\(\int \frac {e^{i n \arctan (a x)}}{x} \, dx\) [172]

Optimal result

Integrand size = 15, antiderivative size = 125 \[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\frac {2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {1-i a x}{1+i a x}\right )}{n}-\frac {2^{1+\frac {n}{2}} (1-i a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{n} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85 \[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\frac {2 (1-i a x)^{-n/2} \left ((1+i a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {i+a x}{i-a x}\right )-2^{n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-i a x)\right )\right )}{n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 125, 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 = {5585, 140, 79, 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.

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

\(\Big \downarrow \) 5585

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 79

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] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 140

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(atan(a*x)*i*n)/x,x)