Integrand size = 34, antiderivative size = 226 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=\frac {3 i e^{\frac {5}{3} (a-i d)-\frac {7}{3} i (d+b x)} (f-i g)^4}{112 b}-\frac {3 i e^{\frac {5}{3} (a-i d)+\frac {17}{3} i (d+b x)} (f+i g)^4}{272 b}+\frac {3 e^{\frac {5}{3} (a-i d)+\frac {11}{3} i (d+b x)} (f+i g)^3 (i f+g)}{44 b}+\frac {3 e^{\frac {5}{3} (a-i d)-\frac {1}{3} i (d+b x)} (f+i g) (i f+g)^3}{4 b}-\frac {9 i e^{\frac {5}{3} (a-i d)+\frac {5}{3} i (d+b x)} \left (f^2+g^2\right )^2}{40 b} \] Output:
3/112*I*exp(5/3*a-5/3*I*d-7/3*I*(b*x+d))*(f-I*g)^4/b-3/272*I*exp(5/3*a-5/3 *I*d+17/3*I*(b*x+d))*(f+I*g)^4/b+3/44*exp(5/3*a-5/3*I*d+11/3*I*(b*x+d))*(f +I*g)^3*(I*f+g)/b+3/4*exp(5/3*a-5/3*I*d-1/3*I*(b*x+d))*(f+I*g)*(I*f+g)^3/b -9/40*I*exp(5/3*a-5/3*I*d+5/3*I*(b*x+d))*(f^2+g^2)^2/b
Time = 2.68 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.67 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=\frac {3 e^{\frac {5 a}{3}-\frac {7 i b x}{3}} \left (-385 i e^{4 i (d+2 b x)} (f+i g)^4+2380 e^{2 i (d+3 b x)} (f+i g)^3 (i f+g)+26180 e^{2 i (-d+b x)} (f+i g) (i f+g)^3+935 i e^{-4 i d} (i f+g)^4-7854 i e^{4 i b x} \left (f^2+g^2\right )^2\right )}{104720 b} \] Input:
Integrate[E^((5*(a + I*b*x))/3)*(g*Cos[d + b*x] + f*Sin[d + b*x])^4,x]
Output:
(3*E^((5*a)/3 - ((7*I)/3)*b*x)*((-385*I)*E^((4*I)*(d + 2*b*x))*(f + I*g)^4 + 2380*E^((2*I)*(d + 3*b*x))*(f + I*g)^3*(I*f + g) + 26180*E^((2*I)*(-d + b*x))*(f + I*g)*(I*f + g)^3 + ((935*I)*(I*f + g)^4)/E^((4*I)*d) - (7854*I )*E^((4*I)*b*x)*(f^2 + g^2)^2))/(104720*b)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(773\) vs. \(2(226)=452\).
Time = 1.29 (sec) , antiderivative size = 773, normalized size of antiderivative = 3.42, 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.
