\(\int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx\) [8]

Optimal result

Integrand size = 32, antiderivative size = 194 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=\frac {i e^{2 (a-i d)-2 i (d+b x)} (f-i g)^4}{32 b}-\frac {i e^{2 (a-i d)+6 i (d+b x)} (f+i g)^4}{96 b}+\frac {e^{2 (a-i d)+4 i (d+b x)} (f+i g)^3 (i f+g)}{16 b}-\frac {3 i e^{2 (a-i d)+2 i (d+b x)} \left (f^2+g^2\right )^2}{16 b}-\frac {1}{4} e^{2 a-2 i d} (f-i g)^3 (f+i g) x \] Output:

1/32*I*exp(2*a-2*I*d-2*I*(b*x+d))*(f-I*g)^4/b-1/96*I*exp(2*a-2*I*d+6*I*(b* 
x+d))*(f+I*g)^4/b+1/16*exp(2*a-2*I*d+4*I*(b*x+d))*(f+I*g)^3*(I*f+g)/b-3/16 
*I*exp(2*a-2*I*d+2*I*(b*x+d))*(f^2+g^2)^2/b-1/4*exp(2*a-2*I*d)*(f-I*g)^3*( 
f+I*g)*x
 

Mathematica [A] (verified)

Time = 2.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.76 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=\frac {e^{2 a} \left (3 i e^{-2 i (2 d+b x)} (f-i g)^4-i e^{4 i d+6 i b x} (f+i g)^4+6 e^{2 i (d+2 b x)} (f+i g)^3 (i f+g)-18 i e^{2 i b x} \left (f^2+g^2\right )^2+24 e^{-2 i d} (-i f+g) (i f+g)^3 (d+b x)\right )}{96 b} \] Input:

Integrate[E^(2*(a + I*b*x))*(g*Cos[d + b*x] + f*Sin[d + b*x])^4,x]
 

Output:

(E^(2*a)*(((3*I)*(f - I*g)^4)/E^((2*I)*(2*d + b*x)) - I*E^((4*I)*d + (6*I) 
*b*x)*(f + I*g)^4 + 6*E^((2*I)*(d + 2*b*x))*(f + I*g)^3*(I*f + g) - (18*I) 
*E^((2*I)*b*x)*(f^2 + g^2)^2 + (24*((-I)*f + g)*(I*f + g)^3*(d + b*x))/E^( 
(2*I)*d)))/(96*b)
 

Rubi [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(625\) vs. \(2(194)=388\).

Time = 1.22 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.22, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {7292, 7293, 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.

\(\displaystyle \int e^{2 (a+i b x)} (f \sin (b x+d)+g \cos (b x+d))^4 \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int e^{2 a+2 i b x} (f \sin (b x+d)+g \cos (b x+d))^4dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (f^4 e^{2 a+2 i b x} \sin ^4(b x+d)+4 f^3 g e^{2 a+2 i b x} \sin ^3(b x+d) \cos (b x+d)+\frac {3}{2} f^2 g^2 e^{2 a+2 i b x} \sin ^2(2 b x+2 d)+4 f g^3 e^{2 a+2 i b x} \sin (b x+d) \cos ^3(b x+d)+g^4 e^{2 a+2 i b x} \cos ^4(b x+d)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i f^4 e^{2 (a+i d)+4 i b x}}{16 b}+\frac {i f^4 e^{2 a+2 i b x} \sin ^4(b x+d)}{6 b}-\frac {f^4 e^{2 a+2 i b x} \sin ^3(b x+d) \cos (b x+d)}{3 b}-\frac {f^3 g e^{2 (a+i d)+4 i b x}}{8 b}-\frac {i f^3 g e^{2 a+2 i b x} \sin (4 b x+4 d)}{12 b}+\frac {f^3 g e^{2 a+2 i b x} \cos (4 b x+4 d)}{6 b}+\frac {i f^2 g^2 e^{2 a+2 i b x} \sin ^2(2 b x+2 d)}{4 b}-\frac {f^2 g^2 e^{2 a+2 i b x} \sin (2 b x+2 d) \cos (2 b x+2 d)}{2 b}-\frac {f g^3 e^{2 (a+i d)+4 i b x}}{8 b}+\frac {i f g^3 e^{2 a+2 i b x} \sin (4 b x+4 d)}{12 b}-\frac {f g^3 e^{2 a+2 i b x} \cos (4 b x+4 d)}{6 b}-\frac {i g^4 e^{2 (a+i d)+4 i b x}}{16 b}+\frac {i g^4 e^{2 a+2 i b x} \cos ^4(b x+d)}{6 b}+\frac {g^4 e^{2 a+2 i b x} \sin (b x+d) \cos ^3(b x+d)}{3 b}-\frac {i f^4 e^{2 a+2 i b x}}{4 b}-\frac {i f^2 g^2 e^{2 a+2 i b x}}{2 b}-\frac {i g^4 e^{2 a+2 i b x}}{4 b}-\frac {1}{4} f^4 x e^{2 a-2 i d}+\frac {1}{2} i f^3 g x e^{2 a-2 i d}+\frac {1}{2} i f g^3 x e^{2 a-2 i d}+\frac {1}{4} g^4 x e^{2 a-2 i d}\)

