Integrand size = 14, antiderivative size = 118 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}+\frac {3 i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {14 a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {4 a^3 \sqrt {1+a^2 x^2}}{i-a x}-\frac {11}{2} i a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.45 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5168, 6874, 277, 270, 272, 44, 65, 214, 665} \[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=\frac {14 a^2 \sqrt {a^2 x^2+1}}{3 x}+\frac {3 i a \sqrt {a^2 x^2+1}}{2 x^2}-\frac {\sqrt {a^2 x^2+1}}{3 x^3}-\frac {11}{2} i a^3 \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-\frac {4 a^3 \sqrt {a^2 x^2+1}}{-a x+i} \]
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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^4 (1+i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = \int \left (\frac {1}{x^4 \sqrt {1+a^2 x^2}}-\frac {3 i a}{x^3 \sqrt {1+a^2 x^2}}-\frac {4 a^2}{x^2 \sqrt {1+a^2 x^2}}+\frac {4 i a^3}{x \sqrt {1+a^2 x^2}}-\frac {4 i a^4}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx \\ & = -\left ((3 i a) \int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx\right )-\left (4 a^2\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (4 i a^3\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx-\left (4 i a^4\right ) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^4 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{3 x^3}+\frac {4 a^2 \sqrt {1+a^2 x^2}}{x}-\frac {4 a^3 \sqrt {1+a^2 x^2}}{i-a x}-\frac {1}{2} (3 i a) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^2\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (2 i a^3\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}}{3 x^3}+\frac {3 i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {14 a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {4 a^3 \sqrt {1+a^2 x^2}}{i-a x}+(4 i a) \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} \left (3 i a^3\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}}{3 x^3}+\frac {3 i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {14 a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {4 a^3 \sqrt {1+a^2 x^2}}{i-a x}-4 i a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+\frac {1}{2} (3 i a) \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}}{3 x^3}+\frac {3 i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {14 a^2 \sqrt {1+a^2 x^2}}{3 x}-\frac {4 a^3 \sqrt {1+a^2 x^2}}{i-a x}-\frac {11}{2} i a^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=\frac {1}{6} \left (\frac {\sqrt {1+a^2 x^2} \left (2 i+7 a x-19 i a^2 x^2+52 a^3 x^3\right )}{x^3 (-i+a x)}+33 i a^3 \log (x)-33 i a^3 \log \left (1+\sqrt {1+a^2 x^2}\right )\right ) \]
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Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {28 a^{4} x^{4}+9 i a^{3} x^{3}+26 a^{2} x^{2}+9 i a x -2}{6 x^{3} \sqrt {a^{2} x^{2}+1}}+\frac {i a^{3} \left (-11 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {8 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a \left (x -\frac {i}{a}\right )}\right )}{2}\) | \(116\) |
default | \(-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}-\frac {16 a^{2} \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 )}{3}-3 i a \left (-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}+\frac {3 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}\right )+10 i a^{3} \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 )-4 a^{2} \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 )+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 )^{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 )-10 i a^{3} \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 )\) | \(829\) |
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Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.18 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=\frac {52 \, a^{4} x^{4} - 52 i \, a^{3} x^{3} - 33 \, {\left (i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 33 \, {\left (-i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + {\left (52 \, a^{3} x^{3} - 19 i \, a^{2} x^{2} + 7 \, a x + 2 i\right )} \sqrt {a^{2} x^{2} + 1}}{6 \, {\left (a x^{4} - i \, x^{3}\right )}} \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{7} - 3 i a^{2} x^{6} - 3 a x^{5} + i x^{4}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{7} - 3 i a^{2} x^{6} - 3 a x^{5} + i x^{4}}\, dx\right ) \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{4}} \,d x } \]
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\[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{4}} \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.99 \[ \int \frac {e^{-3 i \arctan (a x)}}{x^4} \, dx=\frac {14\,a^2\,\sqrt {a^2\,x^2+1}}{3\,x}-\frac {\sqrt {a^2\,x^2+1}}{3\,x^3}+\frac {a\,\sqrt {a^2\,x^2+1}\,3{}\mathrm {i}}{2\,x^2}-\frac {11\,a^3\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )}{2}-\frac {4\,a^4\,\sqrt {a^2\,x^2+1}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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