| \(\displaystyle \int e^{\frac {5}{3} (a+i b x)} (f \sin (b x+d)+g \cos (b x+d))^4 \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int e^{\frac {5 a}{3}+\frac {5 i b x}{3}} (f \sin (b x+d)+g \cos (b x+d))^4dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (f^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^4(b x+d)+4 f^3 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^3(b x+d) \cos (b x+d)+\frac {3}{2} f^2 g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^2(2 b x+2 d)+4 f g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d) \cos ^3(b x+d)+g^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos ^4(b x+d)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {15 i f^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^4(b x+d)}{119 b}+\frac {1620 i f^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^2(b x+d)}{1309 b}-\frac {36 f^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^3(b x+d) \cos (b x+d)}{119 b}-\frac {1944 f^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d) \cos (b x+d)}{1309 b}+\frac {15 i f^3 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (2 b x+2 d)}{11 b}-\frac {15 i f^3 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (4 b x+4 d)}{238 b}-\frac {18 f^3 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (2 b x+2 d)}{11 b}+\frac {18 f^3 g e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (4 b x+4 d)}{119 b}+\frac {45 i f^2 g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin ^2(2 b x+2 d)}{238 b}-\frac {54 f^2 g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (2 b x+2 d) \cos (2 b x+2 d)}{119 b}+\frac {15 i f g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (2 b x+2 d)}{11 b}+\frac {15 i f g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (4 b x+4 d)}{238 b}-\frac {18 f g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (2 b x+2 d)}{11 b}-\frac {18 f g^3 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos (4 b x+4 d)}{119 b}+\frac {15 i g^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos ^4(b x+d)}{119 b}+\frac {1620 i g^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos ^2(b x+d)}{1309 b}+\frac {36 g^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d) \cos ^3(b x+d)}{119 b}+\frac {1944 g^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin (b x+d) \cos (b x+d)}{1309 b}-\frac {5832 i f^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{6545 b}-\frac {324 i f^2 g^2 e^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{595 b}-\frac {5832 i g^4 e^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{6545 b}\) |
Input:
Int[E^((5*(a + I*b*x))/3)*(g*Cos[d + b*x] + f*Sin[d + b*x])^4,x]
Output:
(((-5832*I)/6545)*E^((5*a)/3 + ((5*I)/3)*b*x)*f^4)/b - (((324*I)/595)*E^(( 5*a)/3 + ((5*I)/3)*b*x)*f^2*g^2)/b - (((5832*I)/6545)*E^((5*a)/3 + ((5*I)/ 3)*b*x)*g^4)/b + (((1620*I)/1309)*E^((5*a)/3 + ((5*I)/3)*b*x)*g^4*Cos[d + b*x]^2)/b + (((15*I)/119)*E^((5*a)/3 + ((5*I)/3)*b*x)*g^4*Cos[d + b*x]^4)/ b - (18*E^((5*a)/3 + ((5*I)/3)*b*x)*f^3*g*Cos[2*d + 2*b*x])/(11*b) - (18*E ^((5*a)/3 + ((5*I)/3)*b*x)*f*g^3*Cos[2*d + 2*b*x])/(11*b) + (18*E^((5*a)/3 + ((5*I)/3)*b*x)*f^3*g*Cos[4*d + 4*b*x])/(119*b) - (18*E^((5*a)/3 + ((5*I )/3)*b*x)*f*g^3*Cos[4*d + 4*b*x])/(119*b) - (1944*E^((5*a)/3 + ((5*I)/3)*b *x)*f^4*Cos[d + b*x]*Sin[d + b*x])/(1309*b) + (1944*E^((5*a)/3 + ((5*I)/3) *b*x)*g^4*Cos[d + b*x]*Sin[d + b*x])/(1309*b) + (36*E^((5*a)/3 + ((5*I)/3) *b*x)*g^4*Cos[d + b*x]^3*Sin[d + b*x])/(119*b) + (((1620*I)/1309)*E^((5*a) /3 + ((5*I)/3)*b*x)*f^4*Sin[d + b*x]^2)/b - (36*E^((5*a)/3 + ((5*I)/3)*b*x )*f^4*Cos[d + b*x]*Sin[d + b*x]^3)/(119*b) + (((15*I)/119)*E^((5*a)/3 + (( 5*I)/3)*b*x)*f^4*Sin[d + b*x]^4)/b + (((15*I)/11)*E^((5*a)/3 + ((5*I)/3)*b *x)*f^3*g*Sin[2*d + 2*b*x])/b + (((15*I)/11)*E^((5*a)/3 + ((5*I)/3)*b*x)*f *g^3*Sin[2*d + 2*b*x])/b - (54*E^((5*a)/3 + ((5*I)/3)*b*x)*f^2*g^2*Cos[2*d + 2*b*x]*Sin[2*d + 2*b*x])/(119*b) + (((45*I)/238)*E^((5*a)/3 + ((5*I)/3) *b*x)*f^2*g^2*Sin[2*d + 2*b*x]^2)/b - (((15*I)/238)*E^((5*a)/3 + ((5*I)/3) *b*x)*f^3*g*Sin[4*d + 4*b*x])/b + (((15*I)/238)*E^((5*a)/3 + ((5*I)/3)*b*x )*f*g^3*Sin[4*d + 4*b*x])/b
