Integrand size = 25, antiderivative size = 374 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {1}{16} \left (162 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+18 b^2 c \left (4 c^2+9 d^2\right )+216 \left (2 c^3+3 c d^2\right )\right ) x-\frac {\left (9 b^2 d \left (3 c^2+d^2\right )+27 b c \left (c^2+3 d^2\right )+b^3 c \left (c^2+3 d^2\right )+27 \left (3 c^2 d+d^3\right )\right ) \cos (e+f x)}{f}+\frac {(b c+3 d) \left (24 b c d+9 d^2+b^2 \left (c^2+6 d^2\right )\right ) \cos ^3(e+f x)}{3 f}-\frac {3 b^2 d^2 (b c+3 d) \cos ^5(e+f x)}{5 f}-\frac {\left (648 c d^2+162 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+18 b^2 c \left (4 c^2+9 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{16 f}-\frac {5 b^3 d^3 \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac {3 b d \left (b^2 c^2+9 b c d+9 d^2\right ) \cos (e+f x) \sin ^3(e+f x)}{4 f}-\frac {b^3 d^3 \cos (e+f x) \sin ^5(e+f x)}{6 f} \]
1/16*(18*a^2*b*d*(4*c^2+d^2)+b^3*d*(18*c^2+5*d^2)+6*a*b^2*c*(4*c^2+9*d^2)+ 8*a^3*(2*c^3+3*c*d^2))*x-(3*a*b^2*d*(3*c^2+d^2)+3*a^2*b*c*(c^2+3*d^2)+b^3* c*(c^2+3*d^2)+a^3*(3*c^2*d+d^3))*cos(f*x+e)/f+1/3*(a*d+b*c)*(8*a*b*c*d+a^2 *d^2+b^2*(c^2+6*d^2))*cos(f*x+e)^3/f-3/5*b^2*d^2*(a*d+b*c)*cos(f*x+e)^5/f- 1/16*(24*a^3*c*d^2+18*a^2*b*d*(4*c^2+d^2)+b^3*d*(18*c^2+5*d^2)+6*a*b^2*c*( 4*c^2+9*d^2))*cos(f*x+e)*sin(f*x+e)/f-5/24*b^3*d^3*cos(f*x+e)*sin(f*x+e)^3 /f-3/4*b*d*(a^2*d^2+3*a*b*c*d+b^2*c^2)*cos(f*x+e)*sin(f*x+e)^3/f-1/6*b^3*d ^3*cos(f*x+e)*sin(f*x+e)^5/f
Time = 7.90 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.32 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {25920 c^3 e+4320 b^2 c^3 e+38880 b c^2 d e+1080 b^3 c^2 d e+38880 c d^2 e+9720 b^2 c d^2 e+9720 b d^3 e+300 b^3 d^3 e+25920 c^3 f x+4320 b^2 c^3 f x+38880 b c^2 d f x+1080 b^3 c^2 d f x+38880 c d^2 f x+9720 b^2 c d^2 f x+9720 b d^3 f x+300 b^3 d^3 f x-360 \left (b^3 \left (2 c^3+5 c d^2\right )+18 b \left (4 c^3+9 c d^2\right )+54 \left (4 c^2 d+d^3\right )+3 b^2 \left (18 c^2 d+5 d^3\right )\right ) \cos (e+f x)+20 \left (324 b c d^2+108 d^3+b^3 \left (4 c^3+15 c d^2\right )+9 b^2 \left (12 c^2 d+5 d^3\right )\right ) \cos (3 (e+f x))-36 b^3 c d^2 \cos (5 (e+f x))-108 b^2 d^3 \cos (5 (e+f x))-2160 b^2 c^3 \sin (2 (e+f x))-19440 b c^2 d \sin (2 (e+f x))-720 b^3 c^2 d \sin (2 (e+f x))-19440 c d^2 \sin (2 (e+f x))-6480 b^2 c d^2 \sin (2 (e+f x))-6480 b d^3 \sin (2 (e+f x))-225 b^3 d^3 \sin (2 (e+f x))+90 b^3 c^2 d \sin (4 (e+f x))+810 b^2 c d^2 \sin (4 (e+f x))+810 b d^3 \sin (4 (e+f x))+45 b^3 d^3 \sin (4 (e+f x))-5 b^3 d^3 \sin (6 (e+f x))}{960 f} \]
(25920*c^3*e + 4320*b^2*c^3*e + 38880*b*c^2*d*e + 1080*b^3*c^2*d*e + 38880 *c*d^2*e + 9720*b^2*c*d^2*e + 9720*b*d^3*e + 300*b^3*d^3*e + 25920*c^3*f*x + 4320*b^2*c^3*f*x + 38880*b*c^2*d*f*x + 1080*b^3*c^2*d*f*x + 38880*c*d^2 *f*x + 9720*b^2*c*d^2*f*x + 9720*b*d^3*f*x + 300*b^3*d^3*f*x - 360*(b^3*(2 *c^3 + 5*c*d^2) + 18*b*(4*c^3 + 9*c*d^2) + 54*(4*c^2*d + d^3) + 3*b^2*(18* c^2*d + 5*d^3))*Cos[e + f*x] + 20*(324*b*c*d^2 + 108*d^3 + b^3*(4*c^3 + 15 *c*d^2) + 9*b^2*(12*c^2*d + 5*d^3))*Cos[3*(e + f*x)] - 36*b^3*c*d^2*Cos[5* (e + f*x)] - 108*b^2*d^3*Cos[5*(e + f*x)] - 2160*b^2*c^3*Sin[2*(e + f*x)] - 19440*b*c^2*d*Sin[2*(e + f*x)] - 720*b^3*c^2*d*Sin[2*(e + f*x)] - 19440* c*d^2*Sin[2*(e + f*x)] - 6480*b^2*c*d^2*Sin[2*(e + f*x)] - 6480*b*d^3*Sin[ 2*(e + f*x)] - 225*b^3*d^3*Sin[2*(e + f*x)] + 90*b^3*c^2*d*Sin[4*(e + f*x) ] + 810*b^2*c*d^2*Sin[4*(e + f*x)] + 810*b*d^3*Sin[4*(e + f*x)] + 45*b^3*d ^3*Sin[4*(e + f*x)] - 5*b^3*d^3*Sin[6*(e + f*x)])/(960*f)
