqrt {a^2 x^2+1}}{x} \]
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Rule 44
Rule 65
Rule 214
Rule 270
Rule 272
Rule 277
Rule 665
Rule 5168
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^2}{x^5 (1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = \int \left (\frac {1}{x^5 \sqrt {1+a^2 x^2}}-\frac {3 i a}{x^4 \sqrt {1+a^2 x^2}}-\frac {4 a^2}{x^3 \sqrt {1+a^2 x^2}}+\frac {4 i a^3}{x^2 \sqrt {1+a^2 x^2}}+\frac {4 a^4}{x \sqrt {1+a^2 x^2}}-\frac {4 a^5}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx \\ & = -\left ((3 i a) \int \frac {1}{x^4 \sqrt {1+a^2 x^2}} \, dx\right )-\left (4 a^2\right ) \int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx+\left (4 i a^3\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (4 a^4\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx-\left (4 a^5\right ) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^5 \sqrt {1+a^2 x^2}} \, dx \\ & = \frac {i a \sqrt {1+a^2 x^2}}{x^3}-\frac {4 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+a^2 x}} \, dx,x,x^2\right )-\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )+\left (2 i a^3\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (2 a^4\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}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {1}{8} \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )+\left (4 a^2\right ) \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^4 \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-4 a^4 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+\left (2 a^2\right ) \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}{16} \left (3 a^4\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}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-6 a^4 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+\frac {1}{8} \left (3 a^2\right ) \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}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {51}{8} a^4 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=\frac {1}{8} \left (\frac {\sqrt {1+a^2 x^2} \left (2 i+6 a x-11 i a^2 x^2-29 a^3 x^3-80 i a^4 x^4\right )}{x^4 (-i+a x)}+51 a^4 \log (x)-51 a^4 \log \left (1+\sqrt {1+a^2 x^2}\right )\right ) \]
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Time = 0.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {i \left (48 a^{5} x^{5}+19 i a^{4} x^{4}+40 a^{3} x^{3}+17 i a^{2} x^{2}-8 a x -2 i\right )}{8 x^{4} \sqrt {a^{2} x^{2}+1}}+\frac {a^{4} \left (-51 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {32 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a \left (x -\frac {i}{a}\right )}\right )}{8}\) | \(125\) |
default | \(\text {Expression too large to display}\) | \(937\) |
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Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=\frac {-80 i \, a^{5} x^{5} - 80 \, a^{4} x^{4} - 51 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + 51 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + {\left (-80 i \, a^{4} x^{4} - 29 \, a^{3} x^{3} - 11 i \, a^{2} x^{2} + 6 \, a x + 2 i\right )} \sqrt {a^{2} x^{2} + 1}}{8 \, {\left (a x^{5} - i \, x^{4}\right )}} \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{8} - 3 i a^{2} x^{7} - 3 a x^{6} + i x^{5}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{8} - 3 i a^{2} x^{7} - 3 a x^{6} + i x^{5}}\, dx\right ) \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{5}} \,d x } \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{5}} \,d x } \]
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Time = 0.50 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^5} \, dx=\frac {a^4\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )\,51{}\mathrm {i}}{8}-\frac {\sqrt {a^2\,x^2+1}}{4\,x^4}+\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{x^3}+\frac {19\,a^2\,\sqrt {a^2\,x^2+1}}{8\,x^2}-\frac {a^3\,\sqrt {a^2\,x^2+1}\,6{}\mathrm {i}}{x}+\frac {a^5\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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