Time = 6.04 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.79
| method | result | size |
| parallelrisch | \(-\frac {15 \left (\left (-\frac {1}{4} i f^{4}-\frac {1}{4} i g^{4}+\frac {3}{2} i f^{2} g^{2}-\frac {12}{5} f^{3} g +\frac {12}{5} f \,g^{3}\right ) \cos \left (4 b x +4 d \right )+\left (i f^{3} g -i f \,g^{3}-\frac {3}{5} f^{4}+\frac {18}{5} f^{2} g^{2}-\frac {3}{5} g^{4}\right ) \sin \left (4 b x +4 d \right )+\frac {119 \left (\left (i f^{2}-i g^{2}+\frac {12}{5} f g \right ) \cos \left (2 b x +2 d \right )+\left (-2 i f g +\frac {6}{5} f^{2}-\frac {6}{5} g^{2}\right ) \sin \left (2 b x +2 d \right )+\frac {33 i f^{2}}{100}+\frac {33 i g^{2}}{100}\right ) \left (f^{2}+g^{2}\right )}{11}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{238 b}\) | \(179\) |
| parts | \(f^{4} \left (-\frac {9 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{40 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{952 b}+\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{238 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{22 b}-\frac {9 \sin \left (2 b x +2 d \right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{11 b}\right )+g^{4} \left (-\frac {9 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{40 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{952 b}+\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{238 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{22 b}+\frac {9 \sin \left (2 b x +2 d \right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{11 b}\right )+4 f \,g^{3} \left (-\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{238 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{952 b}-\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{22 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (2 b x +2 d \right )}{44 b}\right )+6 f^{2} g^{2} \left (-\frac {3 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{40 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{952 b}-\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{238 b}\right )+4 f^{3} g \left (-\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{22 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (2 b x +2 d \right )}{44 b}+\frac {9 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{238 b}-\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{952 b}\right )\) | \(524\) |
| norman | \(\frac {-\frac {24 \left (-270 i f^{3} g -325 i f \,g^{3}+162 f^{4}+99 f^{2} g^{2}-195 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )}{1309 b}+\frac {24 \left (-270 i f^{3} g -325 i f \,g^{3}+162 f^{4}+99 f^{2} g^{2}-195 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{7}}{1309 b}-\frac {24 \left (-70 i f^{3} g -15 i f \,g^{3}+42 f^{4}-99 f^{2} g^{2}-9 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}}{187 b}+\frac {24 \left (-70 i f^{3} g -15 i f \,g^{3}+42 f^{4}-99 f^{2} g^{2}-9 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{5}}{187 b}-\frac {3 i \left (-3240 i f^{3} g -3900 i f \,g^{3}+1944 f^{4}+1188 f^{2} g^{2}-1031 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{6545 b}+\frac {12 i \left (-180 i f^{3} g -840 i f \,g^{3}+108 f^{4}+66 f^{2} g^{2}-317 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}}{935 b}+\frac {12 i \left (-180 i f^{3} g -840 i f \,g^{3}+108 f^{4}+66 f^{2} g^{2}-317 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{6}}{935 b}+\frac {6 i \left (-4200 i f^{3} g -900 i f \,g^{3}+1024 f^{4}-1452 f^{2} g^{2}-1101 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}}{935 b}-\frac {3 i \left (-3240 i f^{3} g -3900 i f \,g^{3}+1944 f^{4}+1188 f^{2} g^{2}-1031 g^{4}\right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{8}}{6545 b}}{\left (1+\tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}\right )^{4}}\) | \(541\) |