Input:

Int[E^(2*(a + I*b*x))*(g*Cos[d + b*x] + f*Sin[d + b*x])^4,x]
 

Output:

((-1/4*I)*E^(2*a + (2*I)*b*x)*f^4)/b + ((I/16)*E^(2*(a + I*d) + (4*I)*b*x) 
*f^4)/b - (E^(2*(a + I*d) + (4*I)*b*x)*f^3*g)/(8*b) - ((I/2)*E^(2*a + (2*I 
)*b*x)*f^2*g^2)/b - (E^(2*(a + I*d) + (4*I)*b*x)*f*g^3)/(8*b) - ((I/4)*E^( 
2*a + (2*I)*b*x)*g^4)/b - ((I/16)*E^(2*(a + I*d) + (4*I)*b*x)*g^4)/b - (E^ 
(2*a - (2*I)*d)*f^4*x)/4 + (I/2)*E^(2*a - (2*I)*d)*f^3*g*x + (I/2)*E^(2*a 
- (2*I)*d)*f*g^3*x + (E^(2*a - (2*I)*d)*g^4*x)/4 + ((I/6)*E^(2*a + (2*I)*b 
*x)*g^4*Cos[d + b*x]^4)/b + (E^(2*a + (2*I)*b*x)*f^3*g*Cos[4*d + 4*b*x])/( 
6*b) - (E^(2*a + (2*I)*b*x)*f*g^3*Cos[4*d + 4*b*x])/(6*b) + (E^(2*a + (2*I 
)*b*x)*g^4*Cos[d + b*x]^3*Sin[d + b*x])/(3*b) - (E^(2*a + (2*I)*b*x)*f^4*C 
os[d + b*x]*Sin[d + b*x]^3)/(3*b) + ((I/6)*E^(2*a + (2*I)*b*x)*f^4*Sin[d + 
 b*x]^4)/b - (E^(2*a + (2*I)*b*x)*f^2*g^2*Cos[2*d + 2*b*x]*Sin[2*d + 2*b*x 
])/(2*b) + ((I/4)*E^(2*a + (2*I)*b*x)*f^2*g^2*Sin[2*d + 2*b*x]^2)/b - ((I/ 
12)*E^(2*a + (2*I)*b*x)*f^3*g*Sin[4*d + 4*b*x])/b + ((I/12)*E^(2*a + (2*I) 
*b*x)*f*g^3*Sin[4*d + 4*b*x])/b
 