Time = 1.52 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3272, 3042, 3502, 25, 3042, 3232, 27, 3042, 3232, 3042, 3213}
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.
| \(\displaystyle \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^3 \left (6 d a^3+4 b^2 d a-b^2 (2 b c-13 a d) \sin ^2(e+f x)+b^3 c-b \left (-18 d a^2+b c a-5 b^2 d\right ) \sin (e+f x)\right )dx}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^3 \left (6 d a^3+4 b^2 d a-b^2 (2 b c-13 a d) \sin (e+f x)^2+b^3 c-b \left (-18 d a^2+b c a-5 b^2 d\right ) \sin (e+f x)\right )dx}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\int -(c+d \sin (e+f x))^3 \left (3 d \left (-10 d a^3-24 b^2 d a+b^3 c\right )+b \left (-\left (\left (2 c^2+25 d^2\right ) b^2\right )+18 a c d b-90 a^2 d^2\right ) \sin (e+f x)\right )dx}{5 d}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\int (c+d \sin (e+f x))^3 \left (3 d \left (-10 d a^3-24 b^2 d a+b^3 c\right )+b \left (-\left (\left (2 c^2+25 d^2\right ) b^2\right )+18 a c d b-90 a^2 d^2\right ) \sin (e+f x)\right )dx}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\int (c+d \sin (e+f x))^3 \left (3 d \left (-10 d a^3-24 b^2 d a+b^3 c\right )+b \left (-\left (\left (2 c^2+25 d^2\right ) b^2\right )+18 a c d b-90 a^2 d^2\right ) \sin (e+f x)\right )dx}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {\frac {1}{4} \int -3 (c+d \sin (e+f x))^2 \left (d \left (40 c d a^3+90 b d^2 a^2+78 b^2 c d a-b^3 \left (2 c^2-25 d^2\right )\right )+\left (\left (2 c^3+21 d^2 c\right ) b^3-a \left (18 c^2 d-96 d^3\right ) b^2+90 a^2 c d^2 b+40 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {-\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (d \left (40 c d a^3+90 b d^2 a^2+78 b^2 c d a-b^3 \left (2 c^2-25 d^2\right )\right )+\left (\left (2 c^3+21 d^2 c\right ) b^3-a \left (18 c^2 d-96 d^3\right ) b^2+90 a^2 c d^2 b+40 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {-\frac {3}{4} \int (c+d \sin (e+f x))^2 \left (d \left (40 c d a^3+90 b d^2 a^2+78 b^2 c d a-b^3 \left (2 c^2-25 d^2\right )\right )+\left (\left (2 c^3+21 d^2 c\right ) b^3-a \left (18 c^2 d-96 d^3\right ) b^2+90 a^2 c d^2 b+40 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {-\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (d \left (40 d \left (3 c^2+2 d^2\right ) a^3+450 b c d^2 a^2+6 b^2 d \left (33 c^2+32 d^2\right ) a-b^3 \left (2 c^3-117 c d^2\right )\right )+\left (\left (4 c^4+36 d^2 c^2+75 d^4\right ) b^3-6 a c d \left (6 c^2-71 d^2\right ) b^2+90 a^2 d^2 \left (2 c^2+3 d^2\right ) b+200 a^3 c d^3\right ) \sin (e+f x)\right )dx-\frac {\left (40 a^3 d^3+90 a^2 b c d^2-a b^2 \left (18 c^2 d-96 d^3\right )+b^3 \left (2 c^3+21 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )-\frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {-\frac {3}{4} \left (\frac {1}{3} \int (c+d \sin (e+f x)) \left (d \left (40 d \left (3 c^2+2 d^2\right ) a^3+450 b c d^2 a^2+6 b^2 d \left (33 c^2+32 d^2\right ) a-b^3 \left (2 c^3-117 c d^2\right )\right )+\left (\left (4 c^4+36 d^2 c^2+75 d^4\right ) b^3-6 a c d \left (6 c^2-71 d^2\right ) b^2+90 a^2 d^2 \left (2 c^2+3 d^2\right ) b+200 a^3 c d^3\right ) \sin (e+f x)\right )dx-\frac {\left (40 a^3 d^3+90 a^2 b c d^2-a b^2 \left (18 c^2 d-96 d^3\right )+b^3 \left (2 c^3+21 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )-\frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{5 d f}-\frac {-\frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{4 f}-\frac {3}{4} \left (\frac {1}{3} \left (-\frac {d \left (200 a^3 c d^3+90 a^2 b d^2 \left (2 c^2+3 d^2\right )-6 a b^2 c d \left (6 c^2-71 d^2\right )+b^3 \left (4 c^4+36 c^2 d^2+75 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{2 f}+\frac {15}{2} d^2 x \left (8 a^3 \left (2 c^3+3 c d^2\right )+18 a^2 b d \left (4 c^2+d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+b^3 d \left (18 c^2+5 d^2\right )\right )-\frac {2 \left (40 a^3 d^3 \left (4 c^2+d^2\right )+90 a^2 b c d^2 \left (c^2+4 d^2\right )-6 a b^2 d \left (3 c^4-52 c^2 d^2-16 d^4\right )+b^3 \left (2 c^5+17 c^3 d^2+96 c d^4\right )\right ) \cos (e+f x)}{f}\right )-\frac {\left (40 a^3 d^3+90 a^2 b c d^2-a b^2 \left (18 c^2 d-96 d^3\right )+b^3 \left (2 c^3+21 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f}\right )}{5 d}}{6 d}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}\) |
-1/6*(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(d*f) + ((b^2*(2*b*c - 13*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(5*d*f) - (- 1/4*(b*(18*a*b*c*d - 90*a^2*d^2 - b^2*(2*c^2 + 25*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/f - (3*(-1/3*((90*a^2*b*c*d^2 + 40*a^3*d^3 + b^3*(2*c^3 + 21*c*d^2) - a*b^2*(18*c^2*d - 96*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x] )^2)/f + ((15*d^2*(18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a *b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*x)/2 - (2*(40*a^3*d^3*(4 *c^2 + d^2) + 90*a^2*b*c*d^2*(c^2 + 4*d^2) - 6*a*b^2*d*(3*c^4 - 52*c^2*d^2 - 16*d^4) + b^3*(2*c^5 + 17*c^3*d^2 + 96*c*d^4))*Cos[e + f*x])/f - (d*(20 0*a^3*c*d^3 - 6*a*b^2*c*d*(6*c^2 - 71*d^2) + 90*a^2*b*d^2*(2*c^2 + 3*d^2) + b^3*(4*c^4 + 36*c^2*d^2 + 75*d^4))*Cos[e + f*x]*Sin[e + f*x])/(2*f))/3)) /4)/(5*d))/(6*d)
3.7.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d *(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m ] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 5.35 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.86
| method | result | size |
| parts | \(a^{3} c^{3} x -\frac {\left (3 a \,b^{2} d^{3}+3 b^{3} c \,d^{2}\right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {\left (3 a^{3} c^{2} d +3 a^{2} b \,c^{3}\right ) \cos \left (f x +e \right )}{f}+\frac {\left (3 a^{2} b \,d^{3}+9 a \,b^{2} c \,d^{2}+3 b^{3} c^{2} d \right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {\left (3 a^{3} c \,d^{2}+9 a^{2} b \,c^{2} d +3 a \,b^{2} c^{3}\right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (d^{3} a^{3}+9 c \,d^{2} a^{2} b +9 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {b^{3} d^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\) | \(320\) |
| parallelrisch | \(\frac {45 \left (\left (-16 c^{2} d -5 d^{3}\right ) b^{3}-16 a c \left (c^{2}+3 d^{2}\right ) b^{2}-48 a^{2} \left (c^{2}+\frac {d^{2}}{3}\right ) d b -16 a^{3} c \,d^{2}\right ) \sin \left (2 f x +2 e \right )+80 \left (d a +c b \right ) \left (\left (c^{2}+\frac {15 d^{2}}{4}\right ) b^{2}+8 a b c d +d^{2} a^{2}\right ) \cos \left (3 f x +3 e \right )+45 \left (\left (2 c^{2} d +d^{3}\right ) b^{3}+6 a \,b^{2} c \,d^{2}+2 a^{2} b \,d^{3}\right ) \sin \left (4 f x +4 e \right )+36 \left (-a \,b^{2} d^{3}-b^{3} c \,d^{2}\right ) \cos \left (5 f x +5 e \right )-5 b^{3} d^{3} \sin \left (6 f x +6 e \right )-2880 \left (d a +c b \right ) \left (\frac {\left (c^{2}+\frac {5 d^{2}}{2}\right ) b^{2}}{4}+2 a b c d +a^{2} \left (c^{2}+\frac {d^{2}}{4}\right )\right ) \cos \left (f x +e \right )+4 \left (270 c^{2} d f x +75 d^{3} f x -160 c^{3}-384 c \,d^{2}\right ) b^{3}+1440 a \left (c^{3} f x +\frac {9}{4} c \,d^{2} f x -4 c^{2} d -\frac {16}{15} d^{3}\right ) b^{2}-2880 \left (-\frac {3}{2} c^{2} d f x -\frac {3}{8} d^{3} f x +c^{3}+2 c \,d^{2}\right ) a^{2} b +960 \left (c^{3} f x +\frac {3}{2} c \,d^{2} f x -3 c^{2} d -\frac {2}{3} d^{3}\right ) a^{3}}{960 f}\) | \(388\) |