| default | \(-\frac {9 i f^{4} {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{40 b}-\frac {9 i g^{4} {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{40 b}-\frac {9 i f^{2} g^{2} {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{10 b}+\frac {f^{4} \left (\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{119 b}+\frac {36 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{119 b}\right )}{8}-\frac {f^{4} \left (\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{11 b}+\frac {18 \sin \left (2 b x +2 d \right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{11 b}\right )}{2}+\frac {g^{4} \left (\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{119 b}+\frac {36 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{119 b}\right )}{8}+\frac {g^{4} \left (\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{11 b}+\frac {18 \sin \left (2 b x +2 d \right ) {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{11 b}\right )}{2}+f^{3} g \left (-\frac {18 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{11 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (2 b x +2 d \right )}{11 b}\right )-\frac {f^{3} g \left (-\frac {36 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{119 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{119 b}\right )}{2}+f \,g^{3} \left (-\frac {18 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (2 b x +2 d \right )}{11 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (2 b x +2 d \right )}{11 b}\right )+\frac {f \,g^{3} \left (-\frac {36 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{119 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{119 b}\right )}{2}-\frac {3 f^{2} g^{2} \left (-\frac {3 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}}}{10 b}+\frac {15 i {\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \cos \left (4 b x +4 d \right )}{238 b}+\frac {18 \,{\mathrm e}^{\frac {5 a}{3}+\frac {5 i b x}{3}} \sin \left (4 b x +4 d \right )}{119 b}\right )}{2}\) | \(578\) |
| orering | \(\text {Expression too large to display}\) | \(1416\) |
Input:
int(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x,method=_RETURNVER BOSE)
Output:
-15/238*((-1/4*I*f^4-1/4*I*g^4+3/2*I*f^2*g^2-12/5*f^3*g+12/5*f*g^3)*cos(4* b*x+4*d)+(I*f^3*g-I*f*g^3-3/5*f^4+18/5*f^2*g^2-3/5*g^4)*sin(4*b*x+4*d)+119 /11*((I*f^2-I*g^2+12/5*f*g)*cos(2*b*x+2*d)+(-2*I*f*g+6/5*f^2-6/5*g^2)*sin( 2*b*x+2*d)+33/100*I*f^2+33/100*I*g^2)*(f^2+g^2))*exp(5/3*a+5/3*I*b*x)/b
Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.92 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=-\frac {3 \, {\left (385 \, {\left (i \, f^{4} - 4 \, f^{3} g - 6 i \, f^{2} g^{2} + 4 \, f g^{3} + i \, g^{4}\right )} e^{\left (8 i \, b x + \frac {5}{3} \, a + \frac {19}{3} i \, d\right )} + 2380 \, {\left (-i \, f^{4} + 2 \, f^{3} g + 2 \, f g^{3} + i \, g^{4}\right )} e^{\left (6 i \, b x + \frac {5}{3} \, a + \frac {13}{3} i \, d\right )} + 7854 \, {\left (i \, f^{4} + 2 i \, f^{2} g^{2} + i \, g^{4}\right )} e^{\left (4 i \, b x + \frac {5}{3} \, a + \frac {7}{3} i \, d\right )} + 26180 \, {\left (i \, f^{4} + 2 \, f^{3} g + 2 \, f g^{3} - i \, g^{4}\right )} e^{\left (2 i \, b x + \frac {5}{3} \, a + \frac {1}{3} i \, d\right )} + 935 \, {\left (-i \, f^{4} - 4 \, f^{3} g + 6 i \, f^{2} g^{2} + 4 \, f g^{3} - i \, g^{4}\right )} e^{\left (\frac {5}{3} \, a - \frac {5}{3} i \, d\right )}\right )} e^{\left (-\frac {7}{3} i \, b x - \frac {7}{3} i \, d\right )}}{104720 \, b} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x, algorithm= "fricas")
Output:
-3/104720*(385*(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 + 5/3*a + 19/3*I*d) + 2380*(-I*f^4 + 2*f^3*g + 2*f*g^3 + I*g^4)*e^(6*I*b *x + 5/3*a + 13/3*I*d) + 7854*(I*f^4 + 2*I*f^2*g^2 + I*g^4)*e^(4*I*b*x + 5 /3*a + 7/3*I*d) + 26180*(I*f^4 + 2*f^3*g + 2*f*g^3 - I*g^4)*e^(2*I*b*x + 5 /3*a + 1/3*I*d) + 935*(-I*f^4 - 4*f^3*g + 6*I*f^2*g^2 + 4*f*g^3 - I*g^4)*e ^(5/3*a - 5/3*I*d))*e^(-7/3*I*b*x - 7/3*I*d)/b
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 979 vs. \(2 (192) = 384\).
