\(\int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx\) [58]

Optimal result

Integrand size = 29, antiderivative size = 181 \[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=-\frac {i e^{2 (a-i d)-i (d+b x)}}{32 b}-\frac {i e^{2 (a-i d)+i (d+b x)}}{16 b}-\frac {i e^{2 (a-i d)-3 i (d+b x)}}{96 b}-\frac {i e^{2 (a-i d)+3 i (d+b x)}}{48 b}+\frac {i e^{2 (a-i d)+5 i (d+b x)}}{160 b}+\frac {i e^{2 (a-i d)+7 i (d+b x)}}{224 b} \] Output:

-1/32*I*exp(2*a-2*I*d-I*(b*x+d))/b-1/16*I*exp(2*a-2*I*d+I*(b*x+d))/b-1/96* 
I*exp(2*a-2*I*d-3*I*(b*x+d))/b-1/48*I*exp(2*a-2*I*d+3*I*(b*x+d))/b+1/160*I 
*exp(2*a-2*I*d+5*I*(b*x+d))/b+1/224*I*exp(2*a-2*I*d+7*I*(b*x+d))/b
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.78 \[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=\frac {e^{2 a-3 i b x} \left (5 i \left (-7+3 e^{10 i b x}\right ) \cos (5 d)+70 e^{4 i b x} \left (-i \left (3+e^{2 i b x}\right ) \cos (d)+\left (-3+e^{2 i b x}\right ) \sin (d)\right )-21 e^{2 i b x} \left (-i \left (-5+e^{6 i b x}\right ) \cos (3 d)+\left (5+e^{6 i b x}\right ) \sin (3 d)\right )-5 \left (7+3 e^{10 i b x}\right ) \sin (5 d)\right )}{3360 b} \] Input:

Integrate[E^(2*(a + I*b*x))*Cos[d + b*x]^3*Sin[d + b*x]^2,x]
 

Output:

(E^(2*a - (3*I)*b*x)*((5*I)*(-7 + 3*E^((10*I)*b*x))*Cos[5*d] + 70*E^((4*I) 
*b*x)*((-I)*(3 + E^((2*I)*b*x))*Cos[d] + (-3 + E^((2*I)*b*x))*Sin[d]) - 21 
*E^((2*I)*b*x)*((-I)*(-5 + E^((6*I)*b*x))*Cos[3*d] + (5 + E^((6*I)*b*x))*S 
in[3*d]) - 5*(7 + 3*E^((10*I)*b*x))*Sin[5*d]))/(3360*b)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4972, 2009}

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.

\(\displaystyle \int e^{2 (a+i b x)} \sin ^2(b x+d) \cos ^3(b x+d) \, dx\)

\(\Big \downarrow \) 4972

\(\displaystyle \int \left (\frac {1}{8} e^{2 (a+i b x)} \cos (b x+d)-\frac {1}{16} e^{2 (a+i b x)} \cos (3 b x+3 d)-\frac {1}{16} e^{2 (a+i b x)} \cos (5 b x+5 d)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e^{2 (a+i b x)} \sin (b x+d)}{24 b}-\frac {3 e^{2 (a+i b x)} \sin (3 b x+3 d)}{80 b}-\frac {5 e^{2 (a+i b x)} \sin (5 b x+5 d)}{336 b}-\frac {i e^{2 (a+i b x)} \cos (b x+d)}{12 b}-\frac {i e^{2 (a+i b x)} \cos (3 b x+3 d)}{40 b}-\frac {i e^{2 (a+i b x)} \cos (5 b x+5 d)}{168 b}\)

Input:

Int[E^(2*(a + I*b*x))*Cos[d + b*x]^3*Sin[d + b*x]^2,x]
 

Output:

((-1/12*I)*E^(2*(a + I*b*x))*Cos[d + b*x])/b - ((I/40)*E^(2*(a + I*b*x))*C 
os[3*d + 3*b*x])/b - ((I/168)*E^(2*(a + I*b*x))*Cos[5*d + 5*b*x])/b - (E^( 
2*(a + I*b*x))*Sin[d + b*x])/(24*b) - (3*E^(2*(a + I*b*x))*Sin[3*d + 3*b*x 
])/(80*b) - (5*E^(2*(a + I*b*x))*Sin[5*d + 5*b*x])/(336*b)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4972

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_ 
.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[F^(c*(a + b*x)), 
Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] 
 && IGtQ[m, 0] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.57

method	result	size
parallelrisch	\(-\frac {{\mathrm e}^{2 i b x +2 a} \left (63 \sin \left (3 b x +3 d \right )-192 \sin \left (2 b x +2 d \right )+25 \sin \left (5 b x +5 d \right )+10 i \cos \left (5 b x +5 d \right )-192 i \cos \left (2 b x +2 d \right )+140 i \cos \left (b x +d \right )+42 i \cos \left (3 b x +3 d \right )+70 \sin \left (b x +d \right )\right )}{1680 b}\)	\(103\)
default	\(-\frac {i {\mathrm e}^{2 i b x +2 a} \cos \left (3 b x +3 d \right )}{40 b}-\frac {3 \,{\mathrm e}^{2 i b x +2 a} \sin \left (3 b x +3 d \right )}{80 b}-\frac {i {\mathrm e}^{2 i b x +2 a} \cos \left (5 b x +5 d \right )}{168 b}-\frac {5 \,{\mathrm e}^{2 i b x +2 a} \sin \left (5 b x +5 d \right )}{336 b}-\frac {i {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )}{12 b}-\frac {{\mathrm e}^{2 i b x +2 a} \sin \left (b x +d \right )}{24 b}\)	\(143\)
orering	\(-\frac {12 i {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}}{35 b}-\frac {341 \left (2 i b \,{\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}-3 \,{\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3} b +2 \,{\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{4} \sin \left (b x +d \right ) b \right )}{315 b^{2}}-\frac {8 i \left (-21 b^{2} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}-12 i b^{2} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}+8 i b^{2} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{4} \sin \left (b x +d \right )+6 \,{\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right ) \sin \left (b x +d \right )^{4} b^{2}+2 \,{\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{5} b^{2}\right )}{21 b^{3}}-\frac {5 \left (-110 i b^{3} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}+111 b^{3} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}-68 b^{3} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{4} \sin \left (b x +d \right )+36 i b^{3} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right ) \sin \left (b x +d \right )^{4}+12 i b^{3} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{5}-6 \,{\mathrm e}^{2 i b x +2 a} b^{3} \sin \left (b x +d \right )^{5}\right )}{63 b^{4}}-\frac {4 i \left (825 b^{4} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}-416 i b^{4} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{4} \sin \left (b x +d \right )+696 i b^{4} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}-48 i b^{4} {\mathrm e}^{2 i b x +2 a} \sin \left (b x +d \right )^{5}-324 b^{4} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right ) \sin \left (b x +d \right )^{4}-92 b^{4} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{5}\right )}{105 b^{5}}+\frac {-600 i b^{5} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{5}-5163 b^{5} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}+2942 b^{5} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{4} \sin \left (b x +d \right )+5402 i b^{5} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}-2280 i b^{5} {\mathrm e}^{2 i b x +2 a} \cos \left (b x +d \right ) \sin \left (b x +d \right )^{4}+420 b^{5} {\mathrm e}^{2 i b x +2 a} \sin \left (b x +d \right )^{5}}{315 b^{6}}\)	\(804\)