| derivativedivides | \(\frac {b^{3} d^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {3 a \,b^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-\frac {3 b^{3} c \,d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{2} b \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+9 a \,b^{2} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+3 b^{3} c^{2} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {d^{3} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-3 c \,d^{2} a^{2} b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 a \,b^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-\frac {b^{3} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 a^{3} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+9 a^{2} b \,c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 a \,b^{2} c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c^{2} d \cos \left (f x +e \right )-3 a^{2} b \,c^{3} \cos \left (f x +e \right )+c^{3} a^{3} \left (f x +e \right )}{f}\) | \(489\) |
| default | \(\frac {b^{3} d^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {3 a \,b^{2} d^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-\frac {3 b^{3} c \,d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{2} b \,d^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+9 a \,b^{2} c \,d^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+3 b^{3} c^{2} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {d^{3} a^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-3 c \,d^{2} a^{2} b \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 a \,b^{2} c^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-\frac {b^{3} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+3 a^{3} c \,d^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+9 a^{2} b \,c^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 a \,b^{2} c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-3 a^{3} c^{2} d \cos \left (f x +e \right )-3 a^{2} b \,c^{3} \cos \left (f x +e \right )+c^{3} a^{3} \left (f x +e \right )}{f}\) | \(489\) |
| risch | \(-\frac {3 \cos \left (f x +e \right ) d^{3} a^{3}}{4 f}-\frac {3 \cos \left (f x +e \right ) b^{3} c^{3}}{4 f}+\frac {9 b^{3} c^{2} d x}{8}-\frac {9 \sin \left (2 f x +2 e \right ) a^{2} b \,c^{2} d}{4 f}-\frac {9 \sin \left (2 f x +2 e \right ) a \,b^{2} c \,d^{2}}{4 f}+\frac {9 a^{2} b \,d^{3} x}{8}-\frac {27 \cos \left (f x +e \right ) c \,d^{2} a^{2} b}{4 f}-\frac {27 \cos \left (f x +e \right ) a \,b^{2} c^{2} d}{4 f}+\frac {9 \sin \left (4 f x +4 e \right ) a \,b^{2} c \,d^{2}}{32 f}+\frac {3 \cos \left (3 f x +3 e \right ) c \,d^{2} a^{2} b}{4 f}+\frac {3 \cos \left (3 f x +3 e \right ) a \,b^{2} c^{2} d}{4 f}-\frac {3 d^{3} b^{2} \cos \left (5 f x +5 e \right ) a}{80 f}+\frac {5 \cos \left (3 f x +3 e \right ) b^{3} c \,d^{2}}{16 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{3} c \,d^{2}}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) a^{2} b \,d^{3}}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) a \,b^{2} c^{3}}{4 