Time = 0.88 (sec) , antiderivative size = 979, normalized size of antiderivative = 4.33 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx =\text {Too large to display} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))**4,x)
Output:
Piecewise((((-48254976*I*b**4*f**4*exp(5*a/3)*exp(6*I*d) - 96509952*I*b**4 *f**2*g**2*exp(5*a/3)*exp(6*I*d) - 48254976*I*b**4*g**4*exp(5*a/3)*exp(6*I *d))*exp(5*I*b*x/3) + (-160849920*I*b**4*f**4*exp(5*a/3)*exp(4*I*d) - 3216 99840*b**4*f**3*g*exp(5*a/3)*exp(4*I*d) - 321699840*b**4*f*g**3*exp(5*a/3) *exp(4*I*d) + 160849920*I*b**4*g**4*exp(5*a/3)*exp(4*I*d))*exp(-I*b*x/3) + (14622720*I*b**4*f**4*exp(5*a/3)*exp(8*I*d) - 29245440*b**4*f**3*g*exp(5* a/3)*exp(8*I*d) - 29245440*b**4*f*g**3*exp(5*a/3)*exp(8*I*d) - 14622720*I* b**4*g**4*exp(5*a/3)*exp(8*I*d))*exp(11*I*b*x/3) + (5744640*I*b**4*f**4*ex p(5*a/3)*exp(2*I*d) + 22978560*b**4*f**3*g*exp(5*a/3)*exp(2*I*d) - 3446784 0*I*b**4*f**2*g**2*exp(5*a/3)*exp(2*I*d) - 22978560*b**4*f*g**3*exp(5*a/3) *exp(2*I*d) + 5744640*I*b**4*g**4*exp(5*a/3)*exp(2*I*d))*exp(-7*I*b*x/3) + (-2365440*I*b**4*f**4*exp(5*a/3)*exp(10*I*d) + 9461760*b**4*f**3*g*exp(5* a/3)*exp(10*I*d) + 14192640*I*b**4*f**2*g**2*exp(5*a/3)*exp(10*I*d) - 9461 760*b**4*f*g**3*exp(5*a/3)*exp(10*I*d) - 2365440*I*b**4*g**4*exp(5*a/3)*ex p(10*I*d))*exp(17*I*b*x/3))*exp(-6*I*d)/(214466560*b**5), Ne(b**5*exp(6*I* d), 0)), (x*(f**4*exp(5*a/3)*exp(8*I*d) - 4*f**4*exp(5*a/3)*exp(6*I*d) + 6 *f**4*exp(5*a/3)*exp(4*I*d) - 4*f**4*exp(5*a/3)*exp(2*I*d) + f**4*exp(5*a/ 3) + 4*I*f**3*g*exp(5*a/3)*exp(8*I*d) - 8*I*f**3*g*exp(5*a/3)*exp(6*I*d) + 8*I*f**3*g*exp(5*a/3)*exp(2*I*d) - 4*I*f**3*g*exp(5*a/3) - 6*f**2*g**2*ex p(5*a/3)*exp(8*I*d) + 12*f**2*g**2*exp(5*a/3)*exp(4*I*d) - 6*f**2*g**2*...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (130) = 260\).