Defintions of rubi rules used

rule 2009

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

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 6.81 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.29

method	result	size
parallelrisch	\(-\frac {\left (\left (\left (3 b x -2 i\right ) f^{4}+\left (-6 i b x +2\right ) g \,f^{3}-6 i f^{2} g^{2}+\left (-6 i b x -2\right ) g^{3} f -2 \left (\frac {3 b x}{2}+i\right ) g^{4}\right ) \cos \left (2 b x +2 d \right )+\left (\left (-3 i b x -\frac {1}{2}\right ) f^{4}+g \left (-6 b x +i\right ) f^{3}-6 f^{2} g^{2}+5 \left (-\frac {6 b x}{5}+i\right ) g^{3} f +3 \left (i b x -\frac {7}{6}\right ) g^{4}\right ) \sin \left (2 b x +2 d \right )+\left (2 f \,g^{3}-2 f^{3} g -\frac {1}{4} i f^{4}-\frac {1}{4} i g^{4}+\frac {3}{2} i f^{2} g^{2}\right ) \cos \left (4 b x +4 d \right )+\left (3 f^{2} g^{2}+i f^{3} g -i f \,g^{3}-\frac {1}{2} f^{4}-\frac {1}{2} g^{4}\right ) \sin \left (4 b x +4 d \right )+\frac {9 i \left (f^{2}+g^{2}\right )^{2}}{4}\right ) {\mathrm e}^{2 i b x +2 a}}{12 b}\)	\(250\)
norman	\(\frac {\left (\frac {1}{4} g^{4}+\frac {1}{2} i f^{3} g +\frac {1}{2} i f \,g^{3}-\frac {1}{4} f^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a}+\left (-2 i f^{3} g -2 i f \,g^{3}+f^{4}-g^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}+\left (-2 i f^{3} g -2 i f \,g^{3}+f^{4}-g^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{6}+\left (-\frac {5}{2} g^{4}-5 i f^{3} g -5 i f \,g^{3}+\frac {5}{2} f^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}+\left (\frac {1}{4} g^{4}+\frac {1}{2} i f^{3} g +\frac {1}{2} i f \,g^{3}-\frac {1}{4} f^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{8}+i \left (-2 i f^{3} g -2 i f \,g^{3}+f^{4}-g^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )+i \left (2 i f^{3} g +2 i f \,g^{3}-f^{4}+g^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{5}+i \left (-2 i f^{3} g -2 i f \,g^{3}+f^{4}-g^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}+i \left (2 i f^{3} g +2 i f \,g^{3}-f^{4}+g^{4}\right ) x \,{\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{7}-\frac {2 \left (i f^{4}+i g^{4}+2 f^{3} g -2 f \,g^{3}\right ) {\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}}{b}-\frac {2 \left (i f^{4}+i g^{4}+2 f^{3} g -2 f \,g^{3}\right ) {\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{6}}{b}+\frac {\left (-2 i f^{3} g -2 i f \,g^{3}+f^{4}+3 g^{4}\right ) {\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )}{2 b}-\frac {\left (-2 i f^{3} g -2 i f \,g^{3}+f^{4}+3 g^{4}\right ) {\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{7}}{2 b}-\frac {2 \left (2 i f^{4}+24 i f^{2} g^{2}+2 i g^{4}-20 f^{3} g +20 f \,g^{3}\right ) {\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}}{3 b}-\frac {\left (-26 i f^{3} g +38 i f \,g^{3}+13 f^{4}-96 f^{2} g^{2}+7 g^{4}\right ) {\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}}{6 b}+\frac {\left (-26 i f^{3} g +38 i f \,g^{3}+13 f^{4}-96 f^{2} g^{2}+7 g^{4}\right ) {\mathrm e}^{2 i b x +2 a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{5}}{6 b}}{\left (1+\tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}\right )^{4}}\)	\(809\)
orering	\(\text {Expression too large to display}\)	\(1444\)
parts	\(\text {Expression too large to display}\)	\(1930\)

Input:

int(exp(2*a+2*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x,method=_RETURNVERBOSE 
)
 

Output:

-1/12*(((-2*I+3*b*x)*f^4+(-6*I*b*x+2)*g*f^3-6*I*f^2*g^2+(-6*I*b*x-2)*g^3*f 
-2*(3/2*b*x+I)*g^4)*cos(2*b*x+2*d)+((-3*I*b*x-1/2)*f^4+g*(-6*b*x+I)*f^3-6* 
f^2*g^2+5*(-6/5*b*x+I)*g^3*f+3*(I*b*x-7/6)*g^4)*sin(2*b*x+2*d)+(2*f*g^3-2* 
f^3*g-1/4*I*f^4-1/4*I*g^4+3/2*I*f^2*g^2)*cos(4*b*x+4*d)+(3*f^2*g^2+I*f^3*g 
-I*f*g^3-1/2*f^4-1/2*g^4)*sin(4*b*x+4*d)+9/4*I*(f^2+g^2)^2)*exp(2*a+2*I*b* 
x)/b
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.08 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=-\frac {{\left (24 \, {\left (b f^{4} - 2 i \, b f^{3} g - 2 i \, b f g^{3} - b g^{4}\right )} x e^{\left (2 i \, b x + 2 \, a\right )} - {\left (-i \, f^{4} + 4 \, f^{3} g + 6 i \, f^{2} g^{2} - 4 \, f g^{3} - i \, g^{4}\right )} e^{\left (8 i \, b x + 2 \, a + 6 i \, d\right )} + 6 \, {\left (-i \, f^{4} + 2 \, f^{3} g + 2 \, f g^{3} + i \, g^{4}\right )} e^{\left (6 i \, b x + 2 \, a + 4 i \, d\right )} + 18 \, {\left (i \, f^{4} + 2 i \, f^{2} g^{2} + i \, g^{4}\right )} e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} + 3 \, {\left (-i \, f^{4} - 4 \, f^{3} g + 6 i \, f^{2} g^{2} + 4 \, f g^{3} - i \, g^{4}\right )} e^{\left (2 \, a - 2 i \, d\right )}\right )} e^{\left (-2 i \, b x - 2 i \, d\right )}}{96 \, b} \] Input:

integrate(exp(2*a+2*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x, algorithm="fri 
cas")
 

Output:

-1/96*(24*(b*f^4 - 2*I*b*f^3*g - 2*I*b*f*g^3 - b*g^4)*x*e^(2*I*b*x + 2*a) 
- (-I*f^4 + 4*f^3*g + 6*I*f^2*g^2 - 4*f*g^3 - I*g^4)*e^(8*I*b*x + 2*a + 6* 
I*d) + 6*(-I*f^4 + 2*f^3*g + 2*f*g^3 + I*g^4)*e^(6*I*b*x + 2*a + 4*I*d) + 
18*(I*f^4 + 2*I*f^2*g^2 + I*g^4)*e^(4*I*b*x + 2*a + 2*I*d) + 3*(-I*f^4 - 4 
*f^3*g + 6*I*f^2*g^2 + 4*f*g^3 - I*g^4)*e^(2*a - 2*I*d))*e^(-2*I*b*x - 2*I 
*d)/b
 