f}-\frac {3 \sin \left (2 f x +2 e \right ) b^{3} c^{2} d}{4 f}-\frac {15 \sin \left (2 f x +2 e \right ) b^{3} d^{3}}{64 f}-\frac {3 \cos \left (f x +e \right ) a^{3} c^{2} d}{f}-\frac {3 \cos \left (f x +e \right ) a^{2} b \,c^{3}}{f}-\frac {15 \cos \left (f x +e \right ) a \,b^{2} d^{3}}{8 f}-\frac {15 \cos \left (f x +e \right ) b^{3} c \,d^{2}}{8 f}+\frac {\cos \left (3 f x +3 e \right ) d^{3} a^{3}}{12 f}+\frac {\cos \left (3 f x +3 e \right ) b^{3} c^{3}}{12 f}-\frac {3 d^{2} b^{3} \cos \left (5 f x +5 e \right ) c}{80 f}+\frac {3 x \,a^{3} c \,d^{2}}{2}+\frac {3 x a \,b^{2} c^{3}}{2}+\frac {3 \sin \left (4 f x +4 e \right ) b^{3} d^{3}}{64 f}+\frac {5 x \,b^{3} d^{3}}{16}-\frac {b^{3} d^{3} \sin \left (6 f x +6 e \right )}{192 f}+\frac {9 x \,a^{2} b \,c^{2} d}{2}+\frac {27 x a \,b^{2} c \,d^{2}}{8}+\frac {3 \sin \left (4 f x +4 e \right ) a^{2} b \,d^{3}}{32 f}+\frac {3 \sin \left (4 f x +4 e \right ) b^{3} c^{2} d}{32 f}+\frac {5 \cos \left (3 f x +3 e \right ) a \,b^{2} d^{3}}{16 f}+a^{3} c^{3} x\) | \(645\) |
| norman | \(\text {Expression too large to display}\) | \(1540\) |
a^3*c^3*x-1/5*(3*a*b^2*d^3+3*b^3*c*d^2)/f*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e) ^2)*cos(f*x+e)-(3*a^3*c^2*d+3*a^2*b*c^3)/f*cos(f*x+e)+(3*a^2*b*d^3+9*a*b^2 *c*d^2+3*b^3*c^2*d)/f*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f *x+3/8*e)+(3*a^3*c*d^2+9*a^2*b*c^2*d+3*a*b^2*c^3)/f*(-1/2*sin(f*x+e)*cos(f *x+e)+1/2*f*x+1/2*e)-1/3*(a^3*d^3+9*a^2*b*c*d^2+9*a*b^2*c^2*d+b^3*c^3)/f*( 2+sin(f*x+e)^2)*cos(f*x+e)+b^3*d^3/f*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+ 15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)
Time = 0.32 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {144 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} + {\left (a^{3} + 6 \, a b^{2}\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (8 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{3} + 18 \, {\left (4 \, a^{2} b + b^{3}\right )} c^{2} d + 6 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c d^{2} + {\left (18 \, a^{2} b + 5 \, b^{3}\right )} d^{3}\right )} f x + 240 \, {\left ({\left (3 \, a^{2} b + b^{3}\right )} c^{3} + 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d + 3 \, {\left (3 \, a^{2} b + b^{3}\right )} c d^{2} + {\left (a^{3} + 3 \, a b^{2}\right )} d^{3}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, b^{3} d^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, b^{3} c^{2} d + 54 \, a b^{2} c d^{2} + {\left (18 \, a^{2} b + 13 \, b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (24 \, a b^{2} c^{3} + 6 \, {\left (12 \, a^{2} b + 5 \, b^{3}\right )} c^{2} d + 6 \, {\left (4 \, a^{3} + 15 \, a b^{2}\right )} c d^{2} + {\left (30 \, a^{2} b + 11 \, b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
-1/240*(144*(b^3*c*d^2 + a*b^2*d^3)*cos(f*x + e)^5 - 80*(b^3*c^3 + 9*a*b^2 *c^2*d + 3*(3*a^2*b + 2*b^3)*c*d^2 + (a^3 + 6*a*b^2)*d^3)*cos(f*x + e)^3 - 15*(8*(2*a^3 + 3*a*b^2)*c^3 + 18*(4*a^2*b + b^3)*c^2*d + 6*(4*a^3 + 9*a*b ^2)*c*d^2 + (18*a^2*b + 5*b^3)*d^3)*f*x + 240*((3*a^2*b + b^3)*c^3 + 3*(a^ 3 + 3*a*b^2)*c^2*d + 3*(3*a^2*b + b^3)*c*d^2 + (a^3 + 3*a*b^2)*d^3)*cos(f* x + e) + 5*(8*b^3*d^3*cos(f*x + e)^5 - 2*(18*b^3*c^2*d + 54*a*b^2*c*d^2 + (18*a^2*b + 13*b^3)*d^3)*cos(f*x + e)^3 + 3*(24*a*b^2*c^3 + 6*(12*a^2*b + 5*b^3)*c^2*d + 6*(4*a^3 + 15*a*b^2)*c*d^2 + (30*a^2*b + 11*b^3)*d^3)*cos(f *x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 1217 vs. \(2 (400) = 800\).