Time = 0.08 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.91 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=\text {Too large to display} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x, algorithm= "maxima")
Output:
-3/104720*(7854*I*cos(5/3*b*x)*e^(5/3*a) + 385*I*cos(17/3*b*x + 4*d)*e^(5/ 3*a) - 2380*I*cos(11/3*b*x + 2*d)*e^(5/3*a) - 935*I*cos(7/3*b*x + 4*d)*e^( 5/3*a) + 26180*I*cos(1/3*b*x + 2*d)*e^(5/3*a) - 7854*e^(5/3*a)*sin(5/3*b*x ) - 385*e^(5/3*a)*sin(17/3*b*x + 4*d) + 2380*e^(5/3*a)*sin(11/3*b*x + 2*d) - 935*e^(5/3*a)*sin(7/3*b*x + 4*d) + 26180*e^(5/3*a)*sin(1/3*b*x + 2*d))* f^4/b + 3/5236*(77*cos(17/3*b*x + 4*d)*e^(5/3*a) - 238*cos(11/3*b*x + 2*d) *e^(5/3*a) + 187*cos(7/3*b*x + 4*d)*e^(5/3*a) - 2618*cos(1/3*b*x + 2*d)*e^ (5/3*a) + 77*I*e^(5/3*a)*sin(17/3*b*x + 4*d) - 238*I*e^(5/3*a)*sin(11/3*b* x + 2*d) - 187*I*e^(5/3*a)*sin(7/3*b*x + 4*d) + 2618*I*e^(5/3*a)*sin(1/3*b *x + 2*d))*f^3*g/b + 9/4760*(-238*I*cos(5/3*b*x)*e^(5/3*a) + 35*I*cos(17/3 *b*x + 4*d)*e^(5/3*a) - 85*I*cos(7/3*b*x + 4*d)*e^(5/3*a) + 238*e^(5/3*a)* sin(5/3*b*x) - 35*e^(5/3*a)*sin(17/3*b*x + 4*d) - 85*e^(5/3*a)*sin(7/3*b*x + 4*d))*f^2*g^2/b - 3/5236*(77*cos(17/3*b*x + 4*d)*e^(5/3*a) + 238*cos(11 /3*b*x + 2*d)*e^(5/3*a) + 187*cos(7/3*b*x + 4*d)*e^(5/3*a) + 2618*cos(1/3* b*x + 2*d)*e^(5/3*a) + 77*I*e^(5/3*a)*sin(17/3*b*x + 4*d) + 238*I*e^(5/3*a )*sin(11/3*b*x + 2*d) - 187*I*e^(5/3*a)*sin(7/3*b*x + 4*d) - 2618*I*e^(5/3 *a)*sin(1/3*b*x + 2*d))*f*g^3/b - 3/104720*(7854*I*cos(5/3*b*x)*e^(5/3*a) + 385*I*cos(17/3*b*x + 4*d)*e^(5/3*a) + 2380*I*cos(11/3*b*x + 2*d)*e^(5/3* a) - 935*I*cos(7/3*b*x + 4*d)*e^(5/3*a) - 26180*I*cos(1/3*b*x + 2*d)*e^(5/ 3*a) - 7854*e^(5/3*a)*sin(5/3*b*x) - 385*e^(5/3*a)*sin(17/3*b*x + 4*d) ...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (130) = 260\).