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 877, normalized size of antiderivative = 4.52 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=\frac {x \left (- f^{4} e^{2 a} + 2 i f^{3} g e^{2 a} + 2 i f g^{3} e^{2 a} + g^{4} e^{2 a}\right ) e^{- 2 i d}}{4} + \begin {cases} \frac {\left (\left (- 147456 i b^{3} f^{4} e^{2 a} e^{4 i d} - 294912 i b^{3} f^{2} g^{2} e^{2 a} e^{4 i d} - 147456 i b^{3} g^{4} e^{2 a} e^{4 i d}\right ) e^{2 i b x} + \left (49152 i b^{3} f^{4} e^{2 a} e^{6 i d} - 98304 b^{3} f^{3} g e^{2 a} e^{6 i d} - 98304 b^{3} f g^{3} e^{2 a} e^{6 i d} - 49152 i b^{3} g^{4} e^{2 a} e^{6 i d}\right ) e^{4 i b x} + \left (24576 i b^{3} f^{4} e^{2 a} + 98304 b^{3} f^{3} g e^{2 a} - 147456 i b^{3} f^{2} g^{2} e^{2 a} - 98304 b^{3} f g^{3} e^{2 a} + 24576 i b^{3} g^{4} e^{2 a}\right ) e^{- 2 i b x} + \left (- 8192 i b^{3} f^{4} e^{2 a} e^{8 i d} + 32768 b^{3} f^{3} g e^{2 a} e^{8 i d} + 49152 i b^{3} f^{2} g^{2} e^{2 a} e^{8 i d} - 32768 b^{3} f g^{3} e^{2 a} e^{8 i d} - 8192 i b^{3} g^{4} e^{2 a} e^{8 i d}\right ) e^{6 i b x}\right ) e^{- 4 i d}}{786432 b^{4}} & \text {for}\: b^{4} e^{4 i d} \neq 0 \\x \left (- \frac {\left (- f^{4} e^{2 a} + 2 i f^{3} g e^{2 a} + 2 i f g^{3} e^{2 a} + g^{4} e^{2 a}\right ) e^{- 2 i d}}{4} + \frac {\left (f^{4} e^{2 a} e^{8 i d} - 4 f^{4} e^{2 a} e^{6 i d} + 6 f^{4} e^{2 a} e^{4 i d} - 4 f^{4} e^{2 a} e^{2 i d} + f^{4} e^{2 a} + 4 i f^{3} g e^{2 a} e^{8 i d} - 8 i f^{3} g e^{2 a} e^{6 i d} + 8 i f^{3} g e^{2 a} e^{2 i d} - 4 i f^{3} g e^{2 a} - 6 f^{2} g^{2} e^{2 a} e^{8 i d} + 12 f^{2} g^{2} e^{2 a} e^{4 i d} - 6 f^{2} g^{2} e^{2 a} - 4 i f g^{3} e^{2 a} e^{8 i d} - 8 i f g^{3} e^{2 a} e^{6 i d} + 8 i f g^{3} e^{2 a} e^{2 i d} + 4 i f g^{3} e^{2 a} + g^{4} e^{2 a} e^{8 i d} + 4 g^{4} e^{2 a} e^{6 i d} + 6 g^{4} e^{2 a} e^{4 i d} + 4 g^{4} e^{2 a} e^{2 i d} + g^{4} e^{2 a}\right ) e^{- 4 i d}}{16}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(exp(2*a+2*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))**4,x)
 

Output:

x*(-f**4*exp(2*a) + 2*I*f**3*g*exp(2*a) + 2*I*f*g**3*exp(2*a) + g**4*exp(2 
*a))*exp(-2*I*d)/4 + Piecewise((((-147456*I*b**3*f**4*exp(2*a)*exp(4*I*d) 
- 294912*I*b**3*f**2*g**2*exp(2*a)*exp(4*I*d) - 147456*I*b**3*g**4*exp(2*a 
)*exp(4*I*d))*exp(2*I*b*x) + (49152*I*b**3*f**4*exp(2*a)*exp(6*I*d) - 9830 
4*b**3*f**3*g*exp(2*a)*exp(6*I*d) - 98304*b**3*f*g**3*exp(2*a)*exp(6*I*d) 
- 49152*I*b**3*g**4*exp(2*a)*exp(6*I*d))*exp(4*I*b*x) + (24576*I*b**3*f**4 
*exp(2*a) + 98304*b**3*f**3*g*exp(2*a) - 147456*I*b**3*f**2*g**2*exp(2*a) 
- 98304*b**3*f*g**3*exp(2*a) + 24576*I*b**3*g**4*exp(2*a))*exp(-2*I*b*x) + 
 (-8192*I*b**3*f**4*exp(2*a)*exp(8*I*d) + 32768*b**3*f**3*g*exp(2*a)*exp(8 
*I*d) + 49152*I*b**3*f**2*g**2*exp(2*a)*exp(8*I*d) - 32768*b**3*f*g**3*exp 
(2*a)*exp(8*I*d) - 8192*I*b**3*g**4*exp(2*a)*exp(8*I*d))*exp(6*I*b*x))*exp 
(-4*I*d)/(786432*b**4), Ne(b**4*exp(4*I*d), 0)), (x*(-(-f**4*exp(2*a) + 2* 
I*f**3*g*exp(2*a) + 2*I*f*g**3*exp(2*a) + g**4*exp(2*a))*exp(-2*I*d)/4 + ( 
f**4*exp(2*a)*exp(8*I*d) - 4*f**4*exp(2*a)*exp(6*I*d) + 6*f**4*exp(2*a)*ex 
p(4*I*d) - 4*f**4*exp(2*a)*exp(2*I*d) + f**4*exp(2*a) + 4*I*f**3*g*exp(2*a 
)*exp(8*I*d) - 8*I*f**3*g*exp(2*a)*exp(6*I*d) + 8*I*f**3*g*exp(2*a)*exp(2* 
I*d) - 4*I*f**3*g*exp(2*a) - 6*f**2*g**2*exp(2*a)*exp(8*I*d) + 12*f**2*g** 
2*exp(2*a)*exp(4*I*d) - 6*f**2*g**2*exp(2*a) - 4*I*f*g**3*exp(2*a)*exp(8*I 
*d) - 8*I*f*g**3*exp(2*a)*exp(6*I*d) + 8*I*f*g**3*exp(2*a)*exp(2*I*d) + 4* 
I*f*g**3*exp(2*a) + g**4*exp(2*a)*exp(8*I*d) + 4*g**4*exp(2*a)*exp(6*I*...
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 636 vs. \(2 (124) = 248\).