Input:

int(exp(2*a+2*I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/1680*exp(2*a+2*I*b*x)*(63*sin(3*b*x+3*d)-192*sin(2*b*x+2*d)+25*sin(5*b* 
x+5*d)+10*I*cos(5*b*x+5*d)-192*I*cos(2*b*x+2*d)+140*I*cos(b*x+d)+42*I*cos( 
3*b*x+3*d)+70*sin(b*x+d))/b
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.51 \[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=\frac {{\left (15 i \, e^{\left (10 i \, b x + 2 \, a + 8 i \, d\right )} + 21 i \, e^{\left (8 i \, b x + 2 \, a + 6 i \, d\right )} - 70 i \, e^{\left (6 i \, b x + 2 \, a + 4 i \, d\right )} - 210 i \, e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} - 105 i \, e^{\left (2 i \, b x + 2 \, a\right )} - 35 i \, e^{\left (2 \, a - 2 i \, d\right )}\right )} e^{\left (-3 i \, b x - 3 i \, d\right )}}{3360 \, b} \] Input:

integrate(exp(2*a+2*I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2,x, algorithm="fricas" 
)
 

Output:

1/3360*(15*I*e^(10*I*b*x + 2*a + 8*I*d) + 21*I*e^(8*I*b*x + 2*a + 6*I*d) - 
 70*I*e^(6*I*b*x + 2*a + 4*I*d) - 210*I*e^(4*I*b*x + 2*a + 2*I*d) - 105*I* 
e^(2*I*b*x + 2*a) - 35*I*e^(2*a - 2*I*d))*e^(-3*I*b*x - 3*I*d)/b
 

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.42 \[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=\begin {cases} \frac {\left (377487360 i b^{5} e^{2 a} e^{14 i d} e^{7 i b x} + 528482304 i b^{5} e^{2 a} e^{12 i d} e^{5 i b x} - 1761607680 i b^{5} e^{2 a} e^{10 i d} e^{3 i b x} - 5284823040 i b^{5} e^{2 a} e^{8 i d} e^{i b x} - 2642411520 i b^{5} e^{2 a} e^{6 i d} e^{- i b x} - 880803840 i b^{5} e^{2 a} e^{4 i d} e^{- 3 i b x}\right ) e^{- 9 i d}}{84557168640 b^{6}} & \text {for}\: b^{6} e^{9 i d} \neq 0 \\\frac {x \left (- e^{2 a} e^{10 i d} - e^{2 a} e^{8 i d} + 2 e^{2 a} e^{6 i d} + 2 e^{2 a} e^{4 i d} - e^{2 a} e^{2 i d} - e^{2 a}\right ) e^{- 5 i d}}{32} & \text {otherwise} \end {cases} \] Input:

integrate(exp(2*a+2*I*b*x)*cos(b*x+d)**3*sin(b*x+d)**2,x)
                                                                                    
                                                                                    
 

Output:

Piecewise(((377487360*I*b**5*exp(2*a)*exp(14*I*d)*exp(7*I*b*x) + 528482304 
*I*b**5*exp(2*a)*exp(12*I*d)*exp(5*I*b*x) - 1761607680*I*b**5*exp(2*a)*exp 
(10*I*d)*exp(3*I*b*x) - 5284823040*I*b**5*exp(2*a)*exp(8*I*d)*exp(I*b*x) - 
 2642411520*I*b**5*exp(2*a)*exp(6*I*d)*exp(-I*b*x) - 880803840*I*b**5*exp( 
2*a)*exp(4*I*d)*exp(-3*I*b*x))*exp(-9*I*d)/(84557168640*b**6), Ne(b**6*exp 
(9*I*d), 0)), (x*(-exp(2*a)*exp(10*I*d) - exp(2*a)*exp(8*I*d) + 2*exp(2*a) 
*exp(6*I*d) + 2*exp(2*a)*exp(4*I*d) - exp(2*a)*exp(2*I*d) - exp(2*a))*exp( 
-5*I*d)/32, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.98 \[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=-\frac {-15 i \, \cos \left (7 \, b x + 5 \, d\right ) e^{\left (2 \, a\right )} - 21 i \, \cos \left (5 \, b x + 3 \, d\right ) e^{\left (2 \, a\right )} + 35 i \, \cos \left (3 \, b x + 5 \, d\right ) e^{\left (2 \, a\right )} + 70 i \, \cos \left (3 \, b x + d\right ) e^{\left (2 \, a\right )} + 105 i \, \cos \left (b x + 3 \, d\right ) e^{\left (2 \, a\right )} + 210 i \, \cos \left (b x - d\right ) e^{\left (2 \, a\right )} + 15 \, e^{\left (2 \, a\right )} \sin \left (7 \, b x + 5 \, d\right ) + 21 \, e^{\left (2 \, a\right )} \sin \left (5 \, b x + 3 \, d\right ) + 35 \, e^{\left (2 \, a\right )} \sin \left (3 \, b x + 5 \, d\right ) - 70 \, e^{\left (2 \, a\right )} \sin \left (3 \, b x + d\right ) + 105 \, e^{\left (2 \, a\right )} \sin \left (b x + 3 \, d\right ) - 210 \, e^{\left (2 \, a\right )} \sin \left (b x - d\right )}{3360 \, b} \] Input:

integrate(exp(2*a+2*I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2,x, algorithm="maxima" 
)
 

Output:

-1/3360*(-15*I*cos(7*b*x + 5*d)*e^(2*a) - 21*I*cos(5*b*x + 3*d)*e^(2*a) + 
35*I*cos(3*b*x + 5*d)*e^(2*a) + 70*I*cos(3*b*x + d)*e^(2*a) + 105*I*cos(b* 
x + 3*d)*e^(2*a) + 210*I*cos(b*x - d)*e^(2*a) + 15*e^(2*a)*sin(7*b*x + 5*d 
) + 21*e^(2*a)*sin(5*b*x + 3*d) + 35*e^(2*a)*sin(3*b*x + 5*d) - 70*e^(2*a) 
*sin(3*b*x + d) + 105*e^(2*a)*sin(b*x + 3*d) - 210*e^(2*a)*sin(b*x - d))/b
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (103) = 206\).

Time = 0.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.34 \[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=-\frac {-15 i \, {\left (e^{\left (7 i \, b x + 5 i \, d\right )} + e^{\left (-7 i \, b x - 5 i \, d\right )}\right )} e^{\left (2 \, a\right )} - 21 i \, {\left (e^{\left (5 i \, b x + 3 i \, d\right )} + e^{\left (-5 i \, b x - 3 i \, d\right )}\right )} e^{\left (2 \, a\right )} + 35 i \, {\left (e^{\left (3 i \, b x + 5 i \, d\right )} + e^{\left (-3 i \, b x - 5 i \, d\right )}\right )} e^{\left (2 \, a\right )} + 70 i \, {\left (e^{\left (3 i \, b x + i \, d\right )} + e^{\left (-3 i \, b x - i \, d\right )}\right )} e^{\left (2 \, a\right )} + 105 i \, {\left (e^{\left (i \, b x + 3 i \, d\right )} + e^{\left (-i \, b x - 3 i \, d\right )}\right )} e^{\left (2 \, a\right )} + 210 i \, {\left (e^{\left (i \, b x - i \, d\right )} + e^{\left (-i \, b x + i \, d\right )}\right )} e^{\left (2 \, a\right )} + 70 \, e^{\left (2 \, a\right )} \sin \left (3 \, b x + 5 \, d\right ) + 210 \, e^{\left (2 \, a\right )} \sin \left (b x + 3 \, d\right ) + 420 \, e^{\left (2 \, a\right )} \sin \left (-b x + d\right ) + 140 \, e^{\left (2 \, a\right )} \sin \left (-3 \, b x - d\right ) - 42 \, e^{\left (2 \, a\right )} \sin \left (-5 \, b x - 3 \, d\right ) - 30 \, e^{\left (2 \, a\right )} \sin \left (-7 \, b x - 5 \, d\right )}{6720 \, b} \] Input:

integrate(exp(2*a+2*I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/6720*(-15*I*(e^(7*I*b*x + 5*I*d) + e^(-7*I*b*x - 5*I*d))*e^(2*a) - 21*I 
*(e^(5*I*b*x + 3*I*d) + e^(-5*I*b*x - 3*I*d))*e^(2*a) + 35*I*(e^(3*I*b*x + 
 5*I*d) + e^(-3*I*b*x - 5*I*d))*e^(2*a) + 70*I*(e^(3*I*b*x + I*d) + e^(-3* 
I*b*x - I*d))*e^(2*a) + 105*I*(e^(I*b*x + 3*I*d) + e^(-I*b*x - 3*I*d))*e^( 
2*a) + 210*I*(e^(I*b*x - I*d) + e^(-I*b*x + I*d))*e^(2*a) + 70*e^(2*a)*sin 
(3*b*x + 5*d) + 210*e^(2*a)*sin(b*x + 3*d) + 420*e^(2*a)*sin(-b*x + d) + 1 
40*e^(2*a)*sin(-3*b*x - d) - 42*e^(2*a)*sin(-5*b*x - 3*d) - 30*e^(2*a)*sin 
(-7*b*x - 5*d))/b
 

Mupad [B] (verification not implemented)

Time = 15.67 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.13 \[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=-\frac {{\mathrm {e}}^{2\,a}\,\left (\cos \left (b\,x\right )-\sin \left (b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )-\sin \left (3\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,b}-\frac {{\mathrm {e}}^{2\,a}\,\left (\cos \left (3\,b\,x\right )-\sin \left (3\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (5\,d\right )-\sin \left (5\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{96\,b}+\frac {{\mathrm {e}}^{2\,a}\,\left (\cos \left (5\,b\,x\right )+\sin \left (5\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )+\sin \left (3\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{160\,b}+\frac {{\mathrm {e}}^{2\,a}\,\left (\cos \left (7\,b\,x\right )+\sin \left (7\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (5\,d\right )+\sin \left (5\,d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{224\,b}-\frac {{\mathrm {e}}^{2\,a}\,\left (\cos \left (b\,x\right )+\sin \left (b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )-\sin \left (d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,b}-\frac {{\mathrm {e}}^{2\,a}\,\left (\cos \left (3\,b\,x\right )+\sin \left (3\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )+\sin \left (d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{48\,b} \] Input:

int(cos(d + b*x)^3*exp(2*a + b*x*2i)*sin(d + b*x)^2,x)
 

Output:

(exp(2*a)*(cos(5*b*x) + sin(5*b*x)*1i)*(cos(3*d) + sin(3*d)*1i)*1i)/(160*b 
) - (exp(2*a)*(cos(3*b*x) - sin(3*b*x)*1i)*(cos(5*d) - sin(5*d)*1i)*1i)/(9 
6*b) - (exp(2*a)*(cos(b*x) - sin(b*x)*1i)*(cos(3*d) - sin(3*d)*1i)*1i)/(32 
*b) + (exp(2*a)*(cos(7*b*x) + sin(7*b*x)*1i)*(cos(5*d) + sin(5*d)*1i)*1i)/ 
(224*b) - (exp(2*a)*(cos(b*x) + sin(b*x)*1i)*(cos(d) - sin(d)*1i)*1i)/(16* 
b) - (exp(2*a)*(cos(3*b*x) + sin(3*b*x)*1i)*(cos(d) + sin(d)*1i)*1i)/(48*b 
)
 

Reduce [F]

\[ \int e^{2 (a+i b x)} \cos ^3(d+b x) \sin ^2(d+b x) \, dx=e^{2 a} \left (\int e^{2 b i x} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}d x \right ) \] Input:

int(exp(2*a+2*I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2,x)
 

Output:

e**(2*a)*int(e**(2*b*i*x)*cos(b*x + d)**3*sin(b*x + d)**2,x)