Optimal. Leaf size=171 \[ -\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{3 x^3}-\frac {i a n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{6 x^2}+\frac {2 i a^3 \left (2+n^2\right ) (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-i a x}{1+i a x}\right )}{3 (2-n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 171, normalized size of antiderivative = 1.00, 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 = {5170, 105, 156,
12, 133} \begin {gather*} \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 {(1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{3 x^3}-\frac {i a n (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 105
Rule 133
Rule 156
Rule 5170
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*}
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Mathematica [A]
time = 0.04, size = 119, normalized size = 0.70 \begin {gather*} -\frac {(1-i a x)^{-n/2} (1+i a x)^{\frac {1}{2} (-2+n)} (i+a x) \left (-\left ((-2+n) (-i+a x)^2 (-2 i+a n x)\right )+4 a^3 \left (2+n^2\right ) x^3 \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {i+a x}{i-a x}\right )\right )}{6 (-2+n) x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{i n \arctan \left (a x \right )}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{i n \operatorname {atan}{\left (a x \right )}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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