\(\int e^{i n \arctan (a x)} x^3 \, dx\) [168]

Optimal result

Integrand size = 15, antiderivative size = 207 \[ \int e^{i n \arctan (a x)} x^3 \, dx=-\frac {\left (6+2 n+n^2\right ) (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{24 a^4}+\frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}+\frac {n (1-i a x)^{2-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{12 a^4}-\frac {2^{-2+\frac {n}{2}} n \left (8+n^2\right ) (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)} \] Output:

-1/24*(n^2+2*n+6)*(1-I*a*x)^(1-1/2*n)*(1+I*a*x)^(1+1/2*n)/a^4+1/4*x^2*(1-I 
*a*x)^(1-1/2*n)*(1+I*a*x)^(1+1/2*n)/a^2+1/12*n*(1-I*a*x)^(2-1/2*n)*(1+I*a* 
x)^(1+1/2*n)/a^4-1/3*2^(-2+1/2*n)*n*(n^2+8)*(1-I*a*x)^(1-1/2*n)*hypergeom( 
[-1/2*n, 1-1/2*n],[2-1/2*n],1/2-1/2*I*a*x)/a^4/(2-n)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.01 \[ \int e^{i n \arctan (a x)} x^3 \, dx=\frac {(1-i a x)^{-n/2} (i+a x) \left (-i 2^{3+\frac {n}{2}} n \operatorname {Hypergeometric2F1}\left (-2-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )+i 2^{3+\frac {n}{2}} (-1+n) \operatorname {Hypergeometric2F1}\left (-1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )+(-2+n) \left (a^2 x^2 (1+i a x)^{n/2} (-i+a x)-i 2^{1+\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )\right )\right )}{4 a^4 (-2+n)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5585, 111, 25, 164, 79}

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.

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

\(\Big \downarrow \) 5585

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 164

\(\displaystyle \frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{4 a^2}-\frac {\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}} \left (2 i a n x+n^2+6\right )}{6 a^2}-\frac {i n \left (n^2+8\right ) \int (1-i a x)^{-n/2} (i a x+1)^{n/2}dx}{6 a}}{4 a^2}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{4 a^2}-\frac {\frac {2^{n/2} n \left (n^2+8\right ) (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{3 a^2 (2-n)}+\frac {(1+i a x)^{\frac {n+2}{2}} \left (2 i a n x+n^2+6\right ) (1-i a x)^{1-\frac {n}{2}}}{6 a^2}}{4 a^2}\)

Input:

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

Output:

(x^2*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(4*a^2) - (((1 - I*a*x 
)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2)*(6 + n^2 + (2*I)*a*n*x))/(6*a^2) + (2^ 
(n/2)*n*(8 + n^2)*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 
 2 - n/2, (1 - I*a*x)/2])/(3*a^2*(2 - n)))/(4*a^2)
 

Defintions of rubi rules used

rule 25

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

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 111

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 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(e**(atan(a*x)*i*n)*x**3,x)