Integrand size = 30, antiderivative size = 142 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {e^{a-i d-2 i (d+b x)} (f-i g)^3}{16 b}-\frac {3 e^{a-i d+2 i (d+b x)} (f-i g) (f+i g)^2}{16 b}+\frac {e^{a-i d+4 i (d+b x)} (f+i g)^3}{32 b}+\frac {3}{8} e^{a-i d} (i f-g) (f-i g)^2 x \] Output:
1/16*exp(a-I*d-2*I*(b*x+d))*(f-I*g)^3/b-3/16*exp(a-I*d+2*I*(b*x+d))*(f-I*g )*(f+I*g)^2/b+1/32*exp(a-I*d+4*I*(b*x+d))*(f+I*g)^3/b+3/8*exp(a-I*d)*(I*f- g)*(f-I*g)^2*x
Time = 0.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {e^{a-3 i d} \left (2 e^{-2 i b x} (f-i g)^3-6 e^{2 i (2 d+b x)} (f-i g) (f+i g)^2+e^{6 i d+4 i b x} (f+i g)^3+12 e^{2 i d} (-i f+g) (i f+g)^2 (d+b x)\right )}{32 b} \] Input:
Integrate[E^(a + I*b*x)*(g*Cos[d + b*x] + f*Sin[d + b*x])^3,x]
Output:
(E^(a - (3*I)*d)*((2*(f - I*g)^3)/E^((2*I)*b*x) - 6*E^((2*I)*(2*d + b*x))* (f - I*g)*(f + I*g)^2 + E^((6*I)*d + (4*I)*b*x)*(f + I*g)^3 + 12*E^((2*I)* d)*((-I)*f + g)*(I*f + g)^2*(d + b*x)))/(32*b)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(433\) vs. \(2(142)=284\).
Time = 0.81 (sec) , antiderivative size = 433, normalized size of antiderivative = 3.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.
| \(\displaystyle \int e^{a+i b x} (f \sin (b x+d)+g \cos (b x+d))^3 \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (f^3 e^{a+i b x} \sin ^3(b x+d)+3 f^2 g e^{a+i b x} \sin ^2(b x+d) \cos (b x+d)+3 f g^2 e^{a+i b x} \sin (b x+d) \cos ^2(b x+d)+g^3 e^{a+i b x} \cos ^3(b x+d)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 f^3 e^{a+2 i b x+i d}}{16 b}+\frac {i f^3 e^{a+i b x} \sin ^3(b x+d)}{8 b}-\frac {3 f^3 e^{a+i b x} \sin ^2(b x+d) \cos (b x+d)}{8 b}-\frac {3 i f^2 g e^{a+2 i b x+i d}}{16 b}-\frac {9 f^2 g e^{a+i b x} \sin (3 b x+3 d)}{32 b}-\frac {3 i f^2 g e^{a+i b x} \cos (3 b x+3 d)}{32 b}-\frac {3 f g^2 e^{a+2 i b x+i d}}{16 b}+\frac {3 i f g^2 e^{a+i b x} \sin (3 b x+3 d)}{32 b}-\frac {9 f g^2 e^{a+i b x} \cos (3 b x+3 d)}{32 b}-\frac {3 i g^3 e^{a+2 i b x+i d}}{16 b}+\frac {i g^3 e^{a+i b x} \cos ^3(b x+d)}{8 b}+\frac {3 g^3 e^{a+i b x} \sin (b x+d) \cos ^2(b x+d)}{8 b}+\frac {3}{8} i f^3 x e^{a-i d}+\frac {3}{8} f^2 g x e^{a-i d}+\frac {3}{8} i f g^2 x e^{a-i d}+\frac {3}{8} g^3 x e^{a-i d}\) |
Input:
Int[E^(a + I*b*x)*(g*Cos[d + b*x] + f*Sin[d + b*x])^3,x]
Output:
(-3*E^(a + I*d + (2*I)*b*x)*f^3)/(16*b) - (((3*I)/16)*E^(a + I*d + (2*I)*b *x)*f^2*g)/b - (3*E^(a + I*d + (2*I)*b*x)*f*g^2)/(16*b) - (((3*I)/16)*E^(a + I*d + (2*I)*b*x)*g^3)/b + ((3*I)/8)*E^(a - I*d)*f^3*x + (3*E^(a - I*d)* f^2*g*x)/8 + ((3*I)/8)*E^(a - I*d)*f*g^2*x + (3*E^(a - I*d)*g^3*x)/8 + ((I /8)*E^(a + I*b*x)*g^3*Cos[d + b*x]^3)/b - (((3*I)/32)*E^(a + I*b*x)*f^2*g* Cos[3*d + 3*b*x])/b - (9*E^(a + I*b*x)*f*g^2*Cos[3*d + 3*b*x])/(32*b) + (3 *E^(a + I*b*x)*g^3*Cos[d + b*x]^2*Sin[d + b*x])/(8*b) - (3*E^(a + I*b*x)*f ^3*Cos[d + b*x]*Sin[d + b*x]^2)/(8*b) + ((I/8)*E^(a + I*b*x)*f^3*Sin[d + b *x]^3)/b - (9*E^(a + I*b*x)*f^2*g*Sin[3*d + 3*b*x])/(32*b) + (((3*I)/32)*E ^(a + I*b*x)*f*g^2*Sin[3*d + 3*b*x])/b
Time = 3.16 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.39
| method | result | size |
| parallelrisch | \(-\frac {{\mathrm e}^{i b x +a} \left (\left (3 i f^{2} g -i g^{3}-3 f^{3}+9 f \,g^{2}\right ) \cos \left (3 b x +3 d \right )+\left (i f^{3}-3 i f \,g^{2}+9 f^{2} g -3 g^{3}\right ) \sin \left (3 b x +3 d \right )+\left (\left (-12 i b x -45\right ) f^{3}+45 \left (-\frac {4 b x}{15}+i\right ) g \,f^{2}+\left (-12 i b x +135\right ) g^{2} f -15 \left (\frac {4 b x}{5}+i\right ) g^{3}\right ) \cos \left (b x +d \right )+12 \left (\left (-b x +\frac {19 i}{4}\right ) f^{3}+\left (i b x +\frac {11}{4}\right ) g \,f^{2}-\frac {41 \left (\frac {4 b x}{41}+i\right ) g^{2} f}{4}+\left (i b x -\frac {9}{4}\right ) g^{3}\right ) \sin \left (b x +d \right )\right )}{32 b}\) | \(197\) |
| norman | \(\frac {\left (\frac {3}{8} f^{2} g +\frac {3}{8} g^{3}+\frac {3}{8} i f^{3}+\frac {3}{8} i f \,g^{2}\right ) x \,{\mathrm e}^{i b x +a}+\left (-\frac {3}{8} f^{2} g -\frac {3}{8} g^{3}-\frac {3}{8} i f^{3}-\frac {3}{8} i f \,g^{2}\right ) x \,{\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}+\left (-\frac {3}{8} f^{2} g -\frac {3}{8} g^{3}-\frac {3}{8} i f^{3}-\frac {3}{8} i f \,g^{2}\right ) x \,{\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{6}+\left (\frac {3}{8} f^{2} g +\frac {3}{8} g^{3}+\frac {3}{8} i f^{3}+\frac {3}{8} i f \,g^{2}\right ) x \,{\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}-\frac {\left (-3 i f^{2} g +5 i g^{3}+3 f^{3}+3 f \,g^{2}\right ) {\mathrm e}^{i b x +a}}{8 b}-\frac {\left (-15 i f^{2} g +9 i g^{3}+15 f^{3}-33 f \,g^{2}\right ) {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}}{8 b}+\frac {\left (-15 i f^{2} g +9 i g^{3}+15 f^{3}-33 f \,g^{2}\right ) {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}}{8 b}-\frac {\left (-i f^{3}+3 i f \,g^{2}-9 f^{2} g +3 g^{3}\right ) {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}}{b}+\frac {\left (-3 i f^{2} g +5 i g^{3}+3 f^{3}+3 f \,g^{2}\right ) {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{6}}{8 b}-\frac {3 i \left (i f^{3}+i f \,g^{2}+f^{2} g +g^{3}\right ) x \,{\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )}{4}-\frac {3 i \left (i f^{3}+i f \,g^{2}+f^{2} g +g^{3}\right ) x \,{\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}}{2}-\frac {3 i \left (i f^{3}+i f \,g^{2}+f^{2} g +g^{3}\right ) x \,{\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{5}}{4}}{\left (1+\tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(558\) |
