Optimal. Leaf size=171 \[ \frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}-\frac {2^{-2+\frac {n}{2}} n \left (8+n^2\right ) (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)} \]
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Rubi [A]
time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5170, 102, 152,
71} \begin {gather*} -\frac {2^{\frac {n}{2}-2} n \left (n^2+8\right ) (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)}-\frac {(1+i a x)^{\frac {n+2}{2}} \left (2 i a n x+n^2+6\right ) (1-i a x)^{1-\frac {n}{2}}}{24 a^4}+\frac {x^2 (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 152
Rule 5170
Rubi steps
\begin {align*} \int e^{i n \tan ^{-1}(a x)} x^3 \, dx &=\int x^3 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx\\ &=\frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}+\frac {\int x (1-i a x)^{-n/2} (1+i a x)^{n/2} (-2-i a n x) \, dx}{4 a^2}\\ &=\frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}+\frac {\left (i n \left (8+n^2\right )\right ) \int (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx}{24 a^3}\\ &=\frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}-\frac {2^{-2+\frac {n}{2}} n \left (8+n^2\right ) (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 158, normalized size = 0.92 \begin {gather*} \frac {e^{i n \text {ArcTan}(a x)} \left (-e^{2 i \text {ArcTan}(a x)} n^2 \left (8+n^2\right ) \, _2F_1\left (1,1+\frac {n}{2};2+\frac {n}{2};-e^{2 i \text {ArcTan}(a x)}\right )+(2+n) \left (i a n \left (8+n^2\right ) x-\left (12+n^2+2 i a n x\right ) \left (1+a^2 x^2\right )+6 \left (1+a^2 x^2\right )^2+n \left (8+n^2\right ) \, _2F_1\left (1,\frac {n}{2};1+\frac {n}{2};-e^{2 i \text {ArcTan}(a x)}\right )\right )\right )}{24 a^4 (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{i n \arctan \left (a x \right )} x^{3}\, dx\]
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 x^{3} e^{i n \operatorname {atan}{\left (a x \right )}}\, 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 x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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