Time = 0.49 (sec) , antiderivative size = 1217, normalized size of antiderivative = 3.25 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]
Piecewise((a**3*c**3*x - 3*a**3*c**2*d*cos(e + f*x)/f + 3*a**3*c*d**2*x*si n(e + f*x)**2/2 + 3*a**3*c*d**2*x*cos(e + f*x)**2/2 - 3*a**3*c*d**2*sin(e + f*x)*cos(e + f*x)/(2*f) - a**3*d**3*sin(e + f*x)**2*cos(e + f*x)/f - 2*a **3*d**3*cos(e + f*x)**3/(3*f) - 3*a**2*b*c**3*cos(e + f*x)/f + 9*a**2*b*c **2*d*x*sin(e + f*x)**2/2 + 9*a**2*b*c**2*d*x*cos(e + f*x)**2/2 - 9*a**2*b *c**2*d*sin(e + f*x)*cos(e + f*x)/(2*f) - 9*a**2*b*c*d**2*sin(e + f*x)**2* cos(e + f*x)/f - 6*a**2*b*c*d**2*cos(e + f*x)**3/f + 9*a**2*b*d**3*x*sin(e + f*x)**4/8 + 9*a**2*b*d**3*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 9*a**2* b*d**3*x*cos(e + f*x)**4/8 - 15*a**2*b*d**3*sin(e + f*x)**3*cos(e + f*x)/( 8*f) - 9*a**2*b*d**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*a*b**2*c**3*x* sin(e + f*x)**2/2 + 3*a*b**2*c**3*x*cos(e + f*x)**2/2 - 3*a*b**2*c**3*sin( e + f*x)*cos(e + f*x)/(2*f) - 9*a*b**2*c**2*d*sin(e + f*x)**2*cos(e + f*x) /f - 6*a*b**2*c**2*d*cos(e + f*x)**3/f + 27*a*b**2*c*d**2*x*sin(e + f*x)** 4/8 + 27*a*b**2*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 27*a*b**2*c*d **2*x*cos(e + f*x)**4/8 - 45*a*b**2*c*d**2*sin(e + f*x)**3*cos(e + f*x)/(8 *f) - 27*a*b**2*c*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*a*b**2*d**3* sin(e + f*x)**4*cos(e + f*x)/f - 4*a*b**2*d**3*sin(e + f*x)**2*cos(e + f*x )**3/f - 8*a*b**2*d**3*cos(e + f*x)**5/(5*f) - b**3*c**3*sin(e + f*x)**2*c os(e + f*x)/f - 2*b**3*c**3*cos(e + f*x)**3/(3*f) + 9*b**3*c**2*d*x*sin(e + f*x)**4/8 + 9*b**3*c**2*d*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 9*b**...
