Integrand size = 34, antiderivative size = 179 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {3 e^{\frac {5}{3} (a-i d)-\frac {4}{3} i (d+b x)} (f-i g)^3}{32 b}+\frac {9 e^{\frac {5}{3} (a-i d)+\frac {2}{3} i (d+b x)} (f-i g)^2 (f+i g)}{16 b}-\frac {9 e^{\frac {5}{3} (a-i d)+\frac {8}{3} i (d+b x)} (f-i g) (f+i g)^2}{64 b}+\frac {3 e^{\frac {5}{3} (a-i d)+\frac {14}{3} i (d+b x)} (f+i g)^3}{112 b} \] Output:
3/32*exp(5/3*a-5/3*I*d-4/3*I*(b*x+d))*(f-I*g)^3/b+9/16*exp(5/3*a-5/3*I*d+2 /3*I*(b*x+d))*(f-I*g)^2*(f+I*g)/b-9/64*exp(5/3*a-5/3*I*d+8/3*I*(b*x+d))*(f -I*g)*(f+I*g)^2/b+3/112*exp(5/3*a-5/3*I*d+14/3*I*(b*x+d))*(f+I*g)^3/b
Time = 0.88 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.67 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {3 e^{-\frac {1}{3} i (5 i a+9 d+4 b x)} \left (14 (f-i g)^3+84 e^{2 i (d+b x)} (f-i g)^2 (f+i g)-21 e^{4 i (d+b x)} (f-i g) (f+i g)^2+4 e^{6 i (d+b x)} (f+i g)^3\right )}{448 b} \] Input:
Integrate[E^((5*(a + I*b*x))/3)*(g*Cos[d + b*x] + f*Sin[d + b*x])^3,x]
Output:
(3*(14*(f - I*g)^3 + 84*E^((2*I)*(d + b*x))*(f - I*g)^2*(f + I*g) - 21*E^( (4*I)*(d + b*x))*(f - I*g)*(f + I*g)^2 + 4*E^((6*I)*(d + b*x))*(f + I*g)^3 ))/(448*b*E^((I/3)*((5*I)*a + 9*d + 4*b*x)))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(569\) vs. \(2(179)=358\).
Time = 0.99 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, 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.
| \(\displaystyle \int e^{\frac {5}{3} (a+i b x)} (f \sin (b x+d)+g \cos (b x+d))^3 \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int e^{\frac {5 a}{3}+\frac {5 i b x}{3}} (f \sin (b x+d)+g \cos (b x+d))^3dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (f^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^3(b x+d)+3 f^2 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^2(b x+d) \cos (b x+d)+3 f g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d) \cos ^2(b x+d)+g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos ^3(b x+d)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {15 i f^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^3(b x+d)}{56 b}-\frac {405 i f^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d)}{448 b}+\frac {243 f^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (b x+d)}{448 b}-\frac {27 f^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^2(b x+d) \cos (b x+d)}{56 b}-\frac {27 f^2 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d)}{64 b}-\frac {81 f^2 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (3 b x+3 d)}{224 b}-\frac {45 i f^2 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (b x+d)}{64 b}-\frac {45 i f^2 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (3 b x+3 d)}{224 b}-\frac {45 i f g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d)}{64 b}+\frac {45 i f g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (3 b x+3 d)}{224 b}+\frac {27 f g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (b x+d)}{64 b}-\frac {81 f g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (3 b x+3 d)}{224 b}-\frac {243 g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d)}{448 b}+\frac {15 i g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos ^3(b x+d)}{56 b}-\frac {405 i g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (b x+d)}{448 b}+\frac {27 g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d) \cos ^2(b x+d)}{56 b}\) |
Input:
Int[E^((5*(a + I*b*x))/3)*(g*Cos[d + b*x] + f*Sin[d + b*x])^3,x]
Output:
(243*E^((5*a)/3 + ((5*I)/3)*b*x)*f^3*Cos[d + b*x])/(448*b) - (((45*I)/64)* E^((5*a)/3 + ((5*I)/3)*b*x)*f^2*g*Cos[d + b*x])/b + (27*E^((5*a)/3 + ((5*I )/3)*b*x)*f*g^2*Cos[d + b*x])/(64*b) - (((405*I)/448)*E^((5*a)/3 + ((5*I)/ 3)*b*x)*g^3*Cos[d + b*x])/b + (((15*I)/56)*E^((5*a)/3 + ((5*I)/3)*b*x)*g^3 *Cos[d + b*x]^3)/b - (((45*I)/224)*E^((5*a)/3 + ((5*I)/3)*b*x)*f^2*g*Cos[3 *d + 3*b*x])/b - (81*E^((5*a)/3 + ((5*I)/3)*b*x)*f*g^2*Cos[3*d + 3*b*x])/( 224*b) - (((405*I)/448)*E^((5*a)/3 + ((5*I)/3)*b*x)*f^3*Sin[d + b*x])/b - (27*E^((5*a)/3 + ((5*I)/3)*b*x)*f^2*g*Sin[d + b*x])/(64*b) - (((45*I)/64)* E^((5*a)/3 + ((5*I)/3)*b*x)*f*g^2*Sin[d + b*x])/b - (243*E^((5*a)/3 + ((5* I)/3)*b*x)*g^3*Sin[d + b*x])/(448*b) + (27*E^((5*a)/3 + ((5*I)/3)*b*x)*g^3 *Cos[d + b*x]^2*Sin[d + b*x])/(56*b) - (27*E^((5*a)/3 + ((5*I)/3)*b*x)*f^3 *Cos[d + b*x]*Sin[d + b*x]^2)/(56*b) + (((15*I)/56)*E^((5*a)/3 + ((5*I)/3) *b*x)*f^3*Sin[d + b*x]^3)/b - (81*E^((5*a)/3 + ((5*I)/3)*b*x)*f^2*g*Sin[3* d + 3*b*x])/(224*b) + (((45*I)/224)*E^((5*a)/3 + ((5*I)/3)*b*x)*f*g^2*Sin[ 3*d + 3*b*x])/b
Time = 2.66 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71
| method | result | size |
| parallelrisch | \(-\frac {15 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \left (\left (-i g^{3}+3 i f^{2} g -\frac {9}{5} f^{3}+\frac {27}{5} f \,g^{2}\right ) \cos \left (3 b x +3 d \right )+\left (i f^{3}-3 i f \,g^{2}+\frac {27}{5} f^{2} g -\frac {9}{5} g^{3}\right ) \sin \left (3 b x +3 d \right )+\frac {21 \left (f^{2}+g^{2}\right ) \left (\left (i g -\frac {3 f}{5}\right ) \cos \left (b x +d \right )+\left (i f +\frac {3 g}{5}\right ) \sin \left (b x +d \right )\right )}{2}\right )}{224 b}\) | \(127\) |
| norman | \(\frac {\frac {3 \left (-135 i f^{2} g -95 i g^{3}+81 f^{3}+9 f \,g^{2}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{448 b}-\frac {3 \left (55 i f^{3}+255 i f \,g^{2}-207 f^{2} g +153 g^{3}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}}{112 b}-\frac {9 \left (45 i f^{3}+5 i f \,g^{2}+75 f^{2} g +3 g^{3}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )}{224 b}-\frac {9 \left (45 i f^{3}+5 i f \,g^{2}+75 f^{2} g +3 g^{3}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{5}}{224 b}-\frac {3 \left (-135 i f^{2} g -95 i g^{3}+81 f^{3}+9 f \,g^{2}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{6}}{448 b}-\frac {9 \left (-115 i f^{2} g +85 i g^{3}+69 f^{3}-291 f \,g^{2}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}}{448 b}+\frac {9 \left (-115 i f^{2} g +85 i g^{3}+69 f^{3}-291 f \,g^{2}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}}{448 b}}{\left (1+\tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}\right )^{3}}\) | \(362\) |
| parts | \(g^{3} \left (\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{224 b}+\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{224 b}-\frac {45 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{64 b}-\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{64 b}\right )+f^{3} \left (\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{224 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{224 b}+\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{64 b}-\frac {45 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{64 b}\right )+3 f \,g^{2} \left (-\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{224 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{224 b}+\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{64 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{64 b}\right )+3 f^{2} g \left (-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{224 b}-\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{224 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{64 b}-\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{64 b}\right )\) | \(394\) |
