\(\int e^{-4 i \arctan (a x)} x^m \, dx\) [153]

Optimal result

Integrand size = 14, antiderivative size = 50 \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1+i a x}-4 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x) \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.16 \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\frac {x^{1+m} (-5 i-4 i m+a x-4 (1+m) (-i+a x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x))}{(1+m) (-i+a x)} \] Input:

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

Output:

(x^(1 + m)*(-5*I - (4*I)*m + a*x - 4*(1 + m)*(-I + a*x)*Hypergeometric2F1[ 
1, 1 + m, 2 + m, (-I)*a*x]))/((1 + m)*(-I + a*x))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5585, 100, 25, 27, 90, 74}

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.

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

\(\Big \downarrow \) 5585

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {\int -\frac {a^2 x^m (4 m-i a x+3)}{i a x+1}dx}{a^2}+\frac {4 x^{m+1}}{1+i a x}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 x^{m+1}}{1+i a x}-\frac {\int \frac {a^2 x^m (4 m-i a x+3)}{i a x+1}dx}{a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 x^{m+1}}{1+i a x}-\int \frac {x^m (4 m-i a x+3)}{i a x+1}dx\)

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 74

\(\displaystyle -4 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-i a x)+\frac {4 x^{m+1}}{1+i a x}+\frac {x^{m+1}}{m+1}\)

Input:

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

Output:

x^(1 + m)/(1 + m) + (4*x^(1 + m))/(1 + I*a*x) - 4*x^(1 + m)*Hypergeometric 
2F1[1, 1 + m, 2 + m, (-I)*a*x]
 

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 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 90

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

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5.

Time = 0.47 (sec) , antiderivative size = 428, normalized size of antiderivative = 8.56

method	result	size
meijerg	\(-\frac {i \left (i a \right )^{-m} \left (\frac {x^{m} \left (i a \right )^{m} \left (6 a^{4} x^{4} m +6 i a^{3} m \,x^{3}+a^{2} x^{2} m^{4}+24 i a^{3} x^{3}+11 a^{2} x^{2} m^{3}-2 i a x \,m^{4}+46 a^{2} m^{2} x^{2}-21 i a x \,m^{3}+90 a^{2} m \,x^{2}-79 i a x \,m^{2}+72 a^{2} x^{2}-126 i a m x -m^{4}-72 i a x -10 m^{3}-35 m^{2}-50 m -24\right )}{\left (1+m \right ) m \left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{3}+9 m^{2}+26 m +24\right ) \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{6 a}+\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}-4 a^{2} m \,x^{2}+2 i a x \,m^{2}-6 a^{2} x^{2}+7 i a m x +6 i a x +m^{2}+3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} m \left (m^{2}+3 m +2\right ) \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{3 a}-\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}+2 a^{2} m \,x^{2}+2 i a x \,m^{2}-5 i a m x +m^{2}-3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{2}-3 m +2\right ) m \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{6 a}\)	\(428\)

Input:

int(x^m/(1+I*a*x)^4*(a^2*x^2+1)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/6*I*(I*a)^(-m)/a*(x^m*(I*a)^m*(a^2*x^2*m^4+6*a^4*x^4*m+11*a^2*x^2*m^3-1 
26*I*a*m*x-79*I*a*x*m^2+46*a^2*m^2*x^2+6*I*a^3*m*x^3-m^4-21*I*a*x*m^3+90*a 
^2*m*x^2+24*I*a^3*x^3-10*m^3+72*a^2*x^2-72*I*a*x-35*m^2-2*I*a*x*m^4-50*m-2 
4)/(1+m)/m/(1+I*a*x)^3+x^m*(I*a)^m*(m^3+9*m^2+26*m+24)*LerchPhi(-I*a*x,1,m 
))+1/3*I*(I*a)^(-m)/a*(-x^m*(I*a)^m*(-a^2*m^2*x^2-4*a^2*m*x^2+2*I*a*x*m^2- 
6*a^2*x^2+7*I*a*m*x+m^2+6*I*a*x+3*m+2)/(1+I*a*x)^3+x^m*(I*a)^m*m*(m^2+3*m+ 
2)*LerchPhi(-I*a*x,1,m))-1/6*I*(I*a)^(-m)/a*(-x^m*(I*a)^m*(-a^2*m^2*x^2+2* 
a^2*m*x^2+2*I*a*x*m^2-5*I*a*m*x+m^2-3*m+2)/(1+I*a*x)^3+x^m*(I*a)^m*(m^2-3* 
m+2)*m*LerchPhi(-I*a*x,1,m))
 

Fricas [F]