Time = 0.23 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.28 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {960 \, {\left (f x + e\right )} a^{3} c^{3} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} c^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{3} c^{3} + 2160 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b c^{2} d + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a b^{2} c^{2} d + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{3} c^{2} d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d^{2} + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} b c d^{2} + 270 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} c d^{2} - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} b^{3} c d^{2} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} d^{3} + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b d^{3} - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a b^{2} d^{3} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{3} d^{3} - 2880 \, a^{2} b c^{3} \cos \left (f x + e\right ) - 2880 \, a^{3} c^{2} d \cos \left (f x + e\right )}{960 \, f} \]
1/960*(960*(f*x + e)*a^3*c^3 + 720*(2*f*x + 2*e - sin(2*f*x + 2*e))*a*b^2* c^3 + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*b^3*c^3 + 2160*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*b*c^2*d + 2880*(cos(f*x + e)^3 - 3*cos(f*x + e))*a*b ^2*c^2*d + 90*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*b^3* c^2*d + 720*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^3*c*d^2 + 2880*(cos(f*x + e )^3 - 3*cos(f*x + e))*a^2*b*c*d^2 + 270*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a*b^2*c*d^2 - 192*(3*cos(f*x + e)^5 - 10*cos(f*x + e )^3 + 15*cos(f*x + e))*b^3*c*d^2 + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*a ^3*d^3 + 90*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*b* d^3 - 192*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a*b^2*d ^3 + 5*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin (2*f*x + 2*e))*b^3*d^3 - 2880*a^2*b*c^3*cos(f*x + e) - 2880*a^3*c^2*d*cos( f*x + e))/f
Time = 0.34 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.10 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {b^{3} d^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (16 \, a^{3} c^{3} + 24 \, a b^{2} c^{3} + 72 \, a^{2} b c^{2} d + 18 \, b^{3} c^{2} d + 24 \, a^{3} c d^{2} + 54 \, a b^{2} c d^{2} + 18 \, a^{2} b d^{3} + 5 \, b^{3} d^{3}\right )} x - \frac {3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, b^{3} c^{3} + 36 \, a b^{2} c^{2} d + 36 \, a^{2} b c d^{2} + 15 \, b^{3} c d^{2} + 4 \, a^{3} d^{3} + 15 \, a b^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {3 \, {\left (8 \, a^{2} b c^{3} + 2 \, b^{3} c^{3} + 8 \, a^{3} c^{2} d + 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} + 5 \, b^{3} c d^{2} + 2 \, a^{3} d^{3} + 5 \, a b^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {3 \, {\left (2 \, b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3} + b^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {3 \, {\left (16 \, a b^{2} c^{3} + 48 \, a^{2} b c^{2} d + 16 \, b^{3} c^{2} d + 16 \, a^{3} c d^{2} + 48 \, a b^{2} c d^{2} + 16 \, a^{2} b d^{3} + 5 \, b^{3} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
