Integrand size = 14, antiderivative size = 62 \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=-\frac {1}{3 x^3}-\frac {2 i a}{x^2}+\frac {8 a^2}{x}+\frac {4 a^3}{i+a x}-12 i a^3 \log (x)+12 i a^3 \log (i+a x) \]
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Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5170, 90} \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=\frac {4 a^3}{a x+i}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)+\frac {8 a^2}{x}-\frac {2 i a}{x^2}-\frac {1}{3 x^3} \]
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Rule 90
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a x)^2}{x^4 (1-i a x)^2} \, dx \\ & = \int \left (\frac {1}{x^4}+\frac {4 i a}{x^3}-\frac {8 a^2}{x^2}-\frac {12 i a^3}{x}-\frac {4 a^4}{(i+a x)^2}+\frac {12 i a^4}{i+a x}\right ) \, dx \\ & = -\frac {1}{3 x^3}-\frac {2 i a}{x^2}+\frac {8 a^2}{x}+\frac {4 a^3}{i+a x}-12 i a^3 \log (x)+12 i a^3 \log (i+a x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=-\frac {1}{3 x^3}-\frac {2 i a}{x^2}+\frac {8 a^2}{x}+\frac {4 a^3}{i+a x}-12 i a^3 \log (x)+12 i a^3 \log (i+a x) \]
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Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {1}{3 x^{3}}-12 i a^{3} \ln \left (x \right )-\frac {2 i a}{x^{2}}+\frac {8 a^{2}}{x}+4 a^{4} \left (\frac {1}{a \left (a x +i\right )}+\frac {3 i \ln \left (a x +i\right )}{a}\right )\) | \(61\) |
risch | \(\frac {12 a^{3} x^{3}+6 i a^{2} x^{2}+\frac {5}{3} a x -\frac {1}{3} i}{\left (a x +i\right ) x^{3}}+12 a^{3} \arctan \left (a x \right )+6 i a^{3} \ln \left (a^{2} x^{2}+1\right )-12 i a^{3} \ln \left (-x \right )\) | \(73\) |
parallelrisch | \(-\frac {36 i \ln \left (x \right ) x^{5} a^{5}-36 i \ln \left (a x +i\right ) x^{5} a^{5}-18 i x^{5} a^{5}+36 i a^{3} \ln \left (x \right ) x^{3}-36 i a^{3} \ln \left (a x +i\right ) x^{3}+1-36 a^{4} x^{4}-23 a^{2} x^{2}+6 i a x}{3 \left (a^{2} x^{2}+1\right ) x^{3}}\) | \(103\) |
meijerg | \(\frac {a^{4} \left (-\frac {2 \left (-15 a^{4} x^{4}-10 a^{2} x^{2}+2\right )}{3 x^{3} \left (a^{2}\right )^{\frac {3}{2}} \left (2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \arctan \left (a x \right )}{\left (a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {a^{2}}}+2 i a^{3} \left (\frac {3 a^{2} x^{2}}{3 a^{2} x^{2}+3}+2 \ln \left (a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (a^{2}\right )-\frac {1}{a^{2} x^{2}}\right )-\frac {3 a^{4} \left (-\frac {2 \left (3 a^{2} x^{2}+2\right )}{x \sqrt {a^{2}}\, \left (2 a^{2} x^{2}+2\right )}-\frac {3 a \arctan \left (a x \right )}{\sqrt {a^{2}}}\right )}{\sqrt {a^{2}}}-2 i a^{3} \left (-\frac {2 a^{2} x^{2}}{2 a^{2} x^{2}+2}-\ln \left (a^{2} x^{2}+1\right )+1+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right )+\frac {a^{4} \left (\frac {2 x \sqrt {a^{2}}}{2 a^{2} x^{2}+2}+\frac {\sqrt {a^{2}}\, \arctan \left (a x \right )}{a}\right )}{2 \sqrt {a^{2}}}\) | \(272\) |
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Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.39 \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=\frac {36 \, a^{3} x^{3} + 18 i \, a^{2} x^{2} + 5 \, a x - 36 \, {\left (i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 36 \, {\left (-i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac {a x + i}{a}\right ) - i}{3 \, {\left (a x^{4} + i \, x^{3}\right )}} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13 \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=12 a^{3} \left (- i \log {\left (24 a^{4} x \right )} + i \log {\left (24 a^{4} x + 24 i a^{3} \right )}\right ) + \frac {36 a^{3} x^{3} + 18 i a^{2} x^{2} + 5 a x - i}{3 a x^{4} + 3 i x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.24 \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=12 \, a^{3} \arctan \left (a x\right ) + 6 i \, a^{3} \log \left (a^{2} x^{2} + 1\right ) - 12 i \, a^{3} \log \left (x\right ) + \frac {36 \, a^{4} x^{4} - 18 i \, a^{3} x^{3} + 23 \, a^{2} x^{2} - 6 i \, a x - 1}{3 \, {\left (a^{2} x^{5} + x^{3}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=12 i \, a^{3} \log \left (a x + i\right ) - 12 i \, a^{3} \log \left ({\left | x \right |}\right ) + \frac {36 \, a^{3} x^{3} + 18 i \, a^{2} x^{2} + 5 \, a x - i}{3 \, {\left (a x + i\right )} x^{3}} \]
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Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {e^{4 i \arctan (a x)}}{x^4} \, dx=24\,a^3\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )+\frac {\frac {5\,x}{3}+12\,a^2\,x^3+a\,x^2\,6{}\mathrm {i}-\frac {1{}\mathrm {i}}{3\,a}}{x^4+\frac {x^3\,1{}\mathrm {i}}{a}} \]
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