\(\int e^{i n \arctan (a x)} \, dx\) [157]

Optimal result

Integrand size = 11, antiderivative size = 71 \[ \int e^{i n \arctan (a x)} \, dx=\frac {i 2^{1+\frac {n}{2}} (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{a (2-n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5169, 71} \[ \int e^{i n \arctan (a x)} \, dx=\frac {i 2^{\frac {n}{2}+1} (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{a (2-n)} \]

[In]

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

[Out]

(I*2^(1 + 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*(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 5169

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

Rubi steps \begin{align*} \text {integral}& = \int (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx \\ & = \frac {i 2^{1+\frac {n}{2}} (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-n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.75 \[ \int e^{i n \arctan (a x)} \, dx=-\frac {4 i e^{i (2+n) \arctan (a x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {n}{2},2+\frac {n}{2},-e^{2 i \arctan (a x)}\right )}{a (2+n)} \]

[In]

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

[Out]

((-4*I)*E^(I*(2 + n)*ArcTan[a*x])*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^((2*I)*ArcTan[a*x])])/(a*(2 + n))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(exp(I*n*atan(a*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate(e^(I*n*arctan(a*x)), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

int(exp(n*atan(a*x)*1i), x)