Time = 0.07 (sec) , antiderivative size = 636, normalized size of antiderivative = 3.28 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx =\text {Too large to display} \] Input:

integrate(exp(2*a+2*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x, algorithm="max 
ima")
 

Output:

-1/96*(24*(b*cos(2*d)*e^(2*a) - I*b*e^(2*a)*sin(2*d))*x + 18*I*cos(2*b*x)* 
e^(2*a) + I*cos(6*b*x + 4*d)*e^(2*a) - 6*I*cos(4*b*x + 2*d)*e^(2*a) - 3*I* 
cos(2*b*x + 4*d)*e^(2*a) - 18*e^(2*a)*sin(2*b*x) - e^(2*a)*sin(6*b*x + 4*d 
) + 6*e^(2*a)*sin(4*b*x + 2*d) - 3*e^(2*a)*sin(2*b*x + 4*d))*f^4/b - 1/24* 
(12*(-I*b*cos(2*d)*e^(2*a) - b*e^(2*a)*sin(2*d))*x - cos(6*b*x + 4*d)*e^(2 
*a) + 3*cos(4*b*x + 2*d)*e^(2*a) - 3*cos(2*b*x + 4*d)*e^(2*a) - I*e^(2*a)* 
sin(6*b*x + 4*d) + 3*I*e^(2*a)*sin(4*b*x + 2*d) + 3*I*e^(2*a)*sin(2*b*x + 
4*d))*f^3*g/b - 1/16*(6*I*cos(2*b*x)*e^(2*a) - I*cos(6*b*x + 4*d)*e^(2*a) 
+ 3*I*cos(2*b*x + 4*d)*e^(2*a) - 6*e^(2*a)*sin(2*b*x) + e^(2*a)*sin(6*b*x 
+ 4*d) + 3*e^(2*a)*sin(2*b*x + 4*d))*f^2*g^2/b + 1/24*(12*(I*b*cos(2*d)*e^ 
(2*a) + b*e^(2*a)*sin(2*d))*x - cos(6*b*x + 4*d)*e^(2*a) - 3*cos(4*b*x + 2 
*d)*e^(2*a) - 3*cos(2*b*x + 4*d)*e^(2*a) - I*e^(2*a)*sin(6*b*x + 4*d) - 3* 
I*e^(2*a)*sin(4*b*x + 2*d) + 3*I*e^(2*a)*sin(2*b*x + 4*d))*f*g^3/b + 1/96* 
(24*(b*cos(2*d)*e^(2*a) - I*b*e^(2*a)*sin(2*d))*x - 18*I*cos(2*b*x)*e^(2*a 
) - I*cos(6*b*x + 4*d)*e^(2*a) - 6*I*cos(4*b*x + 2*d)*e^(2*a) + 3*I*cos(2* 
b*x + 4*d)*e^(2*a) + 18*e^(2*a)*sin(2*b*x) + e^(2*a)*sin(6*b*x + 4*d) + 6* 
e^(2*a)*sin(4*b*x + 2*d) + 3*e^(2*a)*sin(2*b*x + 4*d))*g^4/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. 636 vs. \(2 (124) = 248\).

