Integrand size = 14, antiderivative size = 124 \[ \int e^{6 i \arctan (a x)} x^m \, dx=\frac {4 (2+m) x^{1+m}}{(1+m) (1-i a x)^2}-\frac {4 \left (3+3 m+m^2\right ) x^{1+m}}{(1+m) (1-i a x)}-\frac {x^{1+m} (1+i a x)^2}{(1+m) (1-i a x)^2}+\frac {2 \left (3+4 m+2 m^2\right ) x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,i a x)}{1+m} \] Output:
4*(2+m)*x^(1+m)/(1+m)/(1-I*a*x)^2-4*(m^2+3*m+3)*x^(1+m)/(1+m)/(1-I*a*x)-x^ (1+m)*(1+I*a*x)^2/(1+m)/(1-I*a*x)^2+2*(2*m^2+4*m+3)*x^(1+m)*hypergeom([1, 1+m],[2+m],I*a*x)/(1+m)
Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int e^{6 i \arctan (a x)} x^m \, dx=\frac {x^{1+m} \left (5-10 i a x-a^2 x^2+4 m (2-3 i a x)+m^2 (4-4 i a x)+2 \left (3+4 m+2 m^2\right ) (i+a x)^2 \operatorname {Hypergeometric2F1}(1,1+m,2+m,i a x)\right )}{(1+m) (i+a x)^2} \] Input:
Integrate[E^((6*I)*ArcTan[a*x])*x^m,x]
Output:
(x^(1 + m)*(5 - (10*I)*a*x - a^2*x^2 + 4*m*(2 - (3*I)*a*x) + m^2*(4 - (4*I )*a*x) + 2*(3 + 4*m + 2*m^2)*(I + a*x)^2*Hypergeometric2F1[1, 1 + m, 2 + m , I*a*x]))/((1 + m)*(I + a*x)^2)
Time = 0.46 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5585, 111, 27, 162, 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.
| \(\displaystyle \int x^m e^{6 i \arctan (a x)} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {(1+i a x)^3 x^m}{(1-i a x)^3}dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {i \int -\frac {2 a x^m (i a x+1) (i (m+1)-a (m+3) x)}{(1-i a x)^3}dx}{a (m+1)}-\frac {(1+i a x)^2 x^{m+1}}{(m+1) (1-i a x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 i \int \frac {x^m (i a x+1) (i (m+1)-a (m+3) x)}{(1-i a x)^3}dx}{m+1}-\frac {(1+i a x)^2 x^{m+1}}{(m+1) (1-i a x)^2}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle -\frac {2 i \left (i (m+1) \left (2 m^2+4 m+3\right ) \int \frac {x^m}{1-i a x}dx-\frac {2 x^{m+1} \left (a \left (m^2+3 m+3\right ) x+i (m+1)^2\right )}{(1-i a x)^2}\right )}{m+1}-\frac {(1+i a x)^2 x^{m+1}}{(m+1) (1-i a x)^2}\) |
| \(\Big \downarrow \) 74 |
| \(\displaystyle -\frac {2 i \left (i \left (2 m^2+4 m+3\right ) x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,i a x)-\frac {2 x^{m+1} \left (a \left (m^2+3 m+3\right ) x+i (m+1)^2\right )}{(1-i a x)^2}\right )}{m+1}-\frac {(1+i a x)^2 x^{m+1}}{(m+1) (1-i a x)^2}\) |
Input:
Int[E^((6*I)*ArcTan[a*x])*x^m,x]
Output:
-((x^(1 + m)*(1 + I*a*x)^2)/((1 + m)*(1 - I*a*x)^2)) - ((2*I)*((-2*x^(1 + m)*(I*(1 + m)^2 + a*(3 + 3*m + m^2)*x))/(1 - I*a*x)^2 + I*(3 + 4*m + 2*m^2 )*x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, I*a*x]))/(1 + m)
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_))^(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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
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 = 0.68 (sec) , antiderivative size = 748, normalized size of antiderivative = 6.03
| method | result | size |
| meijerg | \(\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {m}{2}+\frac {1}{2}} \left (-a^{2} m^{2} x^{2}+2 a^{2} m \,x^{2}+3 a^{2} x^{2}-m^{2}+4 m +5\right )}{2 \left (1+m \right ) \left (a^{2} x^{2}+1\right )^{2}}+\frac {4 x^{1+m} \left (a^{2}\right )^{\frac {m}{2}+\frac {1}{2}} \left (\frac {1}{16} m^{3}-\frac {3}{16} m^{2}-\frac {1}{16} m +\frac {3}{16}\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{1+m}\right )}{4}+\frac {3 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (a^{2} m \,x^{2}+m -2\right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (-2+m \right ) m \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{4}\right )}{2 a}-\frac {15 \left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (a^{2} m \,x^{2}+a^{2} x^{2}+m -1\right )}{2 \left (a^{2} x^{2}+1\right )^{2} a^{2}}-\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (1+m \right ) \left (m -1\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{4 a^{2}}\right )}{4}-\frac {5 i \left (a^{2}\right )^{-\frac {m}{2}} \left (-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (a^{2} m \,x^{2}+4 a^{2} x^{2}+m +2\right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} m \left (2+m \right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{4}\right )}{a}+\frac {15 \left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (-\frac {x^{1+m} \left (a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \left (a^{2} m \,x^{2}+5 a^{2} x^{2}+m +3\right )}{2 a^{4} \left (a^{2} x^{2}+1\right )^{2}}+\frac {x^{1+m} \left (a^{2}\right )^{\frac {5}{2}+\frac {m}{2}} \left (m^{2}+4 m +3\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{4 a^{4}}\right )}{4}+\frac {3 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (8 a^{4} x^{4}+a^{2} m^{2} x^{2}+8 a^{2} m \,x^{2}+16 a^{2} x^{2}+m^{2}+6 m +8\right )}{2 \left (a^{2} x^{2}+1\right )^{2} m}-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (m^{2}+6 m +8\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{4}\right )}{2 a}-\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {7}{2}+\frac {m}{2}} \left (8 a^{4} x^{4}+a^{2} m^{2} x^{2}+10 a^{2} m \,x^{2}+25 a^{2} x^{2}+m^{2}+8 m +15\right )}{2 \left (a^{2} x^{2}+1\right )^{2} \left (1+m \right ) a^{6}}-\frac {x^{1+m} \left (a^{2}\right )^{\frac {7}{2}+\frac {m}{2}} \left (m^{2}+8 m +15\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}+\frac {1}{2}\right )}{4 a^{6}}\right )}{4}\) | \(748\) |
Input:
int((1+I*a*x)^6/(a^2*x^2+1)^3*x^m,x,method=_RETURNVERBOSE)
Output:
1/4*(a^2)^(-1/2-1/2*m)*(1/2/(1+m)*x^(1+m)*(a^2)^(1/2*m+1/2)*(-a^2*m^2*x^2+ 2*a^2*m*x^2+3*a^2*x^2-m^2+4*m+5)/(a^2*x^2+1)^2+4/(1+m)*x^(1+m)*(a^2)^(1/2* m+1/2)*(1/16*m^3-3/16*m^2-1/16*m+3/16)*LerchPhi(-a^2*x^2,1,1/2*m+1/2))+3/2 *I/a*(a^2)^(-1/2*m)*(1/2*x^m*(a^2)^(1/2*m)*(a^2*m*x^2+m-2)/(a^2*x^2+1)^2-1 /4*x^m*(a^2)^(1/2*m)*(-2+m)*m*LerchPhi(-a^2*x^2,1,1/2*m))-15/4*(a^2)^(-1/2 -1/2*m)*(1/2*x^(1+m)*(a^2)^(3/2+1/2*m)*(a^2*m*x^2+a^2*x^2+m-1)/(a^2*x^2+1) ^2/a^2-1/4*x^(1+m)*(a^2)^(3/2+1/2*m)*(1+m)*(m-1)/a^2*LerchPhi(-a^2*x^2,1,1 /2*m+1/2))-5*I/a*(a^2)^(-1/2*m)*(-1/2*x^m*(a^2)^(1/2*m)*(a^2*m*x^2+4*a^2*x ^2+m+2)/(a^2*x^2+1)^2+1/4*x^m*(a^2)^(1/2*m)*m*(2+m)*LerchPhi(-a^2*x^2,1,1/ 2*m))+15/4*(a^2)^(-1/2-1/2*m)*(-1/2*x^(1+m)*(a^2)^(5/2+1/2*m)*(a^2*m*x^2+5 *a^2*x^2+m+3)/a^4/(a^2*x^2+1)^2+1/4*x^(1+m)*(a^2)^(5/2+1/2*m)*(m^2+4*m+3)/ a^4*LerchPhi(-a^2*x^2,1,1/2*m+1/2))+3/2*I/a*(a^2)^(-1/2*m)*(1/2*x^m*(a^2)^ (1/2*m)*(8*a^4*x^4+a^2*m^2*x^2+8*a^2*m*x^2+16*a^2*x^2+m^2+6*m+8)/(a^2*x^2+ 1)^2/m-1/4*x^m*(a^2)^(1/2*m)*(m^2+6*m+8)*LerchPhi(-a^2*x^2,1,1/2*m))-1/4*( a^2)^(-1/2-1/2*m)*(1/2*x^(1+m)*(a^2)^(7/2+1/2*m)*(8*a^4*x^4+a^2*m^2*x^2+10 *a^2*m*x^2+25*a^2*x^2+m^2+8*m+15)/(a^2*x^2+1)^2/(1+m)/a^6-1/4*x^(1+m)*(a^2 )^(7/2+1/2*m)*(m^2+8*m+15)/a^6*LerchPhi(-a^2*x^2,1,1/2*m+1/2))
\[ \int e^{6 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{6} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((1+I*a*x)^6/(a^2*x^2+1)^3*x^m,x, algorithm="fricas")
Output:
integral(-(a^3*x^3 - 3*I*a^2*x^2 - 3*a*x + I)*x^m/(a^3*x^3 + 3*I*a^2*x^2 - 3*a*x - I), x)