| default | \(-\frac {f^{3} \left (-\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{56 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{56 b}\right )}{4}+\frac {g^{3} \left (\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{56 b}+\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{56 b}\right )}{4}+\frac {3 g^{3} \left (-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{16 b}-\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{16 b}\right )}{4}+\frac {3 f^{3} \left (\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{16 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{16 b}\right )}{4}+\frac {3 f \,g^{2} \left (-\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{56 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{56 b}\right )}{4}+\frac {3 f \,g^{2} \left (\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{16 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{16 b}\right )}{4}+\frac {3 f^{2} g \left (-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (b x +d \right )}{16 b}-\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (b x +d \right )}{16 b}\right )}{4}-\frac {3 f^{2} g \left (\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (3 b x +3 d \right )}{56 b}+\frac {27 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (3 b x +3 d \right )}{56 b}\right )}{4}\) | \(422\) |
| orering | \(-\frac {75 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}}{56 b}-\frac {135 \left (\frac {5 i b \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}}{3}+3 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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 )}{224 b^{2}}-\frac {135 i \left (-\frac {25 b^{2} {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}}{9}+10 i b \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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 )}{224 b^{3}}+\frac {-\frac {375 i b^{3} {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{3}}{896}-\frac {2025 b^{2} {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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 )}{896}+\frac {1215 i b \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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}}{448}+\frac {1215 i b \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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 )}{896}+\frac {243 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )^{3}}{448}+\frac {729 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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 )}{448}+\frac {243 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \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 )}{896}}{b^{4}}\) | \(697\) |
Input:
int(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x,method=_RETURNVER BOSE)
Output:
-15/224*exp(5/3*a+5/3*I*b*x)*((-I*g^3+3*I*f^2*g-9/5*f^3+27/5*f*g^2)*cos(3* b*x+3*d)+(I*f^3-3*I*f*g^2+27/5*f^2*g-9/5*g^3)*sin(3*b*x+3*d)+21/2*(f^2+g^2 )*((I*g-3/5*f)*cos(b*x+d)+(I*f+3/5*g)*sin(b*x+d)))/b
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.83 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {3 \, {\left (4 \, {\left (f^{3} + 3 i \, f^{2} g - 3 \, f g^{2} - i \, g^{3}\right )} e^{\left (6 i \, b x + \frac {5}{3} \, a + \frac {13}{3} i \, d\right )} - 21 \, {\left (f^{3} + i \, f^{2} g + f g^{2} + i \, g^{3}\right )} e^{\left (4 i \, b x + \frac {5}{3} \, a + \frac {7}{3} i \, d\right )} + 84 \, {\left (f^{3} - i \, f^{2} g + f g^{2} - i \, g^{3}\right )} e^{\left (2 i \, b x + \frac {5}{3} \, a + \frac {1}{3} i \, d\right )} + 14 \, {\left (f^{3} - 3 i \, f^{2} g - 3 \, f g^{2} + i \, g^{3}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} i \, d\right )}\right )} e^{\left (-\frac {4}{3} i \, b x - \frac {4}{3} i \, d\right )}}{448 \, b} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm= "fricas")
Output:
3/448*(4*(f^3 + 3*I*f^2*g - 3*f*g^2 - I*g^3)*e^(6*I*b*x + 5/3*a + 13/3*I*d ) - 21*(f^3 + I*f^2*g + f*g^2 + I*g^3)*e^(4*I*b*x + 5/3*a + 7/3*I*d) + 84* (f^3 - I*f^2*g + f*g^2 - I*g^3)*e^(2*I*b*x + 5/3*a + 1/3*I*d) + 14*(f^3 - 3*I*f^2*g - 3*f*g^2 + I*g^3)*e^(5/3*a - 5/3*I*d))*e^(-4/3*I*b*x - 4/3*I*d) /b
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (151) = 302\).