Time = 0.33 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.12 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx =\text {Too large to display} \] Input:
integrate(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x, algorithm= "giac")
Output:
3/209440*(15708*(f^4 + 2*f^2*g^2 + g^4)*e^(5/3*a)*sin(5/3*b*x) - 385*I*((f ^4 + 4*I*f^3*g - 6*f^2*g^2 - 4*I*f*g^3 + g^4)*e^(17/3*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^(-17/3*I*b*x - 4*I*d))*e^(5 /3*a) - 385*((I*f^4 - 4*f^3*g - 6*I*f^2*g^2 + 4*f*g^3 + I*g^4)*e^(17/3*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^(-17/3* I*b*x - 4*I*d))*e^(5/3*a) + 2380*I*((f^4 + 2*I*f^3*g + 2*I*f*g^3 - g^4)*e^ (11/3*I*b*x + 2*I*d) + (f^4 + 2*I*f^3*g + 2*I*f*g^3 - g^4)*e^(-11/3*I*b*x - 2*I*d))*e^(5/3*a) - 2380*((-I*f^4 + 2*f^3*g + 2*f*g^3 + I*g^4)*e^(11/3*I *b*x + 2*I*d) + (I*f^4 - 2*f^3*g - 2*f*g^3 - I*g^4)*e^(-11/3*I*b*x - 2*I*d ))*e^(5/3*a) + 935*I*((f^4 - 4*I*f^3*g - 6*f^2*g^2 + 4*I*f*g^3 + g^4)*e^(7 /3*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^(-7/ 3*I*b*x - 4*I*d))*e^(5/3*a) - 935*((I*f^4 + 4*f^3*g - 6*I*f^2*g^2 - 4*f*g^ 3 + I*g^4)*e^(7/3*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^(-7/3*I*b*x - 4*I*d))*e^(5/3*a) - 26180*I*((f^4 - 2*I*f^3*g - 2*I*f*g^3 - g^4)*e^(1/3*I*b*x + 2*I*d) + (f^4 - 2*I*f^3*g - 2*I*f*g^3 - g^4)*e^(-1/3*I*b*x - 2*I*d))*e^(5/3*a) - 26180*((-I*f^4 - 2*f^3*g - 2*f*g^ 3 + I*g^4)*e^(1/3*I*b*x + 2*I*d) + (I*f^4 + 2*f^3*g + 2*f*g^3 - I*g^4)*e^( -1/3*I*b*x - 2*I*d))*e^(5/3*a) - 7854*I*((f^4 + 2*f^2*g^2 + g^4)*e^(5/3*I* b*x) + (f^4 + 2*f^2*g^2 + g^4)*e^(-5/3*I*b*x))*e^(5/3*a))/b
Time = 18.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.68 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=-\frac {{\mathrm {e}}^{\frac {5\,a}{3}+\frac {b\,x\,5{}\mathrm {i}}{3}}\,{\left (f^2+g^2\right )}^2\,9{}\mathrm {i}}{40\,b}+\frac {{\mathrm {e}}^{\frac {5\,a}{3}-d\,4{}\mathrm {i}-\frac {b\,x\,7{}\mathrm {i}}{3}}\,{\left (g+f\,1{}\mathrm {i}\right )}^4\,3{}\mathrm {i}}{112\,b}-\frac {{\mathrm {e}}^{\frac {5\,a}{3}+d\,4{}\mathrm {i}+\frac {b\,x\,17{}\mathrm {i}}{3}}\,{\left (-g+f\,1{}\mathrm {i}\right )}^4\,3{}\mathrm {i}}{272\,b}-\frac {{\mathrm {e}}^{\frac {5\,a}{3}-d\,2{}\mathrm {i}-\frac {b\,x\,1{}\mathrm {i}}{3}}\,{\left (f-g\,1{}\mathrm {i}\right )}^3\,\left (f+g\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{4\,b}+\frac {{\mathrm {e}}^{\frac {5\,a}{3}+d\,2{}\mathrm {i}+\frac {b\,x\,11{}\mathrm {i}}{3}}\,\left (g+f\,1{}\mathrm {i}\right )\,{\left (-g+f\,1{}\mathrm {i}\right )}^3\,3{}\mathrm {i}}{44\,b} \] Input:
int(exp((5*a)/3 + (b*x*5i)/3)*(g*cos(d + b*x) + f*sin(d + b*x))^4,x)
Output:
(exp((5*a)/3 - d*4i - (b*x*7i)/3)*(f*1i + g)^4*3i)/(112*b) - (exp((5*a)/3 + (b*x*5i)/3)*(f^2 + g^2)^2*9i)/(40*b) - (exp((5*a)/3 + d*4i + (b*x*17i)/3 )*(f*1i - g)^4*3i)/(272*b) - (exp((5*a)/3 - d*2i - (b*x*1i)/3)*(f - g*1i)^ 3*(f + g*1i)*3i)/(4*b) + (exp((5*a)/3 + d*2i + (b*x*11i)/3)*(f*1i + g)*(f* 1i - g)^3*3i)/(44*b)
