\(\int \frac {e^{i n \arctan (a x)}}{x^4} \, dx\) [175]

Optimal result

Integrand size = 15, antiderivative size = 171 \[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{3 x^3}-\frac {i a n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{6 x^2}+\frac {2 i a^3 \left (2+n^2\right ) (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-i a x}{1+i a x}\right )}{3 (2-n)} \] Output:

-1/3*(1-I*a*x)^(1-1/2*n)*(1+I*a*x)^(1+1/2*n)/x^3-1/6*I*a*n*(1-I*a*x)^(1-1/ 
2*n)*(1+I*a*x)^(1+1/2*n)/x^2+2/3*I*a^3*(n^2+2)*(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 = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.70 \[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=-\frac {(1-i a x)^{-n/2} (1+i a x)^{\frac {1}{2} (-2+n)} (i+a x) \left (-\left ((-2+n) (-i+a x)^2 (-2 i+a n x)\right )+4 a^3 \left (2+n^2\right ) x^3 \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {i+a x}{i-a x}\right )\right )}{6 (-2+n) x^3} \] Input:

Integrate[E^(I*n*ArcTan[a*x])/x^4,x]
 

Output:

-1/6*((1 + I*a*x)^((-2 + n)/2)*(I + a*x)*(-((-2 + n)*(-I + a*x)^2*(-2*I + 
a*n*x)) + 4*a^3*(2 + n^2)*x^3*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (I + 
a*x)/(I - a*x)]))/((-2 + n)*x^3*(1 - I*a*x)^(n/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5585, 114, 25, 27, 168, 27, 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.

\(\displaystyle \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx\)

\(\Big \downarrow \) 5585

\(\displaystyle \int \frac {(1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^4}dx\)

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

\(\displaystyle \frac {1}{3} a \left (\frac {2 i a^2 \left (n^2+2\right ) (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n-2}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-i a x}{i a x+1}\right )}{2-n}-\frac {i n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{2 x^2}\right )-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{3 x^3}\)

Input:

Int[E^(I*n*ArcTan[a*x])/x^4,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 114

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[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*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

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]

\[\int \frac {{\mathrm e}^{i n \arctan \left (a x \right )}}{x^{4}}d x\]

Input:

int(exp(I*n*arctan(a*x))/x^4,x)
 

Output:

int(exp(I*n*arctan(a*x))/x^4,x)
 

Fricas [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{4}} \,d x } \] Input:

integrate(exp(I*n*arctan(a*x))/x^4,x, algorithm="fricas")
 

Output:

integral(1/(x^4*(-(a*x + I)/(a*x - I))^(1/2*n)), x)
 

Sympy [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=\int \frac {e^{i n \operatorname {atan}{\left (a x \right )}}}{x^{4}}\, dx \] Input:

integrate(exp(I*n*atan(a*x))/x**4,x)
 

Output:

Integral(exp(I*n*atan(a*x))/x**4, x)
 

Maxima [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{4}} \,d x } \] Input:

integrate(exp(I*n*arctan(a*x))/x^4,x, algorithm="maxima")
 

Output:

integrate(e^(I*n*arctan(a*x))/x^4, x)
 

Giac [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{4}} \,d x } \] Input:

integrate(exp(I*n*arctan(a*x))/x^4,x, algorithm="giac")
 

Output:

integrate(e^(I*n*arctan(a*x))/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{x^4} \,d x \] Input:

int(exp(n*atan(a*x)*1i)/x^4,x)
 

Output:

int(exp(n*atan(a*x)*1i)/x^4, x)
 

Reduce [F]

\[ \int \frac {e^{i n \arctan (a x)}}{x^4} \, dx=\frac {-e^{\mathit {atan} \left (a x \right ) i n} a^{3} i n \,x^{3}+e^{\mathit {atan} \left (a x \right ) i n} a^{2} n^{2} x^{2}-e^{\mathit {atan} \left (a x \right ) i n} a i n x -2 e^{\mathit {atan} \left (a x \right ) i n}-\left (\int \frac {e^{\mathit {atan} \left (a x \right ) i n}}{a^{2} x^{3}+x}d x \right ) a^{3} i \,n^{3} x^{3}-2 \left (\int \frac {e^{\mathit {atan} \left (a x \right ) i n}}{a^{2} x^{3}+x}d x \right ) a^{3} i n \,x^{3}}{6 x^{3}} \] Input:

int(exp(I*n*atan(a*x))/x^4,x)
                                                                                    
                                                                                    
 

Output:

( - e**(atan(a*x)*i*n)*a**3*i*n*x**3 + e**(atan(a*x)*i*n)*a**2*n**2*x**2 - 
 e**(atan(a*x)*i*n)*a*i*n*x - 2*e**(atan(a*x)*i*n) - int(e**(atan(a*x)*i*n 
)/(a**2*x**3 + x),x)*a**3*i*n**3*x**3 - 2*int(e**(atan(a*x)*i*n)/(a**2*x** 
3 + x),x)*a**3*i*n*x**3)/(6*x**3)