Optimal. Leaf size=159 \[ -\frac {i 2^{n/2} \left (n^2+2\right ) (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{3 a^3 (2-n)}-\frac {i n (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{6 a^3}+\frac {x (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{3 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 159, 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, 90, 80, 69} \[ -\frac {i 2^{n/2} \left (n^2+2\right ) (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{3 a^3 (2-n)}-\frac {i n (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{6 a^3}+\frac {x (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{3 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 80
Rule 90
Rule 5062
Rubi steps
\begin {align*} \int e^{i n \tan ^{-1}(a x)} x^2 \, dx &=\int x^2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx\\ &=\frac {x (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{3 a^2}+\frac {\int (1-i a x)^{-n/2} (1+i a x)^{n/2} (-1-i a n x) \, dx}{3 a^2}\\ &=-\frac {i n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{6 a^3}+\frac {x (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{3 a^2}-\frac {\left (2+n^2\right ) \int (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx}{6 a^2}\\ &=-\frac {i n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{6 a^3}+\frac {x (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{3 a^2}-\frac {i 2^{n/2} \left (2+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^3 (2-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 116, normalized size = 0.73 \[ \frac {(a x+i) (1-i a x)^{-n/2} \left (2^{\frac {n}{2}+1} \left (n^2+2\right ) \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )+(n-2) (a x-i) (2 a x-i n) (1+i a x)^{n/2}\right )}{6 a^3 (n-2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{i n \arctan \left (a x \right )} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________