\(\int \frac {e^{i n \arctan (a x)}}{x^3} \, dx\) [174]

Optimal result

Integrand size = 15, antiderivative size = 120 \[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 x^2}+\frac {2 a^2 n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-i a x}{1+i a x}\right )}{2-n} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\frac {(1-i a x)^{-n/2} (1+i a x)^{n/2} (i+a x) \left (-\left ((-2+n) (-i+a x)^2\right )+4 a^2 n x^2 \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {i+a x}{i-a x}\right )\right )}{2 (-2+n) x^2 (-i+a x)} \] Input:

Integrate[E^(I*n*ArcTan[a*x])/x^3,x]
 

Output:

((1 + I*a*x)^(n/2)*(I + a*x)*(-((-2 + n)*(-I + a*x)^2) + 4*a^2*n*x^2*Hyper 
geometric2F1[2, 1 - n/2, 2 - n/2, (I + a*x)/(I - a*x)]))/(2*(-2 + n)*x^2*( 
1 - I*a*x)^(n/2)*(-I + a*x))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5585, 107, 141}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx\)

\(\Big \downarrow \) 5585

\(\displaystyle \int \frac {(1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^3}dx\)

\(\Big \downarrow \) 107

\(\displaystyle \frac {1}{2} i a n \int \frac {(1-i a x)^{-n/2} (i a x+1)^{n/2}}{x^2}dx-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{2 x^2}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {2 a^2 n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n-2}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-i a x}{i a x+1}\right )}{2-n}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{2 x^2}\)

Input:

Int[E^(I*n*ArcTan[a*x])/x^3,x]
 

Output:

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

Defintions of rubi rules used

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 141

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

rule 5585

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] &&  !Intege 
rQ[(I*n - 1)/2]
 

Maple [F]

\[\int \frac {{\mathrm e}^{i n \arctan \left (a x \right )}}{x^{3}}d x\]

Input:

int(exp(I*n*arctan(a*x))/x^3,x)
 

Output:

int(exp(I*n*arctan(a*x))/x^3,x)
 

Fricas [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{3}} \,d x } \] Input:

integrate(exp(I*n*arctan(a*x))/x^3,x, algorithm="fricas")
 

Output:

integral(1/(x^3*(-(a*x + I)/(a*x - I))^(1/2*n)), x)
 

Sympy [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int \frac {e^{i n \operatorname {atan}{\left (a x \right )}}}{x^{3}}\, dx \] Input:

integrate(exp(I*n*atan(a*x))/x**3,x)
 

Output:

Integral(exp(I*n*atan(a*x))/x**3, x)
 

Maxima [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{3}} \,d x } \] Input:

integrate(exp(I*n*arctan(a*x))/x^3,x, algorithm="maxima")
 

Output:

integrate(e^(I*n*arctan(a*x))/x^3, x)
 

Giac [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{3}} \,d x } \] Input:

integrate(exp(I*n*arctan(a*x))/x^3,x, algorithm="giac")
 

Output:

integrate(e^(I*n*arctan(a*x))/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{x^3} \,d x \] Input:

int(exp(n*atan(a*x)*1i)/x^3,x)
 

Output:

int(exp(n*atan(a*x)*1i)/x^3, x)
 

Reduce [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\frac {-e^{\mathit {atan} \left (a x \right ) i n} a^{2} x^{2}-e^{\mathit {atan} \left (a x \right ) i n} a i n x -e^{\mathit {atan} \left (a x \right ) i n}-\left (\int \frac {e^{\mathit {atan} \left (a x \right ) i n}}{a^{2} x^{3}+x}d x \right ) a^{2} n^{2} x^{2}}{2 x^{2}} \] Input:

int(exp(I*n*atan(a*x))/x^3,x)
 

Output:

( - (e**(atan(a*x)*i*n)*a**2*x**2 + e**(atan(a*x)*i*n)*a*i*n*x + e**(atan( 
a*x)*i*n) + int(e**(atan(a*x)*i*n)/(a**2*x**3 + x),x)*a**2*n**2*x**2))/(2* 
x**2)