Time = 0.65 (sec) , antiderivative size = 738, normalized size of antiderivative = 4.12 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\begin {cases} \frac {\left (\left (344064 b^{3} f^{3} e^{\frac {5 a}{3}} e^{i d} - 1032192 i b^{3} f^{2} g e^{\frac {5 a}{3}} e^{i d} - 1032192 b^{3} f g^{2} e^{\frac {5 a}{3}} e^{i d} + 344064 i b^{3} g^{3} e^{\frac {5 a}{3}} e^{i d}\right ) e^{- \frac {4 i b x}{3}} + \left (2064384 b^{3} f^{3} e^{\frac {5 a}{3}} e^{3 i d} - 2064384 i b^{3} f^{2} g e^{\frac {5 a}{3}} e^{3 i d} + 2064384 b^{3} f g^{2} e^{\frac {5 a}{3}} e^{3 i d} - 2064384 i b^{3} g^{3} e^{\frac {5 a}{3}} e^{3 i d}\right ) e^{\frac {2 i b x}{3}} + \left (- 516096 b^{3} f^{3} e^{\frac {5 a}{3}} e^{5 i d} - 516096 i b^{3} f^{2} g e^{\frac {5 a}{3}} e^{5 i d} - 516096 b^{3} f g^{2} e^{\frac {5 a}{3}} e^{5 i d} - 516096 i b^{3} g^{3} e^{\frac {5 a}{3}} e^{5 i d}\right ) e^{\frac {8 i b x}{3}} + \left (98304 b^{3} f^{3} e^{\frac {5 a}{3}} e^{7 i d} + 294912 i b^{3} f^{2} g e^{\frac {5 a}{3}} e^{7 i d} - 294912 b^{3} f g^{2} e^{\frac {5 a}{3}} e^{7 i d} - 98304 i b^{3} g^{3} e^{\frac {5 a}{3}} e^{7 i d}\right ) e^{\frac {14 i b x}{3}}\right ) e^{- 4 i d}}{3670016 b^{4}} & \text {for}\: b^{4} e^{4 i d} \neq 0 \\\frac {x \left (i f^{3} e^{\frac {5 a}{3}} e^{6 i d} - 3 i f^{3} e^{\frac {5 a}{3}} e^{4 i d} + 3 i f^{3} e^{\frac {5 a}{3}} e^{2 i d} - i f^{3} e^{\frac {5 a}{3}} - 3 f^{2} g e^{\frac {5 a}{3}} e^{6 i d} + 3 f^{2} g e^{\frac {5 a}{3}} e^{4 i d} + 3 f^{2} g e^{\frac {5 a}{3}} e^{2 i d} - 3 f^{2} g e^{\frac {5 a}{3}} - 3 i f g^{2} e^{\frac {5 a}{3}} e^{6 i d} - 3 i f g^{2} e^{\frac {5 a}{3}} e^{4 i d} + 3 i f g^{2} e^{\frac {5 a}{3}} e^{2 i d} + 3 i f g^{2} e^{\frac {5 a}{3}} + g^{3} e^{\frac {5 a}{3}} e^{6 i d} + 3 g^{3} e^{\frac {5 a}{3}} e^{4 i d} + 3 g^{3} e^{\frac {5 a}{3}} e^{2 i d} + g^{3} e^{\frac {5 a}{3}}\right ) e^{- 3 i d}}{8} & \text {otherwise} \end {cases} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))**3,x)
Output:
Piecewise((((344064*b**3*f**3*exp(5*a/3)*exp(I*d) - 1032192*I*b**3*f**2*g* exp(5*a/3)*exp(I*d) - 1032192*b**3*f*g**2*exp(5*a/3)*exp(I*d) + 344064*I*b **3*g**3*exp(5*a/3)*exp(I*d))*exp(-4*I*b*x/3) + (2064384*b**3*f**3*exp(5*a /3)*exp(3*I*d) - 2064384*I*b**3*f**2*g*exp(5*a/3)*exp(3*I*d) + 2064384*b** 3*f*g**2*exp(5*a/3)*exp(3*I*d) - 2064384*I*b**3*g**3*exp(5*a/3)*exp(3*I*d) )*exp(2*I*b*x/3) + (-516096*b**3*f**3*exp(5*a/3)*exp(5*I*d) - 516096*I*b** 3*f**2*g*exp(5*a/3)*exp(5*I*d) - 516096*b**3*f*g**2*exp(5*a/3)*exp(5*I*d) - 516096*I*b**3*g**3*exp(5*a/3)*exp(5*I*d))*exp(8*I*b*x/3) + (98304*b**3*f **3*exp(5*a/3)*exp(7*I*d) + 294912*I*b**3*f**2*g*exp(5*a/3)*exp(7*I*d) - 2 94912*b**3*f*g**2*exp(5*a/3)*exp(7*I*d) - 98304*I*b**3*g**3*exp(5*a/3)*exp (7*I*d))*exp(14*I*b*x/3))*exp(-4*I*d)/(3670016*b**4), Ne(b**4*exp(4*I*d), 0)), (x*(I*f**3*exp(5*a/3)*exp(6*I*d) - 3*I*f**3*exp(5*a/3)*exp(4*I*d) + 3 *I*f**3*exp(5*a/3)*exp(2*I*d) - I*f**3*exp(5*a/3) - 3*f**2*g*exp(5*a/3)*ex p(6*I*d) + 3*f**2*g*exp(5*a/3)*exp(4*I*d) + 3*f**2*g*exp(5*a/3)*exp(2*I*d) - 3*f**2*g*exp(5*a/3) - 3*I*f*g**2*exp(5*a/3)*exp(6*I*d) - 3*I*f*g**2*exp (5*a/3)*exp(4*I*d) + 3*I*f*g**2*exp(5*a/3)*exp(2*I*d) + 3*I*f*g**2*exp(5*a /3) + g**3*exp(5*a/3)*exp(6*I*d) + 3*g**3*exp(5*a/3)*exp(4*I*d) + 3*g**3*e xp(5*a/3)*exp(2*I*d) + g**3*exp(5*a/3))*exp(-3*I*d)/8, True))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (107) = 214\).
Time = 0.07 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.81 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx =\text {Too large to display} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm= "maxima")
Output:
3/448*(4*cos(14/3*b*x + 3*d)*e^(5/3*a) - 21*cos(8/3*b*x + d)*e^(5/3*a) + 1 4*cos(4/3*b*x + 3*d)*e^(5/3*a) + 84*cos(2/3*b*x - d)*e^(5/3*a) + 4*I*e^(5/ 3*a)*sin(14/3*b*x + 3*d) - 21*I*e^(5/3*a)*sin(8/3*b*x + d) - 14*I*e^(5/3*a )*sin(4/3*b*x + 3*d) + 84*I*e^(5/3*a)*sin(2/3*b*x - d))*f^3/b + 9/448*(4*I *cos(14/3*b*x + 3*d)*e^(5/3*a) - 7*I*cos(8/3*b*x + d)*e^(5/3*a) - 14*I*cos (4/3*b*x + 3*d)*e^(5/3*a) - 28*I*cos(2/3*b*x - d)*e^(5/3*a) - 4*e^(5/3*a)* sin(14/3*b*x + 3*d) + 7*e^(5/3*a)*sin(8/3*b*x + d) - 14*e^(5/3*a)*sin(4/3* b*x + 3*d) + 28*e^(5/3*a)*sin(2/3*b*x - d))*f^2*g/b - 9/448*(4*cos(14/3*b* x + 3*d)*e^(5/3*a) + 7*cos(8/3*b*x + d)*e^(5/3*a) + 14*cos(4/3*b*x + 3*d)* e^(5/3*a) - 28*cos(2/3*b*x - d)*e^(5/3*a) + 4*I*e^(5/3*a)*sin(14/3*b*x + 3 *d) + 7*I*e^(5/3*a)*sin(8/3*b*x + d) - 14*I*e^(5/3*a)*sin(4/3*b*x + 3*d) - 28*I*e^(5/3*a)*sin(2/3*b*x - d))*f*g^2/b - 3/448*(4*I*cos(14/3*b*x + 3*d) *e^(5/3*a) + 21*I*cos(8/3*b*x + d)*e^(5/3*a) - 14*I*cos(4/3*b*x + 3*d)*e^( 5/3*a) + 84*I*cos(2/3*b*x - d)*e^(5/3*a) - 4*e^(5/3*a)*sin(14/3*b*x + 3*d) - 21*e^(5/3*a)*sin(8/3*b*x + d) - 14*e^(5/3*a)*sin(4/3*b*x + 3*d) - 84*e^ (5/3*a)*sin(2/3*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. 568 vs. \(2 (107) = 214\).
