\(\int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx\) [8]

Optimal result

Integrand size = 27, antiderivative size = 142 \[ \int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx=-\frac {e^{a-i d-2 i (d+b x)}}{64 b}-\frac {e^{a-i d+2 i (d+b x)}}{32 b}+\frac {e^{a-i d-4 i (d+b x)}}{128 b}-\frac {e^{a-i d+4 i (d+b x)}}{128 b}+\frac {e^{a-i d+6 i (d+b x)}}{192 b}+\frac {1}{16} i e^{a-i d} x \] Output:

-1/64*exp(a-I*d-2*I*(b*x+d))/b-1/32*exp(a-I*d+2*I*(b*x+d))/b+1/128*exp(a-I 
*d-4*I*(b*x+d))/b-1/128*exp(a-I*d+4*I*(b*x+d))/b+1/192*exp(a-I*d+6*I*(b*x+ 
d))/b+1/16*I*exp(a-I*d)*x
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.01 \[ \int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx=\frac {e^a \left (-12 \left (\left (e^{2 i b x}-2 i b x\right ) \cos (d)+i \left (e^{2 i b x}+2 i b x\right ) \sin (d)\right )-3 e^{-2 i b x} \left (\left (2+e^{6 i b x}\right ) \cos (3 d)+i \left (-2+e^{6 i b x}\right ) \sin (3 d)\right )+e^{-4 i b x} \left (\left (3+2 e^{10 i b x}\right ) \cos (5 d)+i \left (-3+2 e^{10 i b x}\right ) \sin (5 d)\right )\right )}{384 b} \] Input:

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

Output:

(E^a*(-12*((E^((2*I)*b*x) - (2*I)*b*x)*Cos[d] + I*(E^((2*I)*b*x) + (2*I)*b 
*x)*Sin[d]) - (3*((2 + E^((6*I)*b*x))*Cos[3*d] + I*(-2 + E^((6*I)*b*x))*Si 
n[3*d]))/E^((2*I)*b*x) + ((3 + 2*E^((10*I)*b*x))*Cos[5*d] + I*(-3 + 2*E^(( 
10*I)*b*x))*Sin[5*d])/E^((4*I)*b*x)))/(384*b)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.

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

\(\Big \downarrow \) 4972

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e^{a+2 i b x+i d}}{32 b}+\frac {i e^{a+i b x} \sin (3 b x+3 d)}{128 b}-\frac {i e^{a+i b x} \sin (5 b x+5 d)}{384 b}-\frac {3 e^{a+i b x} \cos (3 b x+3 d)}{128 b}+\frac {5 e^{a+i b x} \cos (5 b x+5 d)}{384 b}+\frac {1}{16} i x e^{a-i d}\)

Input:

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

Output:

-1/32*E^(a + I*d + (2*I)*b*x)/b + (I/16)*E^(a - I*d)*x - (3*E^(a + I*b*x)* 
Cos[3*d + 3*b*x])/(128*b) + (5*E^(a + I*b*x)*Cos[5*d + 5*b*x])/(384*b) + ( 
(I/128)*E^(a + I*b*x)*Sin[3*d + 3*b*x])/b - ((I/384)*E^(a + I*b*x)*Sin[5*d 
 + 5*b*x])/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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (112 ) = 224\).

Time = 3.92 (sec) , antiderivative size = 786, normalized size of antiderivative = 5.54

method	result	size
orering	\(-\frac {\left (-6 b x +i\right ) {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}}{6 b}+\frac {i \left (8 b x +15 i\right ) \left (i b \,{\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}-2 \,{\mathrm e}^{i b x +a} \cos \left (b x +d \right ) \sin \left (b x +d \right )^{4} b +3 \,{\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2} b \right )}{48 b^{2}}-\frac {5 \left (-6 b x +i\right ) \left (-18 b^{2} {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}-4 i b^{2} {\mathrm e}^{i b x +a} \cos \left (b x +d \right ) \sin \left (b x +d \right )^{4}+6 i b^{2} {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}+2 \,{\mathrm e}^{i b x +a} b^{2} \sin \left (b x +d \right )^{5}+6 \,{\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{4} \sin \left (b x +d \right ) b^{2}\right )}{96 b^{3}}+\frac {i \left (10 b x +3 i\right ) \left (-52 i b^{3} {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}+50 \,{\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{4} \cos \left (b x +d \right ) b^{3}-84 \,{\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{2} \cos \left (b x +d \right )^{3} b^{3}+6 i b^{3} {\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{5}+18 i b^{3} {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{4} \sin \left (b x +d \right )+6 \,{\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{5} b^{3}\right )}{192 b^{4}}-\frac {\left (-6 b x +i\right ) \left (504 \,{\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{3} \cos \left (b x +d \right )^{2} b^{4}+184 i b^{4} {\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{4} \cos \left (b x +d \right )-312 i b^{4} {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{3} \sin \left (b x +d \right )^{2}+24 i {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{5} b^{4}-56 \,{\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{5} b^{4}-216 \,{\mathrm e}^{i b x +a} \sin \left (b x +d \right ) \cos \left (b x +d \right )^{4} b^{4}\right )}{384 b^{5}}+\frac {i x \left (-240 i b^{5} {\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{5}+2688 \,{\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{2} \cos \left (b x +d \right )^{3} b^{5}-1472 \,{\mathrm e}^{i b x +a} \sin \left (b x +d \right )^{4} \cos \left (b x +d \right ) b^{5}+2176 i b^{5} {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{3}-960 i b^{5} {\mathrm e}^{i b x +a} \sin \left (b x +d \right ) \cos \left (b x +d \right )^{4}-240 b^{5} {\mathrm e}^{i b x +a} \cos \left (b x +d \right )^{5}\right )}{384 b^{5}}\)	\(786\)