| orering | \(-\frac {\left (-4 b x +i\right ) {\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}}{4 b}+\frac {i \left (b x +i\right ) \left (i b \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}+3 \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2} \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )\right )}{4 b^{2}}-\frac {\left (-4 b x +i\right ) \left (-b^{2} {\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}+6 i b \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2} \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )+6 \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right ) \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )^{2}+3 \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2} \left (-g \,b^{2} \cos \left (b x +d \right )-f \,b^{2} \sin \left (b x +d \right )\right )\right )}{16 b^{3}}+\frac {i x \left (-i b^{3} {\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}-9 b^{2} {\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2} \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )+18 i b \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right ) \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )^{2}+9 i b \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2} \left (-g \,b^{2} \cos \left (b x +d \right )-f \,b^{2} \sin \left (b x +d \right )\right )+6 \,{\mathrm e}^{i b x +a} \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )^{3}+18 \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right ) \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right ) \left (-g \,b^{2} \cos \left (b x +d \right )-f \,b^{2} \sin \left (b x +d \right )\right )+3 \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2} \left (g \,b^{3} \sin \left (b x +d \right )-f \,b^{3} \cos \left (b x +d \right )\right )\right )}{16 b^{3}}\) | \(690\) |
| parts | \(\text {Expression too large to display}\) | \(1262\) |
Input:
int(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x,method=_RETURNVERBOSE)
Output:
-1/32*exp(a+I*b*x)*((3*I*f^2*g-I*g^3-3*f^3+9*f*g^2)*cos(3*b*x+3*d)+(-3*I*f *g^2+I*f^3+9*f^2*g-3*g^3)*sin(3*b*x+3*d)+((-12*I*x*b-45)*f^3+45*(-4/15*b*x +I)*g*f^2+(-12*I*x*b+135)*g^2*f-15*(4/5*b*x+I)*g^3)*cos(b*x+d)+12*((-b*x+1 9/4*I)*f^3+(I*b*x+11/4)*g*f^2-41/4*(4/41*b*x+I)*g^2*f+(I*b*x-9/4)*g^3)*sin (b*x+d))/b