Time = 0.38 (sec) , antiderivative size = 568, normalized size of antiderivative = 3.17 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx =\text {Too large to display} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm= "giac")
Output:
3/896*(4*((f^3 + 3*I*f^2*g - 3*f*g^2 - I*g^3)*e^(14/3*I*b*x + 3*I*d) - (f^ 3 + 3*I*f^2*g - 3*f*g^2 - I*g^3)*e^(-14/3*I*b*x - 3*I*d))*e^(5/3*a) + 4*I* ((-I*f^3 + 3*f^2*g + 3*I*f*g^2 - g^3)*e^(14/3*I*b*x + 3*I*d) - (I*f^3 - 3* f^2*g - 3*I*f*g^2 + g^3)*e^(-14/3*I*b*x - 3*I*d))*e^(5/3*a) - 21*((f^3 + I *f^2*g + f*g^2 + I*g^3)*e^(8/3*I*b*x + I*d) - (f^3 + I*f^2*g + f*g^2 + I*g ^3)*e^(-8/3*I*b*x - I*d))*e^(5/3*a) + 21*I*((I*f^3 - f^2*g + I*f*g^2 - g^3 )*e^(8/3*I*b*x + I*d) - (-I*f^3 + f^2*g - I*f*g^2 + g^3)*e^(-8/3*I*b*x - I *d))*e^(5/3*a) - 14*((f^3 - 3*I*f^2*g - 3*f*g^2 + I*g^3)*e^(4/3*I*b*x + 3* I*d) - (f^3 - 3*I*f^2*g - 3*f*g^2 + I*g^3)*e^(-4/3*I*b*x - 3*I*d))*e^(5/3* a) + 14*I*((-I*f^3 - 3*f^2*g + 3*I*f*g^2 + g^3)*e^(4/3*I*b*x + 3*I*d) - (I *f^3 + 3*f^2*g - 3*I*f*g^2 - g^3)*e^(-4/3*I*b*x - 3*I*d))*e^(5/3*a) + 84*( (f^3 - I*f^2*g + f*g^2 - I*g^3)*e^(2/3*I*b*x - I*d) - (f^3 - I*f^2*g + f*g ^2 - I*g^3)*e^(-2/3*I*b*x + I*d))*e^(5/3*a) - 84*I*((I*f^3 + f^2*g + I*f*g ^2 + g^3)*e^(2/3*I*b*x - I*d) - (-I*f^3 - f^2*g - I*f*g^2 - g^3)*e^(-2/3*I *b*x + I*d))*e^(5/3*a))/b
Time = 1.79 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.69 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}-d\,3{}\mathrm {i}-\frac {b\,x\,4{}\mathrm {i}}{3}}\,{\left (f-g\,1{}\mathrm {i}\right )}^3}{32\,b}+\frac {3\,{\mathrm {e}}^{\frac {5\,a}{3}+d\,3{}\mathrm {i}+\frac {b\,x\,14{}\mathrm {i}}{3}}\,{\left (f+g\,1{}\mathrm {i}\right )}^3}{112\,b}-\frac {9\,{\mathrm {e}}^{\frac {5\,a}{3}-d\,1{}\mathrm {i}+\frac {b\,x\,2{}\mathrm {i}}{3}}\,\left (f+g\,1{}\mathrm {i}\right )\,{\left (g+f\,1{}\mathrm {i}\right )}^2}{16\,b}-\frac {{\mathrm {e}}^{\frac {5\,a}{3}+d\,1{}\mathrm {i}+\frac {b\,x\,8{}\mathrm {i}}{3}}\,\left (g+f\,1{}\mathrm {i}\right )\,{\left (-g+f\,1{}\mathrm {i}\right )}^2\,9{}\mathrm {i}}{64\,b} \] Input:
int(exp((5*a)/3 + (b*x*5i)/3)*(g*cos(d + b*x) + f*sin(d + b*x))^3,x)
Output:
(3*exp((5*a)/3 - d*3i - (b*x*4i)/3)*(f - g*1i)^3)/(32*b) + (3*exp((5*a)/3 + d*3i + (b*x*14i)/3)*(f + g*1i)^3)/(112*b) - (9*exp((5*a)/3 - d*1i + (b*x *2i)/3)*(f + g*1i)*(f*1i + g)^2)/(16*b) - (exp((5*a)/3 + d*1i + (b*x*8i)/3 )*(f*1i + g)*(f*1i - g)^2*9i)/(64*b)
Time = 0.16 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.61 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^3 \, dx=\frac {3 e^{\frac {5 b i x}{3}+\frac {5 a}{3}} \left (81 \cos \left (b x +d \right )^{3} f^{3}-135 \cos \left (b x +d \right )^{3} f^{2} g i +9 \cos \left (b x +d \right )^{3} f \,g^{2}-95 \cos \left (b x +d \right )^{3} g^{3} i -135 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) f^{3} i -225 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) f^{2} g -15 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) f \,g^{2} i -9 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right ) g^{3}+9 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} f^{3}-15 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} f^{2} g i +225 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} f \,g^{2}-135 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{2} g^{3} i -95 \sin \left (b x +d \right )^{3} f^{3} i -9 \sin \left (b x +d \right )^{3} f^{2} g -135 \sin \left (b x +d \right )^{3} f \,g^{2} i -81 \sin \left (b x +d \right )^{3} g^{3}\right )}{448 b} \] Input:
int(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^3,x)
Output:
(3*e**((5*a + 5*b*i*x)/3)*(81*cos(b*x + d)**3*f**3 - 135*cos(b*x + d)**3*f **2*g*i + 9*cos(b*x + d)**3*f*g**2 - 95*cos(b*x + d)**3*g**3*i - 135*cos(b *x + d)**2*sin(b*x + d)*f**3*i - 225*cos(b*x + d)**2*sin(b*x + d)*f**2*g - 15*cos(b*x + d)**2*sin(b*x + d)*f*g**2*i - 9*cos(b*x + d)**2*sin(b*x + d) *g**3 + 9*cos(b*x + d)*sin(b*x + d)**2*f**3 - 15*cos(b*x + d)*sin(b*x + d) **2*f**2*g*i + 225*cos(b*x + d)*sin(b*x + d)**2*f*g**2 - 135*cos(b*x + d)* sin(b*x + d)**2*g**3*i - 95*sin(b*x + d)**3*f**3*i - 9*sin(b*x + d)**3*f** 2*g - 135*sin(b*x + d)**3*f*g**2*i - 81*sin(b*x + d)**3*g**3))/(448*b)