\(\int e^{i n \arctan (a x)} x^2 \, dx\) [155]

Optimal result

Integrand size = 15, antiderivative size = 159 \[ \int e^{i n \arctan (a x)} x^2 \, dx=-\frac {i n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{6 a^3}+\frac {x (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{3 a^2}-\frac {i 2^{n/2} \left (2+n^2\right ) (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{3 a^3 (2-n)} \]

[Out]

-1/6*I*n*(1-I*a*x)^(1-1/2*n)*(1+I*a*x)^(1+1/2*n)/a^3+1/3*x*(1-I*a*x)^(1-1/2*n)*(1+I*a*x)^(1+1/2*n)/a^2-1/3*I*2
^(1/2*n)*(n^2+2)*(1-I*a*x)^(1-1/2*n)*hypergeom([-1/2*n, 1-1/2*n],[2-1/2*n],1/2-1/2*I*a*x)/a^3/(2-n)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5170, 92, 81, 71} \[ \int e^{i n \arctan (a x)} x^2 \, dx=-\frac {i 2^{n/2} \left (n^2+2\right ) (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{3 a^3 (2-n)}-\frac {i n (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{6 a^3}+\frac {x (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{3 a^2} \]

[In]

Int[E^(I*n*ArcTan[a*x])*x^2,x]

[Out]

((-1/6*I)*n*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/a^3 + (x*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)
/2))/(3*a^2) - ((I/3)*2^(n/2)*(2 + n^2)*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 -
I*a*x)/2])/(a^3*(2 - n))

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 5170

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a*x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))
), x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[(I*n - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \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}} \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.73 \[ \int e^{i n \arctan (a x)} x^2 \, dx=\frac {(1-i a x)^{-n/2} (i+a x) \left ((-2+n) (1+i a x)^{n/2} (-i+a x) (-i n+2 a x)+2^{1+\frac {n}{2}} \left (2+n^2\right ) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )\right )}{6 a^3 (-2+n)} \]

[In]

Integrate[E^(I*n*ArcTan[a*x])*x^2,x]

[Out]

((I + a*x)*((-2 + n)*(1 + I*a*x)^(n/2)*(-I + a*x)*((-I)*n + 2*a*x) + 2^(1 + n/2)*(2 + n^2)*Hypergeometric2F1[1
- n/2, -1/2*n, 2 - n/2, (1 - I*a*x)/2]))/(6*a^3*(-2 + n)*(1 - I*a*x)^(n/2))

Maple [F]

\[\int {\mathrm e}^{i n \arctan \left (a x \right )} x^{2}d x\]

[In]

int(exp(I*n*arctan(a*x))*x^2,x)

[Out]

int(exp(I*n*arctan(a*x))*x^2,x)

Fricas [F]

\[ \int e^{i n \arctan (a x)} x^2 \, dx=\int { x^{2} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]

[In]

integrate(exp(I*n*arctan(a*x))*x^2,x, algorithm="fricas")

[Out]

integral(x^2/(-(a*x + I)/(a*x - I))^(1/2*n), x)

Sympy [F]

\[ \int e^{i n \arctan (a x)} x^2 \, dx=\int x^{2} e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

integrate(exp(I*n*atan(a*x))*x**2,x)

[Out]

Integral(x**2*exp(I*n*atan(a*x)), x)

Maxima [F]

\[ \int e^{i n \arctan (a x)} x^2 \, dx=\int { x^{2} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]

[In]

integrate(exp(I*n*arctan(a*x))*x^2,x, algorithm="maxima")

[Out]

integrate(x^2*e^(I*n*arctan(a*x)), x)

Giac [F]

\[ \int e^{i n \arctan (a x)} x^2 \, dx=\int { x^{2} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]

[In]

integrate(exp(I*n*arctan(a*x))*x^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int e^{i n \arctan (a x)} x^2 \, dx=\int x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \]

[In]

int(x^2*exp(n*atan(a*x)*1i),x)

[Out]

int(x^2*exp(n*atan(a*x)*1i), x)