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.05 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=-\frac {{\left (12 \, {\left (-i \, b f^{3} - b f^{2} g - i \, b f g^{2} - b g^{3}\right )} x e^{\left (2 i \, b x + a + i \, d\right )} - {\left (f^{3} + 3 i \, f^{2} g - 3 \, f g^{2} - i \, g^{3}\right )} e^{\left (6 i \, b x + a + 5 i \, d\right )} + 6 \, {\left (f^{3} + i \, f^{2} g + f g^{2} + i \, g^{3}\right )} e^{\left (4 i \, b x + a + 3 i \, d\right )} - 2 \, {\left (f^{3} - 3 i \, f^{2} g - 3 \, f g^{2} + i \, g^{3}\right )} e^{\left (a - i \, d\right )}\right )} e^{\left (-2 i \, b x - 2 i \, d\right )}}{32 \, b} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm="fricas" )
Output:
-1/32*(12*(-I*b*f^3 - b*f^2*g - I*b*f*g^2 - b*g^3)*x*e^(2*I*b*x + a + I*d) - (f^3 + 3*I*f^2*g - 3*f*g^2 - I*g^3)*e^(6*I*b*x + a + 5*I*d) + 6*(f^3 + I*f^2*g + f*g^2 + I*g^3)*e^(4*I*b*x + a + 3*I*d) - 2*(f^3 - 3*I*f^2*g - 3* f*g^2 + I*g^3)*e^(a - I*d))*e^(-2*I*b*x - 2*I*d)/b
Time = 0.41 (sec) , antiderivative size = 609, normalized size of antiderivative = 4.29 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {x \left (3 i f^{3} e^{a} + 3 f^{2} g e^{a} + 3 i f g^{2} e^{a} + 3 g^{3} e^{a}\right ) e^{- i d}}{8} + \begin {cases} \frac {\left (\left (512 b^{2} f^{3} e^{a} - 1536 i b^{2} f^{2} g e^{a} - 1536 b^{2} f g^{2} e^{a} + 512 i b^{2} g^{3} e^{a}\right ) e^{- 2 i b x} + \left (- 1536 b^{2} f^{3} e^{a} e^{4 i d} - 1536 i b^{2} f^{2} g e^{a} e^{4 i d} - 1536 b^{2} f g^{2} e^{a} e^{4 i d} - 1536 i b^{2} g^{3} e^{a} e^{4 i d}\right ) e^{2 i b x} + \left (256 b^{2} f^{3} e^{a} e^{6 i d} + 768 i b^{2} f^{2} g e^{a} e^{6 i d} - 768 b^{2} f g^{2} e^{a} e^{6 i d} - 256 i b^{2} g^{3} e^{a} e^{6 i d}\right ) e^{4 i b x}\right ) e^{- 3 i d}}{8192 b^{3}} & \text {for}\: b^{3} e^{3 i d} \neq 0 \\x \left (- \frac {\left (3 i f^{3} e^{a} + 3 f^{2} g e^{a} + 3 i f g^{2} e^{a} + 3 g^{3} e^{a}\right ) e^{- i d}}{8} + \frac {\left (i f^{3} e^{a} e^{6 i d} - 3 i f^{3} e^{a} e^{4 i d} + 3 i f^{3} e^{a} e^{2 i d} - i f^{3} e^{a} - 3 f^{2} g e^{a} e^{6 i d} + 3 f^{2} g e^{a} e^{4 i d} + 3 f^{2} g e^{a} e^{2 i d} - 3 f^{2} g e^{a} - 3 i f g^{2} e^{a} e^{6 i d} - 3 i f g^{2} e^{a} e^{4 i d} + 3 i f g^{2} e^{a} e^{2 i d} + 3 i f g^{2} e^{a} + g^{3} e^{a} e^{6 i d} + 3 g^{3} e^{a} e^{4 i d} + 3 g^{3} e^{a} e^{2 i d} + g^{3} e^{a}\right ) e^{- 3 i d}}{8}\right ) & \text {otherwise} \end {cases} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))**3,x)
Output:
x*(3*I*f**3*exp(a) + 3*f**2*g*exp(a) + 3*I*f*g**2*exp(a) + 3*g**3*exp(a))* exp(-I*d)/8 + Piecewise((((512*b**2*f**3*exp(a) - 1536*I*b**2*f**2*g*exp(a ) - 1536*b**2*f*g**2*exp(a) + 512*I*b**2*g**3*exp(a))*exp(-2*I*b*x) + (-15 36*b**2*f**3*exp(a)*exp(4*I*d) - 1536*I*b**2*f**2*g*exp(a)*exp(4*I*d) - 15 36*b**2*f*g**2*exp(a)*exp(4*I*d) - 1536*I*b**2*g**3*exp(a)*exp(4*I*d))*exp (2*I*b*x) + (256*b**2*f**3*exp(a)*exp(6*I*d) + 768*I*b**2*f**2*g*exp(a)*ex p(6*I*d) - 768*b**2*f*g**2*exp(a)*exp(6*I*d) - 256*I*b**2*g**3*exp(a)*exp( 6*I*d))*exp(4*I*b*x))*exp(-3*I*d)/(8192*b**3), Ne(b**3*exp(3*I*d), 0)), (x *(-(3*I*f**3*exp(a) + 3*f**2*g*exp(a) + 3*I*f*g**2*exp(a) + 3*g**3*exp(a)) *exp(-I*d)/8 + (I*f**3*exp(a)*exp(6*I*d) - 3*I*f**3*exp(a)*exp(4*I*d) + 3* I*f**3*exp(a)*exp(2*I*d) - I*f**3*exp(a) - 3*f**2*g*exp(a)*exp(6*I*d) + 3* f**2*g*exp(a)*exp(4*I*d) + 3*f**2*g*exp(a)*exp(2*I*d) - 3*f**2*g*exp(a) - 3*I*f*g**2*exp(a)*exp(6*I*d) - 3*I*f*g**2*exp(a)*exp(4*I*d) + 3*I*f*g**2*e xp(a)*exp(2*I*d) + 3*I*f*g**2*exp(a) + g**3*exp(a)*exp(6*I*d) + 3*g**3*exp (a)*exp(4*I*d) + 3*g**3*exp(a)*exp(2*I*d) + g**3*exp(a))*exp(-3*I*d)/8), T rue))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (95) = 190\).
Time = 0.06 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.84 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=-\frac {{\left (12 \, {\left (-i \, b \cos \left (d\right ) e^{a} - b e^{a} \sin \left (d\right )\right )} x - \cos \left (4 \, b x + 3 \, d\right ) e^{a} - 2 \, \cos \left (2 \, b x + 3 \, d\right ) e^{a} + 6 \, \cos \left (2 \, b x + d\right ) e^{a} - i \, e^{a} \sin \left (4 \, b x + 3 \, d\right ) + 2 i \, e^{a} \sin \left (2 \, b x + 3 \, d\right ) + 6 i \, e^{a} \sin \left (2 \, b x + d\right )\right )} f^{3}}{32 \, b} + \frac {3 \, {\left (4 \, {\left (b \cos \left (d\right ) e^{a} - i \, b e^{a} \sin \left (d\right )\right )} x + i \, \cos \left (4 \, b x + 3 \, d\right ) e^{a} - 2 i \, \cos \left (2 \, b x + 3 \, d\right ) e^{a} - 2 i \, \cos \left (2 \, b x + d\right ) e^{a} - e^{a} \sin \left (4 \, b x + 3 \, d\right ) - 2 \, e^{a} \sin \left (2 \, b x + 3 \, d\right ) + 2 \, e^{a} \sin \left (2 \, b x + d\right )\right )} f^{2} g}{32 \, b} + \frac {3 \, {\left (4 \, {\left (i \, b \cos \left (d\right ) e^{a} + b e^{a} \sin \left (d\right )\right )} x - \cos \left (4 \, b x + 3 \, d\right ) e^{a} - 2 \, \cos \left (2 \, b x + 3 \, d\right ) e^{a} - 2 \, \cos \left (2 \, b x + d\right ) e^{a} - i \, e^{a} \sin \left (4 \, b x + 3 \, d\right ) + 2 i \, e^{a} \sin \left (2 \, b x + 3 \, d\right ) - 2 i \, e^{a} \sin \left (2 \, b x + d\right )\right )} f g^{2}}{32 \, b} + \frac {{\left (12 \, {\left (b \cos \left (d\right ) e^{a} - i \, b e^{a} \sin \left (d\right )\right )} x - i \, \cos \left (4 \, b x + 3 \, d\right ) e^{a} + 2 i \, \cos \left (2 \, b x + 3 \, d\right ) e^{a} - 6 i \, \cos \left (2 \, b x + d\right ) e^{a} + e^{a} \sin \left (4 \, b x + 3 \, d\right ) + 2 \, e^{a} \sin \left (2 \, b x + 3 \, d\right ) + 6 \, e^{a} \sin \left (2 \, b x + d\right )\right )} g^{3}}{32 \, b} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm="maxima" )