Input:

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

Output:

-1/6*(-6*b*x+I)/b*exp(a+I*b*x)*cos(b*x+d)^2*sin(b*x+d)^3+1/48*I*(8*b*x+15* 
I)/b^2*(I*b*exp(a+I*b*x)*cos(b*x+d)^2*sin(b*x+d)^3-2*exp(a+I*b*x)*cos(b*x+ 
d)*sin(b*x+d)^4*b+3*exp(a+I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2*b)-5/96*(-6*b*x 
+I)/b^3*(-18*b^2*exp(a+I*b*x)*cos(b*x+d)^2*sin(b*x+d)^3-4*I*b^2*exp(a+I*b* 
x)*cos(b*x+d)*sin(b*x+d)^4+6*I*b^2*exp(a+I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2+ 
2*exp(a+I*b*x)*b^2*sin(b*x+d)^5+6*exp(a+I*b*x)*cos(b*x+d)^4*sin(b*x+d)*b^2 
)+1/192*I*(10*b*x+3*I)/b^4*(-52*I*b^3*exp(a+I*b*x)*cos(b*x+d)^2*sin(b*x+d) 
^3+50*exp(a+I*b*x)*sin(b*x+d)^4*cos(b*x+d)*b^3-84*exp(a+I*b*x)*sin(b*x+d)^ 
2*cos(b*x+d)^3*b^3+6*I*b^3*exp(a+I*b*x)*sin(b*x+d)^5+18*I*b^3*exp(a+I*b*x) 
*cos(b*x+d)^4*sin(b*x+d)+6*exp(a+I*b*x)*cos(b*x+d)^5*b^3)-1/384*(-6*b*x+I) 
/b^5*(504*exp(a+I*b*x)*sin(b*x+d)^3*cos(b*x+d)^2*b^4+184*I*b^4*exp(a+I*b*x 
)*sin(b*x+d)^4*cos(b*x+d)-312*I*b^4*exp(a+I*b*x)*cos(b*x+d)^3*sin(b*x+d)^2 
+24*I*exp(a+I*b*x)*cos(b*x+d)^5*b^4-56*exp(a+I*b*x)*sin(b*x+d)^5*b^4-216*e 
xp(a+I*b*x)*sin(b*x+d)*cos(b*x+d)^4*b^4)+1/384*I/b^5*x*(-240*I*b^5*exp(a+I 
*b*x)*sin(b*x+d)^5+2688*exp(a+I*b*x)*sin(b*x+d)^2*cos(b*x+d)^3*b^5-1472*ex 
p(a+I*b*x)*sin(b*x+d)^4*cos(b*x+d)*b^5+2176*I*b^5*exp(a+I*b*x)*cos(b*x+d)^ 
2*sin(b*x+d)^3-960*I*b^5*exp(a+I*b*x)*sin(b*x+d)*cos(b*x+d)^4-240*b^5*exp( 
a+I*b*x)*cos(b*x+d)^5)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.60 \[ \int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx=\frac {{\left (24 i \, b x e^{\left (4 i \, b x + a + 3 i \, d\right )} + 2 \, e^{\left (10 i \, b x + a + 9 i \, d\right )} - 3 \, e^{\left (8 i \, b x + a + 7 i \, d\right )} - 12 \, e^{\left (6 i \, b x + a + 5 i \, d\right )} - 6 \, e^{\left (2 i \, b x + a + i \, d\right )} + 3 \, e^{\left (a - i \, d\right )}\right )} e^{\left (-4 i \, b x - 4 i \, d\right )}}{384 \, b} \] Input:

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

Output:

1/384*(24*I*b*x*e^(4*I*b*x + a + 3*I*d) + 2*e^(10*I*b*x + a + 9*I*d) - 3*e 
^(8*I*b*x + a + 7*I*d) - 12*e^(6*I*b*x + a + 5*I*d) - 6*e^(2*I*b*x + a + I 
*d) + 3*e^(a - I*d))*e^(-4*I*b*x - 4*I*d)/b
 

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.70 \[ \int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx=\frac {i x e^{a} e^{- i d}}{16} + \begin {cases} \frac {\left (33554432 b^{4} e^{a} e^{13 i d} e^{6 i b x} - 50331648 b^{4} e^{a} e^{11 i d} e^{4 i b x} - 201326592 b^{4} e^{a} e^{9 i d} e^{2 i b x} - 100663296 b^{4} e^{a} e^{5 i d} e^{- 2 i b x} + 50331648 b^{4} e^{a} e^{3 i d} e^{- 4 i b x}\right ) e^{- 8 i d}}{6442450944 b^{5}} & \text {for}\: b^{5} e^{8 i d} \neq 0 \\x \left (\frac {\left (i e^{a} e^{10 i d} - i e^{a} e^{8 i d} - 2 i e^{a} e^{6 i d} + 2 i e^{a} e^{4 i d} + i e^{a} e^{2 i d} - i e^{a}\right ) e^{- 5 i d}}{32} - \frac {i e^{a} e^{- i d}}{16}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

I*x*exp(a)*exp(-I*d)/16 + Piecewise(((33554432*b**4*exp(a)*exp(13*I*d)*exp 
(6*I*b*x) - 50331648*b**4*exp(a)*exp(11*I*d)*exp(4*I*b*x) - 201326592*b**4 
*exp(a)*exp(9*I*d)*exp(2*I*b*x) - 100663296*b**4*exp(a)*exp(5*I*d)*exp(-2* 
I*b*x) + 50331648*b**4*exp(a)*exp(3*I*d)*exp(-4*I*b*x))*exp(-8*I*d)/(64424 
50944*b**5), Ne(b**5*exp(8*I*d), 0)), (x*((I*exp(a)*exp(10*I*d) - I*exp(a) 
*exp(8*I*d) - 2*I*exp(a)*exp(6*I*d) + 2*I*exp(a)*exp(4*I*d) + I*exp(a)*exp 
(2*I*d) - I*exp(a))*exp(-5*I*d)/32 - I*exp(a)*exp(-I*d)/16), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06 \[ \int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx=-\frac {24 \, {\left (-i \, b \cos \left (d\right ) e^{a} - b e^{a} \sin \left (d\right )\right )} x - 2 \, \cos \left (6 \, b x + 5 \, d\right ) e^{a} - 3 \, \cos \left (4 \, b x + 5 \, d\right ) e^{a} + 3 \, \cos \left (4 \, b x + 3 \, d\right ) e^{a} + 6 \, \cos \left (2 \, b x + 3 \, d\right ) e^{a} + 12 \, \cos \left (2 \, b x + d\right ) e^{a} - 2 i \, e^{a} \sin \left (6 \, b x + 5 \, d\right ) + 3 i \, e^{a} \sin \left (4 \, b x + 5 \, d\right ) + 3 i \, e^{a} \sin \left (4 \, b x + 3 \, d\right ) - 6 i \, e^{a} \sin \left (2 \, b x + 3 \, d\right ) + 12 i \, e^{a} \sin \left (2 \, b x + d\right )}{384 \, b} \] Input:

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

Output:

-1/384*(24*(-I*b*cos(d)*e^a - b*e^a*sin(d))*x - 2*cos(6*b*x + 5*d)*e^a - 3 
*cos(4*b*x + 5*d)*e^a + 3*cos(4*b*x + 3*d)*e^a + 6*cos(2*b*x + 3*d)*e^a + 
12*cos(2*b*x + d)*e^a - 2*I*e^a*sin(6*b*x + 5*d) + 3*I*e^a*sin(4*b*x + 5*d 
) + 3*I*e^a*sin(4*b*x + 3*d) - 6*I*e^a*sin(2*b*x + 3*d) + 12*I*e^a*sin(2*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. 218 vs. \(2 (85) = 170\).

