Integrand size = 13, antiderivative size = 107 \[ \int e^{i n \arctan (a x)} x \, dx=\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 a^2}+\frac {2^{n/2} n (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{a^2 (2-n)} \] Output:
1/2*(1-I*a*x)^(1-1/2*n)*(1+I*a*x)^(1+1/2*n)/a^2+2^(1/2*n)*n*(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^2/(2-n)
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.98 \[ \int e^{i n \arctan (a x)} x \, dx=\frac {(1-i a x)^{-n/2} (i+a x) \left ((-2+n) (1+i a x)^{n/2} (-i+a x)+i 2^{1+\frac {n}{2}} n \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )\right )}{2 a^2 (-2+n)} \] Input:
Integrate[E^(I*n*ArcTan[a*x])*x,x]
Output:
((I + a*x)*((-2 + n)*(1 + I*a*x)^(n/2)*(-I + a*x) + I*2^(1 + n/2)*n*Hyperg eometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - I*a*x)/2]))/(2*a^2*(-2 + n)*(1 - I*a*x)^(n/2))
Time = 0.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5585, 90, 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.
\(\displaystyle \int x e^{i n \arctan (a x)} \, dx\) |
\(\Big \downarrow \) 5585 |
\(\displaystyle \int x (1-i a x)^{-n/2} (1+i a x)^{n/2}dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{2 a^2}-\frac {i n \int (1-i a x)^{-n/2} (i a x+1)^{n/2}dx}{2 a}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {2^{n/2} n (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{a^2 (2-n)}+\frac {(1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{2 a^2}\) |
Input:
Int[E^(I*n*ArcTan[a*x])*x,x]
Output:
((1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^((2 + n)/2))/(2*a^2) + (2^(n/2)*n*(1 - I*a*x)^(1 - n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - I*a*x)/2 ])/(a^2*(2 - n))
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]))
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]
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]
\[\int {\mathrm e}^{i n \arctan \left (a x \right )} x d x\]
Input:
int(exp(I*n*arctan(a*x))*x,x)
Output:
int(exp(I*n*arctan(a*x))*x,x)
\[ \int e^{i n \arctan (a x)} x \, dx=\int { x e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(I*n*arctan(a*x))*x,x, algorithm="fricas")
Output:
integral(x/(-(a*x + I)/(a*x - I))^(1/2*n), x)
\[ \int e^{i n \arctan (a x)} x \, dx=\int x e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \] Input:
integrate(exp(I*n*atan(a*x))*x,x)
Output:
Integral(x*exp(I*n*atan(a*x)), x)
\[ \int e^{i n \arctan (a x)} x \, dx=\int { x e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(I*n*arctan(a*x))*x,x, algorithm="maxima")
Output:
integrate(x*e^(I*n*arctan(a*x)), x)
\[ \int e^{i n \arctan (a x)} x \, dx=\int { x e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(I*n*arctan(a*x))*x,x, algorithm="giac")
Output:
integrate(x*e^(I*n*arctan(a*x)), x)
Timed out. \[ \int e^{i n \arctan (a x)} x \, dx=\int x\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \] Input:
int(x*exp(n*atan(a*x)*1i),x)
Output:
int(x*exp(n*atan(a*x)*1i), x)
\[ \int e^{i n \arctan (a x)} x \, dx=\int e^{\mathit {atan} \left (a x \right ) i n} x d x \] Input:
int(exp(I*n*atan(a*x))*x,x)
Output:
int(e**(atan(a*x)*i*n)*x,x)