Time = 0.20 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.12 \[ \int e^{\frac {5}{3} (a+i b x)} (g \cos (d+b x)+f \sin (d+b x))^4 \, dx=\frac {3 e^{\frac {5 b i x}{3}+\frac {5 a}{3}} \left (5400 \cos \left (b x +d \right )^{3} \sin \left (b x +d \right ) f^{3} g i +6500 \cos \left (b x +d \right )^{3} \sin \left (b x +d \right ) f \,g^{3} i -726 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{2} f^{2} g^{2} i +6500 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{3} f^{3} g i +5400 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{3} f \,g^{3} i +1031 \sin \left (b x +d \right )^{4} f^{4} i -3240 \cos \left (b x +d \right )^{4} f^{3} g -3240 \cos \left (b x +d \right )^{3} \sin \left (b x +d \right ) f^{4}+3900 \sin \left (b x +d \right )^{4} f^{3} g +3900 \cos \left (b x +d \right )^{3} \sin \left (b x +d \right ) g^{4}+3240 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{3} g^{4}-1944 \cos \left (b x +d \right )^{4} f^{4} i -3900 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{3} f^{4}+1031 \cos \left (b x +d \right )^{4} g^{4} i -1944 \sin \left (b x +d \right )^{4} g^{4} i -3900 \cos \left (b x +d \right )^{4} f \,g^{3}+3240 \sin \left (b x +d \right )^{4} f \,g^{3}-1188 \cos \left (b x +d \right )^{4} f^{2} g^{2} i -1980 \cos \left (b x +d \right )^{3} \sin \left (b x +d \right ) f^{2} g^{2}-1188 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{2} f^{4} i -1980 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{2} f^{3} g +1980 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{2} f \,g^{3}-1188 \cos \left (b x +d \right )^{2} \sin \left (b x +d \right )^{2} g^{4} i +1980 \cos \left (b x +d \right ) \sin \left (b x +d \right )^{3} f^{2} g^{2}-1188 \sin \left (b x +d \right )^{4} f^{2} g^{2} i \right )}{6545 b} \] Input:
int(exp(5/3*a+5/3*I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^4,x)
Output:
(3*e**((5*a + 5*b*i*x)/3)*( - 1944*cos(b*x + d)**4*f**4*i - 3240*cos(b*x + d)**4*f**3*g - 1188*cos(b*x + d)**4*f**2*g**2*i - 3900*cos(b*x + d)**4*f* g**3 + 1031*cos(b*x + d)**4*g**4*i - 3240*cos(b*x + d)**3*sin(b*x + d)*f** 4 + 5400*cos(b*x + d)**3*sin(b*x + d)*f**3*g*i - 1980*cos(b*x + d)**3*sin( b*x + d)*f**2*g**2 + 6500*cos(b*x + d)**3*sin(b*x + d)*f*g**3*i + 3900*cos (b*x + d)**3*sin(b*x + d)*g**4 - 1188*cos(b*x + d)**2*sin(b*x + d)**2*f**4 *i - 1980*cos(b*x + d)**2*sin(b*x + d)**2*f**3*g - 726*cos(b*x + d)**2*sin (b*x + d)**2*f**2*g**2*i + 1980*cos(b*x + d)**2*sin(b*x + d)**2*f*g**3 - 1 188*cos(b*x + d)**2*sin(b*x + d)**2*g**4*i - 3900*cos(b*x + d)*sin(b*x + d )**3*f**4 + 6500*cos(b*x + d)*sin(b*x + d)**3*f**3*g*i + 1980*cos(b*x + d) *sin(b*x + d)**3*f**2*g**2 + 5400*cos(b*x + d)*sin(b*x + d)**3*f*g**3*i + 3240*cos(b*x + d)*sin(b*x + d)**3*g**4 + 1031*sin(b*x + d)**4*f**4*i + 390 0*sin(b*x + d)**4*f**3*g - 1188*sin(b*x + d)**4*f**2*g**2*i + 3240*sin(b*x + d)**4*f*g**3 - 1944*sin(b*x + d)**4*g**4*i))/(6545*b)