\(\int e^{i n \arctan (a x)} x \, dx\) [156]

Optimal result

Integrand size = 13, antiderivative size = 107 \[ \int e^{i n \arctan (a x)} x \, dx=\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 a^2}+\frac {2^{n/2} n (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{a^2 (2-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5170, 81, 71} \[ \int e^{i n \arctan (a x)} x \, dx=\frac {2^{n/2} n (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{a^2 (2-n)}+\frac {(1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{2 a^2} \]

[In]

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

[Out]

((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(2*a^2) + (2^(n/2)*n*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1
- n/2, -1/2*n, 2 - n/2, (1 - I*a*x)/2])/(a^2*(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 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 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx \\ & = \frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 a^2}-\frac {(i n) \int (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx}{2 a} \\ & = \frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 a^2}+\frac {2^{n/2} n (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 )}{a^2 (2-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.98 \[ \int e^{i n \arctan (a x)} x \, dx=\frac {(1-i a x)^{-n/2} (i+a x) \left ((-2+n) (1+i a x)^{n/2} (-i+a x)+i 2^{1+\frac {n}{2}} n \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )\right )}{2 a^2 (-2+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x*exp(I*n*atan(a*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x*e^(I*n*arctan(a*x)), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

int(x*exp(n*atan(a*x)*1i), x)