Optimal. Leaf size=171 \[ \frac{2 i a^3 \left (n^2+2\right ) (1+i a x)^{\frac{n-2}{2}} (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right )}{3 (2-n)}-\frac{i a n (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{6 x^2}-\frac{(1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{3 x^3} \]
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Rubi [A] time = 0.0720683, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5062, 129, 151, 12, 131} \[ \frac{2 i a^3 \left (n^2+2\right ) (1+i a x)^{\frac{n-2}{2}} (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right )}{3 (2-n)}-\frac{i a n (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{6 x^2}-\frac{(1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 129
Rule 151
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{i n \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^4} \, dx\\ &=-\frac{(1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{1}{3} \int \frac{(1-i a x)^{-n/2} (1+i a x)^{n/2} \left (-i a n+a^2 x\right )}{x^3} \, dx\\ &=-\frac{(1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{i a n (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{6 x^2}-\frac{1}{6} \int \frac{a^2 \left (2+n^2\right ) (1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^2} \, dx\\ &=-\frac{(1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{i a n (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{6 x^2}-\frac{1}{6} \left (a^2 \left (2+n^2\right )\right ) \int \frac{(1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^2} \, dx\\ &=-\frac{(1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{3 x^3}-\frac{i a n (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{2+n}{2}}}{6 x^2}+\frac{2 i a^3 \left (2+n^2\right ) (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{1}{2} (-2+n)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{1+i a x}\right )}{3 (2-n)}\\ \end{align*}
Mathematica [A] time = 0.0545292, size = 119, normalized size = 0.7 \[ -\frac{(a x+i) (1-i a x)^{-n/2} (1+i a x)^{\frac{n-2}{2}} \left (4 a^3 \left (n^2+2\right ) x^3 \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a x+i}{i-a x}\right )-(n-2) (a x-i)^2 (a n x-2 i)\right )}{6 (n-2) x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{in\arctan \left ( ax \right ) }}}{{x}^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x^{4} \left (-\frac{a x + i}{a x - i}\right )^{\frac{1}{2} \, n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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