Optimal. Leaf size=171 \[ -\frac {2^{\frac {n}{2}-2} n \left (n^2+8\right ) (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)}-\frac {(1+i a x)^{\frac {n+2}{2}} \left (2 i a n x+n^2+6\right ) (1-i a x)^{1-\frac {n}{2}}}{24 a^4}+\frac {x^2 (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{4 a^2} \]
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Rubi [A] time = 0.11, 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 = {5062, 100, 147, 69} \[ -\frac {2^{\frac {n}{2}-2} n \left (n^2+8\right ) (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)}-\frac {(1+i a x)^{\frac {n+2}{2}} \left (2 i a n x+n^2+6\right ) (1-i a x)^{1-\frac {n}{2}}}{24 a^4}+\frac {x^2 (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 147
Rule 5062
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 = 0.20, size = 210, normalized size = 1.23 \[ \frac {(a x+i) (1-i a x)^{-n/2} \left ((n-2) \left (a^2 x^2 (a x-i) (1+i a x)^{n/2}-i 2^{\frac {n}{2}+1} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )\right )-i 2^{\frac {n}{2}+3} n \, _2F_1\left (-\frac {n}{2}-2,1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )+i 2^{\frac {n}{2}+3} (n-1) \, _2F_1\left (-\frac {n}{2}-1,1-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )\right )}{4 a^4 (n-2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{\left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, 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 \[ \int x^{3} e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \]
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 \[ \int x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \]
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 \[ \int x^{3} e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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