\[ \int e^{6 i \arctan (a x)} x^m \, dx=- \int \left (- \frac {x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx - \int \frac {15 a^{2} x^{2} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx - \int \left (- \frac {15 a^{4} x^{4} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx - \int \frac {a^{6} x^{6} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx - \int \left (- \frac {6 i a x x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx - \int \frac {20 i a^{3} x^{3} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx - \int \left (- \frac {6 i a^{5} x^{5} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx \] Input:
integrate((1+I*a*x)**6/(a**2*x**2+1)**3*x**m,x)
Output:
-Integral(-x**m/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x) - Integral (15*a**2*x**2*x**m/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x) - Integ ral(-15*a**4*x**4*x**m/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x) - I ntegral(a**6*x**6*x**m/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x) - I ntegral(-6*I*a*x*x**m/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x) - In tegral(20*I*a**3*x**3*x**m/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x) - Integral(-6*I*a**5*x**5*x**m/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1 ), x)
\[ \int e^{6 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{6} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((1+I*a*x)^6/(a^2*x^2+1)^3*x^m,x, algorithm="maxima")
Output:
integrate((I*a*x + 1)^6*x^m/(a^2*x^2 + 1)^3, x)
\[ \int e^{6 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{6} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((1+I*a*x)^6/(a^2*x^2+1)^3*x^m,x, algorithm="giac")
Output:
integrate((I*a*x + 1)^6*x^m/(a^2*x^2 + 1)^3, x)
Timed out. \[ \int e^{6 i \arctan (a x)} x^m \, dx=\int \frac {x^m\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^6}{{\left (a^2\,x^2+1\right )}^3} \,d x \] Input:
int((x^m*(a*x*1i + 1)^6)/(a^2*x^2 + 1)^3,x)
Output:
int((x^m*(a*x*1i + 1)^6)/(a^2*x^2 + 1)^3, x)
\[ \int e^{6 i \arctan (a x)} x^m \, dx=\text {too large to display} \] Input:
int((1+I*a*x)^6/(a^2*x^2+1)^3*x^m,x)
Output:
( - x**m*a**5*m**5*x**5 + 10*x**m*a**5*m**4*x**5 - 35*x**m*a**5*m**3*x**5 + 50*x**m*a**5*m**2*x**5 - 24*x**m*a**5*m*x**5 + 6*x**m*a**4*i*m**5*x**4 - 54*x**m*a**4*i*m**4*x**4 + 150*x**m*a**4*i*m**3*x**4 - 90*x**m*a**4*i*m** 2*x**4 - 156*x**m*a**4*i*m*x**4 + 144*x**m*a**4*i*x**4 + 16*x**m*a**3*m**5 *x**3 - 124*x**m*a**3*m**4*x**3 + 236*x**m*a**3*m**3*x**3 + 136*x**m*a**3* m**2*x**3 - 480*x**m*a**3*m*x**3 - 26*x**m*a**2*i*m**5*x**2 + 158*x**m*a** 2*i*m**4*x**2 - 118*x**m*a**2*i*m**3*x**2 - 446*x**m*a**2*i*m**2*x**2 + 14 4*x**m*a**2*i*m*x**2 + 288*x**m*a**2*i*x**2 - 31*x**m*a*m**5*x + 118*x**m* a*m**4*x + 115*x**m*a*m**3*x - 274*x**m*a*m**2*x - 360*x**m*a*m*x + 32*x** m*i*m**5 - 32*x**m*i*m**4 - 176*x**m*i*m**3 - 112*x**m*i*m**2 + 144*x**m*i *m + 144*x**m*i - 32*int(x**m/(a**6*m**4*x**7 - 10*a**6*m**3*x**7 + 35*a** 6*m**2*x**7 - 50*a**6*m*x**7 + 24*a**6*x**7 + 3*a**4*m**4*x**5 - 30*a**4*m **3*x**5 + 105*a**4*m**2*x**5 - 150*a**4*m*x**5 + 72*a**4*x**5 + 3*a**2*m* *4*x**3 - 30*a**2*m**3*x**3 + 105*a**2*m**2*x**3 - 150*a**2*m*x**3 + 72*a* *2*x**3 + m**4*x - 10*m**3*x + 35*m**2*x - 50*m*x + 24*x),x)*a**4*i*m**10* x**4 + 352*int(x**m/(a**6*m**4*x**7 - 10*a**6*m**3*x**7 + 35*a**6*m**2*x** 7 - 50*a**6*m*x**7 + 24*a**6*x**7 + 3*a**4*m**4*x**5 - 30*a**4*m**3*x**5 + 105*a**4*m**2*x**5 - 150*a**4*m*x**5 + 72*a**4*x**5 + 3*a**2*m**4*x**3 - 30*a**2*m**3*x**3 + 105*a**2*m**2*x**3 - 150*a**2*m*x**3 + 72*a**2*x**3 + m**4*x - 10*m**3*x + 35*m**2*x - 50*m*x + 24*x),x)*a**4*i*m**9*x**4 - 1...