3.159 \(\int \frac{e^{i n \tan ^{-1}(a x)}}{x^2} \, dx\)

Optimal. Leaf size=79 \[ -\frac{4 i a (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right )}{2-n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0307808, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5062, 131} \[ -\frac{4 i a (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right )}{2-n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5062

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]

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{e^{i n \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac{(1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^2} \, dx\\ &=-\frac{4 i a (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{1}{2} (-2+n)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{1+i a x}\right )}{2-n}\\ \end{align*}

Mathematica [A]  time = 0.0154725, size = 82, normalized size = 1.04 \[ -\frac{2 i a (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{n}{2}-1} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};-\frac{1-i a x}{-i a x-1}\right )}{1-\frac{n}{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{in\arctan \left ( ax \right ) }}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(I*n*arctan(a*x))/x^2, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x^{2} \left (-\frac{a x + i}{a x - i}\right )^{\frac{1}{2} \, n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{i n \operatorname{atan}{\left (a x \right )}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(I*n*atan(a*x))/x**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(I*n*arctan(a*x))/x^2, x)