Time = 0.34 (sec) , antiderivative size = 636, normalized size of antiderivative = 3.28 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx =\text {Too large to display} \] Input:

integrate(exp(2*a+2*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x, algorithm="gia 
c")
 

Output:

-1/192*(48*(f^4 - 2*I*f^3*g - 2*I*f*g^3 - g^4)*(b*x + d)*cos(2*d)*e^(2*a) 
- 48*(I*f^4 + 2*f^3*g + 2*f*g^3 - I*g^4)*(b*x + d)*e^(2*a)*sin(2*d) - 36*( 
f^4 + 2*f^2*g^2 + g^4)*e^(2*a)*sin(2*b*x) + I*((f^4 + 4*I*f^3*g - 6*f^2*g^ 
2 - 4*I*f*g^3 + g^4)*e^(6*I*b*x + 4*I*d) + (f^4 + 4*I*f^3*g - 6*f^2*g^2 - 
4*I*f*g^3 + g^4)*e^(-6*I*b*x - 4*I*d))*e^(2*a) + ((I*f^4 - 4*f^3*g - 6*I*f 
^2*g^2 + 4*f*g^3 + I*g^4)*e^(6*I*b*x + 4*I*d) + (-I*f^4 + 4*f^3*g + 6*I*f^ 
2*g^2 - 4*f*g^3 - I*g^4)*e^(-6*I*b*x - 4*I*d))*e^(2*a) - 6*I*((f^4 + 2*I*f 
^3*g + 2*I*f*g^3 - g^4)*e^(4*I*b*x + 2*I*d) + (f^4 + 2*I*f^3*g + 2*I*f*g^3 
 - g^4)*e^(-4*I*b*x - 2*I*d))*e^(2*a) + 6*((-I*f^4 + 2*f^3*g + 2*f*g^3 + I 
*g^4)*e^(4*I*b*x + 2*I*d) + (I*f^4 - 2*f^3*g - 2*f*g^3 - I*g^4)*e^(-4*I*b* 
x - 2*I*d))*e^(2*a) - 3*I*((f^4 - 4*I*f^3*g - 6*f^2*g^2 + 4*I*f*g^3 + g^4) 
*e^(2*I*b*x + 4*I*d) + (f^4 - 4*I*f^3*g - 6*f^2*g^2 + 4*I*f*g^3 + g^4)*e^( 
-2*I*b*x - 4*I*d))*e^(2*a) + 3*((I*f^4 + 4*f^3*g - 6*I*f^2*g^2 - 4*f*g^3 + 
 I*g^4)*e^(2*I*b*x + 4*I*d) + (-I*f^4 - 4*f^3*g + 6*I*f^2*g^2 + 4*f*g^3 - 
I*g^4)*e^(-2*I*b*x - 4*I*d))*e^(2*a) + 18*I*((f^4 + 2*f^2*g^2 + g^4)*e^(2* 
I*b*x) + (f^4 + 2*f^2*g^2 + g^4)*e^(-2*I*b*x))*e^(2*a))/b
 

Mupad [B] (verification not implemented)