Output:
-1/32*(12*(-I*b*cos(d)*e^a - b*e^a*sin(d))*x - cos(4*b*x + 3*d)*e^a - 2*co s(2*b*x + 3*d)*e^a + 6*cos(2*b*x + d)*e^a - I*e^a*sin(4*b*x + 3*d) + 2*I*e ^a*sin(2*b*x + 3*d) + 6*I*e^a*sin(2*b*x + d))*f^3/b + 3/32*(4*(b*cos(d)*e^ a - I*b*e^a*sin(d))*x + I*cos(4*b*x + 3*d)*e^a - 2*I*cos(2*b*x + 3*d)*e^a - 2*I*cos(2*b*x + d)*e^a - e^a*sin(4*b*x + 3*d) - 2*e^a*sin(2*b*x + 3*d) + 2*e^a*sin(2*b*x + d))*f^2*g/b + 3/32*(4*(I*b*cos(d)*e^a + b*e^a*sin(d))*x - cos(4*b*x + 3*d)*e^a - 2*cos(2*b*x + 3*d)*e^a - 2*cos(2*b*x + d)*e^a - I*e^a*sin(4*b*x + 3*d) + 2*I*e^a*sin(2*b*x + 3*d) - 2*I*e^a*sin(2*b*x + d) )*f*g^2/b + 1/32*(12*(b*cos(d)*e^a - I*b*e^a*sin(d))*x - I*cos(4*b*x + 3*d )*e^a + 2*I*cos(2*b*x + 3*d)*e^a - 6*I*cos(2*b*x + d)*e^a + e^a*sin(4*b*x + 3*d) + 2*e^a*sin(2*b*x + 3*d) + 6*e^a*sin(2*b*x + d))*g^3/b
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (95) = 190\).
Time = 0.28 (sec) , antiderivative size = 482, normalized size of antiderivative = 3.39 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=-\frac {24 \, {\left (-i \, f^{3} - f^{2} g - i \, f g^{2} - g^{3}\right )} {\left (b x + d\right )} \cos \left (d\right ) e^{a} - 24 \, {\left (f^{3} - i \, f^{2} g + f g^{2} - i \, g^{3}\right )} {\left (b x + d\right )} e^{a} \sin \left (d\right ) - {\left ({\left (f^{3} + 3 i \, f^{2} g - 3 \, f g^{2} - i \, g^{3}\right )} e^{\left (4 i \, b x + 3 i \, d\right )} - {\left (f^{3} + 3 i \, f^{2} g - 3 \, f g^{2} - i \, g^{3}\right )} e^{\left (-4 i \, b x - 3 i \, d\right )}\right )} e^{a} + i \, {\left ({\left (i \, f^{3} - 3 \, f^{2} g - 3 i \, f g^{2} + g^{3}\right )} e^{\left (4 i \, b x + 3 i \, d\right )} - {\left (-i \, f^{3} + 3 \, f^{2} g + 3 i \, f g^{2} - g^{3}\right )} e^{\left (-4 i \, b x - 3 i \, d\right )}\right )} e^{a} + 2 \, {\left ({\left (f^{3} - 3 i \, f^{2} g - 3 \, f g^{2} + i \, g^{3}\right )} e^{\left (2 i \, b x + 3 i \, d\right )} - {\left (f^{3} - 3 i \, f^{2} g - 3 \, f g^{2} + i \, g^{3}\right )} e^{\left (-2 i \, b x - 3 i \, d\right )}\right )} e^{a} - 2 i \, {\left ({\left (-i \, f^{3} - 3 \, f^{2} g + 3 i \, f g^{2} + g^{3}\right )} e^{\left (2 i \, b x + 3 i \, d\right )} - {\left (i \, f^{3} + 3 \, f^{2} g - 3 i \, f g^{2} - g^{3}\right )} e^{\left (-2 i \, b x - 3 i \, d\right )}\right )} e^{a} + 