Time = 0.15 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.54 \[ \int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx=-\frac {-48 i \, {\left (b x + d\right )} \cos \left (d\right ) e^{a} - 48 \, {\left (b x + d\right )} e^{a} \sin \left (d\right ) - 2 \, {\left (e^{\left (6 i \, b x + 5 i \, d\right )} - e^{\left (-6 i \, b x - 5 i \, d\right )}\right )} e^{a} + 3 \, {\left (e^{\left (4 i \, b x + 5 i \, d\right )} - e^{\left (-4 i \, b x - 5 i \, d\right )}\right )} e^{a} + 3 \, {\left (e^{\left (4 i \, b x + 3 i \, d\right )} - e^{\left (-4 i \, b x - 3 i \, d\right )}\right )} e^{a} - 6 \, {\left (e^{\left (2 i \, b x + 3 i \, d\right )} - e^{\left (-2 i \, b x - 3 i \, d\right )}\right )} e^{a} + 12 \, {\left (e^{\left (2 i \, b x + i \, d\right )} - e^{\left (-2 i \, b x - i \, d\right )}\right )} e^{a} - 6 \, \cos \left (4 \, b x + 5 \, d\right ) e^{a} + 12 \, \cos \left (2 \, b x + 3 \, d\right ) e^{a} + 24 \, \cos \left (-2 \, b x - d\right ) e^{a} + 6 \, \cos \left (-4 \, b x - 3 \, d\right ) e^{a} - 4 \, \cos \left (-6 \, b x - 5 \, d\right ) e^{a}}{768 \, b} \] Input:

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

Output:

-1/768*(-48*I*(b*x + d)*cos(d)*e^a - 48*(b*x + d)*e^a*sin(d) - 2*(e^(6*I*b 
*x + 5*I*d) - e^(-6*I*b*x - 5*I*d))*e^a + 3*(e^(4*I*b*x + 5*I*d) - e^(-4*I 
*b*x - 5*I*d))*e^a + 3*(e^(4*I*b*x + 3*I*d) - e^(-4*I*b*x - 3*I*d))*e^a - 
6*(e^(2*I*b*x + 3*I*d) - e^(-2*I*b*x - 3*I*d))*e^a + 12*(e^(2*I*b*x + I*d) 
 - e^(-2*I*b*x - I*d))*e^a - 6*cos(4*b*x + 5*d)*e^a + 12*cos(2*b*x + 3*d)* 
e^a + 24*cos(-2*b*x - d)*e^a + 6*cos(-4*b*x - 3*d)*e^a - 4*cos(-6*b*x - 5* 
d)*e^a)/b
 

Mupad [B] (verification not implemented)

Time = 16.22 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.24 \[ \int e^{a+i b x} \cos ^2(d+b x) \sin ^3(d+b x) \, dx=\frac {x\,{\mathrm {e}}^a\,\left (\cos \left (d\right )-\sin \left (d\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16}-\frac {{\mathrm {e}}^a\,\left (\cos \left (2\,b\,x\right )-\sin \left (2\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )-\sin \left (3\,d\right )\,1{}\mathrm {i}\right )}{64\,b}-\frac {{\mathrm {e}}^a\,\left (\cos \left (4\,b\,x\right )+\sin \left (4\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (3\,d\right )+\sin \left (3\,d\right )\,1{}\mathrm {i}\right )}{128\,b}+\frac {{\mathrm {e}}^a\,\left (\cos \left (4\,b\,x\right )-\sin \left (4\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (5\,d\right )-\sin \left (5\,d\right )\,1{}\mathrm {i}\right )}{128\,b}+\frac {{\mathrm {e}}^a\,\left (\cos \left (6\,b\,x\right )+\sin \left (6\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (5\,d\right )+\sin \left (5\,d\right )\,1{}\mathrm {i}\right )}{192\,b}-\frac {{\mathrm {e}}^a\,\left (\cos \left (2\,b\,x\right )+\sin \left (2\,b\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (d\right )+\sin \left (d\right )\,1{}\mathrm {i}\right )}{32\,b} \] Input:

int(cos(d + b*x)^2*exp(a + b*x*1i)*sin(d + b*x)^3,x)
 

Output:

(x*exp(a)*(cos(d) - sin(d)*1i)*1i)/16 - (exp(a)*(cos(2*b*x) - sin(2*b*x)*1 
i)*(cos(3*d) - sin(3*d)*1i))/(64*b) - (exp(a)*(cos(4*b*x) + sin(4*b*x)*1i) 
*(cos(3*d) + sin(3*d)*1i))/(128*b) + (exp(a)*(cos(4*b*x) - sin(4*b*x)*1i)* 
(cos(5*d) - sin(5*d)*1i))/(128*b) + (exp(a)*(cos(6*b*x) + sin(6*b*x)*1i)*( 
cos(5*d) + sin(5*d)*1i))/(192*b) - (exp(a)*(cos(2*b*x) + sin(2*b*x)*1i)*(c 
os(d) + sin(d)*1i))/(32*b)
 

Reduce [F]

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

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

Output:

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