Time = 16.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.75 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=-\frac {{\mathrm {e}}^{2\,a+b\,x\,2{}\mathrm {i}}\,{\left (f^2+g^2\right )}^2\,3{}\mathrm {i}}{16\,b}+\frac {{\mathrm {e}}^{2\,a-d\,4{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,{\left (g+f\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{32\,b}-\frac {{\mathrm {e}}^{2\,a+d\,4{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,{\left (-g+f\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{96\,b}+\frac {{\mathrm {e}}^{2\,a+d\,2{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,\left (g+f\,1{}\mathrm {i}\right )\,{\left (-g+f\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{16\,b}-\frac {x\,{\mathrm {e}}^{2\,a-d\,2{}\mathrm {i}}\,{\left (f-g\,1{}\mathrm {i}\right )}^3\,\left (f+g\,1{}\mathrm {i}\right )}{4} \] Input:

int(exp(2*a + b*x*2i)*(g*cos(d + b*x) + f*sin(d + b*x))^4,x)
 

Output:

(exp(2*a - d*4i - b*x*2i)*(f*1i + g)^4*1i)/(32*b) - (exp(2*a + b*x*2i)*(f^ 
2 + g^2)^2*3i)/(16*b) - (exp(2*a + d*4i + b*x*6i)*(f*1i - g)^4*1i)/(96*b) 
+ (exp(2*a + d*2i + b*x*4i)*(f*1i + g)*(f*1i - g)^3*1i)/(16*b) - (x*exp(2* 
a - d*2i)*(f - g*1i)^3*(f + g*1i))/4
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 639, normalized size of antiderivative = 3.29 \[ \int e^{2 (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx =\text {Too large to display} \] Input:

int(exp(2*a+2*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x)
 

Output:

(e**(2*a + 2*b*i*x)*( - 6*cos(b*x + d)**4*b*f**4*x + 12*cos(b*x + d)**4*b* 
f**3*g*i*x + 12*cos(b*x + d)**4*b*f*g**3*i*x + 6*cos(b*x + d)**4*b*g**4*x 
- 3*cos(b*x + d)**4*f**4*i - 6*cos(b*x + d)**4*f**3*g - 6*cos(b*x + d)**4* 
f*g**3 - 9*cos(b*x + d)**4*g**4*i + 12*cos(b*x + d)**3*sin(b*x + d)*b*f**4 
*i*x + 24*cos(b*x + d)**3*sin(b*x + d)*b*f**3*g*x + 24*cos(b*x + d)**3*sin 
(b*x + d)*b*f*g**3*x - 12*cos(b*x + d)**3*sin(b*x + d)*b*g**4*i*x - 12*cos 
(b*x + d)**2*sin(b*x + d)**2*f**4*i - 24*cos(b*x + d)**2*sin(b*x + d)**2*f 
**3*g + 24*cos(b*x + d)**2*sin(b*x + d)**2*f*g**3 - 12*cos(b*x + d)**2*sin 
(b*x + d)**2*g**4*i + 12*cos(b*x + d)*sin(b*x + d)**3*b*f**4*i*x + 24*cos( 
b*x + d)*sin(b*x + d)**3*b*f**3*g*x + 24*cos(b*x + d)*sin(b*x + d)**3*b*f* 
g**3*x - 12*cos(b*x + d)*sin(b*x + d)**3*b*g**4*i*x - 8*cos(b*x + d)*sin(b 
*x + d)**3*f**4 + 16*cos(b*x + d)*sin(b*x + d)**3*f**3*g*i + 48*cos(b*x + 
d)*sin(b*x + d)**3*f**2*g**2 - 16*cos(b*x + d)*sin(b*x + d)**3*f*g**3*i - 
8*cos(b*x + d)*sin(b*x + d)**3*g**4 + 6*sin(b*x + d)**4*b*f**4*x - 12*sin( 
b*x + d)**4*b*f**3*g*i*x - 12*sin(b*x + d)**4*b*f*g**3*i*x - 6*sin(b*x + d 
)**4*b*g**4*x - 5*sin(b*x + d)**4*f**4*i + 14*sin(b*x + d)**4*f**3*g - 24* 
sin(b*x + d)**4*f**2*g**2*i - 2*sin(b*x + d)**4*f*g**3 + sin(b*x + d)**4*g 
**4*i))/(24*b)