6 \, {\left ({\left (f^{3} + i \, f^{2} g + f g^{2} + i \, g^{3}\right )} e^{\left (2 i \, b x + i \, d\right )} - {\left (f^{3} + i \, f^{2} g + f g^{2} + i \, g^{3}\right )} e^{\left (-2 i \, b x - i \, d\right )}\right )} e^{a} + 6 i \, {\left ({\left (-i \, f^{3} + f^{2} g - i \, f g^{2} + g^{3}\right )} e^{\left (2 i \, b x + i \, d\right )} - {\left (i \, f^{3} - f^{2} g + i \, f g^{2} - g^{3}\right )} e^{\left (-2 i \, b x - i \, d\right )}\right )} e^{a}}{64 \, b} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm="giac")
Output:
-1/64*(24*(-I*f^3 - f^2*g - I*f*g^2 - g^3)*(b*x + d)*cos(d)*e^a - 24*(f^3 - I*f^2*g + f*g^2 - I*g^3)*(b*x + d)*e^a*sin(d) - ((f^3 + 3*I*f^2*g - 3*f* g^2 - I*g^3)*e^(4*I*b*x + 3*I*d) - (f^3 + 3*I*f^2*g - 3*f*g^2 - I*g^3)*e^( -4*I*b*x - 3*I*d))*e^a + I*((I*f^3 - 3*f^2*g - 3*I*f*g^2 + g^3)*e^(4*I*b*x + 3*I*d) - (-I*f^3 + 3*f^2*g + 3*I*f*g^2 - g^3)*e^(-4*I*b*x - 3*I*d))*e^a + 2*((f^3 - 3*I*f^2*g - 3*f*g^2 + I*g^3)*e^(2*I*b*x + 3*I*d) - (f^3 - 3*I *f^2*g - 3*f*g^2 + I*g^3)*e^(-2*I*b*x - 3*I*d))*e^a - 2*I*((-I*f^3 - 3*f^2 *g + 3*I*f*g^2 + g^3)*e^(2*I*b*x + 3*I*d) - (I*f^3 + 3*f^2*g - 3*I*f*g^2 - g^3)*e^(-2*I*b*x - 3*I*d))*e^a + 6*((f^3 + I*f^2*g + f*g^2 + I*g^3)*e^(2* I*b*x + I*d) - (f^3 + I*f^2*g + f*g^2 + I*g^3)*e^(-2*I*b*x - I*d))*e^a + 6 *I*((-I*f^3 + f^2*g - I*f*g^2 + g^3)*e^(2*I*b*x + I*d) - (I*f^3 - f^2*g + I*f*g^2 - g^3)*e^(-2*I*b*x - I*d))*e^a)/b
Time = 16.57 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.77 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {{\mathrm {e}}^{a-d\,3{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,{\left (f-g\,1{}\mathrm {i}\right )}^3}{16\,b}+\frac {{\mathrm {e}}^{a+d\,3{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,{\left (f+g\,1{}\mathrm {i}\right )}^3}{32\,b}-\frac {{\mathrm {e}}^{a+d\,1{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,\left (g+f\,1{}\mathrm {i}\right )\,{\left (-g+f\,1{}\mathrm {i}\right )}^2\,3{}\mathrm {i}}{16\,b}-\frac {x\,{\mathrm {e}}^{a-d\,1{}\mathrm {i}}\,\left (f+g\,1{}\mathrm {i}\right )\,{\left (g+f\,1{}\mathrm {i}\right )}^2\,3{}\mathrm {i}}{8} \] Input:
int(exp(a + b*x*1i)*(g*cos(d + b*x) + f*sin(d + b*x))^3,x)
Output:
(exp(a - d*3i - b*x*2i)*(f - g*1i)^3)/(16*b) + (exp(a + d*3i + b*x*4i)*(f + g*1i)^3)/(32*b) - (exp(a + d*1i + b*x*2i)*(f*1i + g)*(f*1i - g)^2*3i)/(1 6*b) - (x*exp(a - d*1i)*(f + g*1i)*(f*1i + g)^2*3i)/8