\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \] Input:

integrate(x^m/(1+I*a*x)^4*(a^2*x^2+1)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int \frac {x^{m} \left (a^{2} x^{2} + 1\right )^{2}}{\left (a x - i\right )^{4}}\, dx \] Input:

integrate(x**m/(1+I*a*x)**4*(a**2*x**2+1)**2,x)
 

Output:

Integral(x**m*(a**2*x**2 + 1)**2/(a*x - I)**4, x)
 

Maxima [F]

\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \] Input:

integrate(x^m/(1+I*a*x)^4*(a^2*x^2+1)^2,x, algorithm="maxima")
 

Output:

integrate((a^2*x^2 + 1)^2*x^m/(I*a*x + 1)^4, x)
 

Giac [F]

\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \] Input:

integrate(x^m/(1+I*a*x)^4*(a^2*x^2+1)^2,x, algorithm="giac")
 

Output:

integrate((a^2*x^2 + 1)^2*x^m/(I*a*x + 1)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int \frac {x^m\,{\left (a^2\,x^2+1\right )}^2}{{\left (1+a\,x\,1{}\mathrm {i}\right )}^4} \,d x \] Input:

int((x^m*(a^2*x^2 + 1)^2)/(a*x*1i + 1)^4,x)
 

Output:

int((x^m*(a^2*x^2 + 1)^2)/(a*x*1i + 1)^4, x)
 

Reduce [F]

\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\frac {x^{m} a m x +2 x^{m} i m +2 x^{m} i +2 \left (\int \frac {x^{m}}{a^{4} i \,x^{5}+4 a^{3} x^{4}-6 a^{2} i \,x^{3}-4 a \,x^{2}+i x}d x \right ) m^{2}+2 \left (\int \frac {x^{m}}{a^{4} i \,x^{5}+4 a^{3} x^{4}-6 a^{2} i \,x^{3}-4 a \,x^{2}+i x}d x \right ) m +8 \left (\int \frac {x^{m}}{a^{4} i \,x^{4}+4 a^{3} x^{3}-6 a^{2} i \,x^{2}-4 a x +i}d x \right ) a i \,m^{2}+8 \left (\int \frac {x^{m}}{a^{4} i \,x^{4}+4 a^{3} x^{3}-6 a^{2} i \,x^{2}-4 a x +i}d x \right ) a i m +8 \left (\int -\frac {x^{m} x}{a^{4} i \,x^{4}+4 a^{3} x^{3}-6 a^{2} i \,x^{2}-4 a x +i}d x \right ) a^{2} m^{2}+8 \left (\int -\frac {x^{m} x}{a^{4} i \,x^{4}+4 a^{3} x^{3}-6 a^{2} i \,x^{2}-4 a x +i}d x \right ) a^{2} m -2 \left (\int \frac {x^{m} x^{3}}{a^{4} i \,x^{4}+4 a^{3} x^{3}-6 a^{2} i \,x^{2}-4 a x +i}d x \right ) a^{4} m^{2}-2 \left (\int \frac {x^{m} x^{3}}{a^{4} i \,x^{4}+4 a^{3} x^{3}-6 a^{2} i \,x^{2}-4 a x +i}d x \right ) a^{4} m}{a m \left (m +1\right )} \] Input:

int(x^m/(1+I*a*x)^4*(a^2*x^2+1)^2,x)
 

Output:

(x**m*a*m*x + 2*x**m*i*m + 2*x**m*i + 2*int(x**m/(a**4*i*x**5 + 4*a**3*x** 
4 - 6*a**2*i*x**3 - 4*a*x**2 + i*x),x)*m**2 + 2*int(x**m/(a**4*i*x**5 + 4* 
a**3*x**4 - 6*a**2*i*x**3 - 4*a*x**2 + i*x),x)*m + 8*int(x**m/(a**4*i*x**4 
 + 4*a**3*x**3 - 6*a**2*i*x**2 - 4*a*x + i),x)*a*i*m**2 + 8*int(x**m/(a**4 
*i*x**4 + 4*a**3*x**3 - 6*a**2*i*x**2 - 4*a*x + i),x)*a*i*m + 8*int(( - x* 
*m*x)/(a**4*i*x**4 + 4*a**3*x**3 - 6*a**2*i*x**2 - 4*a*x + i),x)*a**2*m**2 
 + 8*int(( - x**m*x)/(a**4*i*x**4 + 4*a**3*x**3 - 6*a**2*i*x**2 - 4*a*x + 
i),x)*a**2*m - 2*int((x**m*x**3)/(a**4*i*x**4 + 4*a**3*x**3 - 6*a**2*i*x** 
2 - 4*a*x + i),x)*a**4*m**2 - 2*int((x**m*x**3)/(a**4*i*x**4 + 4*a**3*x**3 
 - 6*a**2*i*x**2 - 4*a*x + i),x)*a**4*m)/(a*m*(m + 1))