Integrand size = 14, antiderivative size = 50 \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1+i a x}-4 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x) \]
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.16 \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\frac {x^{1+m} (-5 i-4 i m+a x-4 (1+m) (-i+a x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x))}{(1+m) (-i+a x)} \]
(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))
Time = 0.21 (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 |
\(\displaystyle -4 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-i a x)+\frac {4 x^{m+1}}{1+i a x}+\frac {x^{m+1}}{m+1}\) |
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]
3.2.38.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])))
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[((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])))
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]
Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.57 (sec) , antiderivative size = 428, normalized size of antiderivative = 8.56
method | result | size |
meijerg | \(-\frac {i \left (i a \right )^{-m} \left (\frac {x^{m} \left (i a \right )^{m} \left (6 a^{4} x^{4} m +6 i a^{3} x^{3} m +a^{2} x^{2} m^{4}+24 i a^{3} x^{3}+11 a^{2} x^{2} m^{3}-2 i a x \,m^{4}+46 a^{2} m^{2} x^{2}-21 i a x \,m^{3}+90 a^{2} m \,x^{2}-79 i a x \,m^{2}+72 a^{2} x^{2}-126 i a m x -m^{4}-72 i a x -10 m^{3}-35 m^{2}-50 m -24\right )}{\left (1+m \right ) m \left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{3}+9 m^{2}+26 m +24\right ) \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{6 a}+\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}-4 a^{2} m \,x^{2}+2 i a x \,m^{2}-6 a^{2} x^{2}+7 i a m x +6 i a x +m^{2}+3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} m \left (m^{2}+3 m +2\right ) \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{3 a}-\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}+2 a^{2} m \,x^{2}+2 i a x \,m^{2}-5 i a m x +m^{2}-3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{2}-3 m +2\right ) m \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{6 a}\) | \(428\) |
-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^3-2 *I*a*x*m^4+6*I*a^3*x^3*m+46*a^2*m^2*x^2-72*I*a*x-m^4-79*I*a*x*m^2+90*a^2*m *x^2+24*I*a^3*x^3-10*m^3+72*a^2*x^2-21*I*a*x*m^3-35*m^2-126*I*a*m*x-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))
\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \]
\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int \frac {x^{m} \left (a^{2} x^{2} + 1\right )^{2}}{\left (a x - i\right )^{4}}\, dx \]
\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \]
\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \]
Timed out. \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int \frac {x^m\,{\left (a^2\,x^2+1\right )}^2}{{\left (1+a\,x\,1{}\mathrm {i}\right )}^4} \,d x \]