Time = 0.16 (sec) , antiderivative size = 509, normalized size of antiderivative = 3.58 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {e^{b i x +a} \left (-3 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) b \,f^{2} g i x +3 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} b f \,g^{2} i x -3 \cos \left (b x +d \right )^{3} f^{3}-3 \sin \left (b x +d \right )^{3} g^{3}+3 \cos \left (b x +d \right )^{3} b \,f^{3} i x +3 \cos \left (b x +d \right )^{3} b \,f^{2} g x +3 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) b \,f^{3} x +3 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} b \,g^{3} x +6 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} f^{2} g i +3 \sin \left (b x +d \right )^{3} b f \,g^{2} x -3 \sin \left (b x +d \right )^{3} b \,g^{3} i x +3 \cos \left (b x +d \right )^{3} b \,g^{3} x +3 \cos \left (b x +d \right )^{3} f^{2} g i +6 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} f \,g^{2}-6 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} g^{3} i +3 \sin \left (b x +d \right )^{3} b \,f^{3} x -3 \sin \left (b x +d \right )^{3} f \,g^{2} i +3 \cos \left (b x +d \right )^{3} b f \,g^{2} i x +3 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) b f \,g^{2} x -3 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) b \,g^{3} i x +3 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} b \,f^{3} i x +3 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} b \,f^{2} g x -3 \sin \left (b x +d \right )^{3} b \,f^{2} g i x -3 \cos \left (b x +d \right )^{3} f \,g^{2}-5 \cos \left (b x +d \right )^{3} g^{3} i -6 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} f^{3}+9 \sin \left (b x +d \right )^{3} f^{2} g +\sin \left (b x +d \right )^{3} f^{3} i \right )}{8 b} \] Input:
int(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x)
Output:
(e**(a + b*i*x)*(3*cos(b*x + d)**3*b*f**3*i*x + 3*cos(b*x + d)**3*b*f**2*g *x + 3*cos(b*x + d)**3*b*f*g**2*i*x + 3*cos(b*x + d)**3*b*g**3*x - 3*cos(b *x + d)**3*f**3 + 3*cos(b*x + d)**3*f**2*g*i - 3*cos(b*x + d)**3*f*g**2 - 5*cos(b*x + d)**3*g**3*i + 3*cos(b*x + d)**2*sin(b*x + d)*b*f**3*x - 3*cos (b*x + d)**2*sin(b*x + d)*b*f**2*g*i*x + 3*cos(b*x + d)**2*sin(b*x + d)*b* f*g**2*x - 3*cos(b*x + d)**2*sin(b*x + d)*b*g**3*i*x + 3*cos(b*x + d)*sin( b*x + d)**2*b*f**3*i*x + 3*cos(b*x + d)*sin(b*x + d)**2*b*f**2*g*x + 3*cos (b*x + d)*sin(b*x + d)**2*b*f*g**2*i*x + 3*cos(b*x + d)*sin(b*x + d)**2*b* g**3*x - 6*cos(b*x + d)*sin(b*x + d)**2*f**3 + 6*cos(b*x + d)*sin(b*x + d) **2*f**2*g*i + 6*cos(b*x + d)*sin(b*x + d)**2*f*g**2 - 6*cos(b*x + d)*sin( b*x + d)**2*g**3*i + 3*sin(b*x + d)**3*b*f**3*x - 3*sin(b*x + d)**3*b*f**2 *g*i*x + 3*sin(b*x + d)**3*b*f*g**2*x - 3*sin(b*x + d)**3*b*g**3*i*x + sin (b*x + d)**3*f**3*i + 9*sin(b*x + d)**3*f**2*g - 3*sin(b*x + d)**3*f*g**2* i - 3*sin(b*x + d)**3*g**3))/(8*b)