-1/192*b^3*d^3*sin(6*f*x + 6*e)/f + 1/16*(16*a^3*c^3 + 24*a*b^2*c^3 + 72*a ^2*b*c^2*d + 18*b^3*c^2*d + 24*a^3*c*d^2 + 54*a*b^2*c*d^2 + 18*a^2*b*d^3 + 5*b^3*d^3)*x - 3/80*(b^3*c*d^2 + a*b^2*d^3)*cos(5*f*x + 5*e)/f + 1/48*(4* b^3*c^3 + 36*a*b^2*c^2*d + 36*a^2*b*c*d^2 + 15*b^3*c*d^2 + 4*a^3*d^3 + 15* a*b^2*d^3)*cos(3*f*x + 3*e)/f - 3/8*(8*a^2*b*c^3 + 2*b^3*c^3 + 8*a^3*c^2*d + 18*a*b^2*c^2*d + 18*a^2*b*c*d^2 + 5*b^3*c*d^2 + 2*a^3*d^3 + 5*a*b^2*d^3 )*cos(f*x + e)/f + 3/64*(2*b^3*c^2*d + 6*a*b^2*c*d^2 + 2*a^2*b*d^3 + b^3*d ^3)*sin(4*f*x + 4*e)/f - 3/64*(16*a*b^2*c^3 + 48*a^2*b*c^2*d + 16*b^3*c^2* d + 16*a^3*c*d^2 + 48*a*b^2*c*d^2 + 16*a^2*b*d^3 + 5*b^3*d^3)*sin(2*f*x + 2*e)/f
Time = 9.52 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.53 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {180\,a^3\,d^3\,\cos \left (e+f\,x\right )+180\,b^3\,c^3\,\cos \left (e+f\,x\right )-20\,a^3\,d^3\,\cos \left (3\,e+3\,f\,x\right )-20\,b^3\,c^3\,\cos \left (3\,e+3\,f\,x\right )+\frac {225\,b^3\,d^3\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {45\,b^3\,d^3\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {5\,b^3\,d^3\,\sin \left (6\,e+6\,f\,x\right )}{4}-75\,a\,b^2\,d^3\,\cos \left (3\,e+3\,f\,x\right )+9\,a\,b^2\,d^3\,\cos \left (5\,e+5\,f\,x\right )-75\,b^3\,c\,d^2\,\cos \left (3\,e+3\,f\,x\right )+9\,b^3\,c\,d^2\,\cos \left (5\,e+5\,f\,x\right )+180\,a\,b^2\,c^3\,\sin \left (2\,e+2\,f\,x\right )+180\,a^2\,b\,d^3\,\sin \left (2\,e+2\,f\,x\right )-\frac {45\,a^2\,b\,d^3\,\sin \left (4\,e+4\,f\,x\right )}{2}+180\,a^3\,c\,d^2\,\sin \left (2\,e+2\,f\,x\right )+180\,b^3\,c^2\,d\,\sin \left (2\,e+2\,f\,x\right )-\frac {45\,b^3\,c^2\,d\,\sin \left (4\,e+4\,f\,x\right )}{2}+720\,a^2\,b\,c^3\,\cos \left (e+f\,x\right )+450\,a\,b^2\,d^3\,\cos \left (e+f\,x\right )+720\,a^3\,c^2\,d\,\cos \left (e+f\,x\right )+450\,b^3\,c\,d^2\,\cos \left (e+f\,x\right )-240\,a^3\,c^3\,f\,x-75\,b^3\,d^3\,f\,x+1620\,a\,b^2\,c^2\,d\,\cos \left (e+f\,x\right )+1620\,a^2\,b\,c\,d^2\,\cos \left (e+f\,x\right )-360\,a\,b^2\,c^3\,f\,x-270\,a^2\,b\,d^3\,f\,x-360\,a^3\,c\,d^2\,f\,x-270\,b^3\,c^2\,d\,f\,x-180\,a\,b^2\,c^2\,d\,\cos \left (3\,e+3\,f\,x\right )-180\,a^2\,b\,c\,d^2\,\cos \left (3\,e+3\,f\,x\right )+540\,a\,b^2\,c\,d^2\,\sin \left (2\,e+2\,f\,x\right )+540\,a^2\,b\,c^2\,d\,\sin \left (2\,e+2\,f\,x\right )-\frac {135\,a\,b^2\,c\,d^2\,\sin \left (4\,e+4\,f\,x\right )}{2}-810\,a\,b^2\,c\,d^2\,f\,x-1080\,a^2\,b\,c^2\,d\,f\,x}{240\,f} \]
-(180*a^3*d^3*cos(e + f*x) + 180*b^3*c^3*cos(e + f*x) - 20*a^3*d^3*cos(3*e + 3*f*x) - 20*b^3*c^3*cos(3*e + 3*f*x) + (225*b^3*d^3*sin(2*e + 2*f*x))/4 - (45*b^3*d^3*sin(4*e + 4*f*x))/4 + (5*b^3*d^3*sin(6*e + 6*f*x))/4 - 75*a *b^2*d^3*cos(3*e + 3*f*x) + 9*a*b^2*d^3*cos(5*e + 5*f*x) - 75*b^3*c*d^2*co s(3*e + 3*f*x) + 9*b^3*c*d^2*cos(5*e + 5*f*x) + 180*a*b^2*c^3*sin(2*e + 2* f*x) + 180*a^2*b*d^3*sin(2*e + 2*f*x) - (45*a^2*b*d^3*sin(4*e + 4*f*x))/2 + 180*a^3*c*d^2*sin(2*e + 2*f*x) + 180*b^3*c^2*d*sin(2*e + 2*f*x) - (45*b^ 3*c^2*d*sin(4*e + 4*f*x))/2 + 720*a^2*b*c^3*cos(e + f*x) + 450*a*b^2*d^3*c os(e + f*x) + 720*a^3*c^2*d*cos(e + f*x) + 450*b^3*c*d^2*cos(e + f*x) - 24 0*a^3*c^3*f*x - 75*b^3*d^3*f*x + 1620*a*b^2*c^2*d*cos(e + f*x) + 1620*a^2* b*c*d^2*cos(e + f*x) - 360*a*b^2*c^3*f*x - 270*a^2*b*d^3*f*x - 360*a^3*c*d ^2*f*x - 270*b^3*c^2*d*f*x - 180*a*b^2*c^2*d*cos(3*e + 3*f*x) - 180*a^2*b* c*d^2*cos(3*e + 3*f*x) + 540*a*b^2*c*d^2*sin(2*e + 2*f*x) + 540*a^2*b*c^2* d*sin(2*e + 2*f*x) - (135*a*b^2*c*d^2*sin(4*e + 4*f*x))/2 - 810*a*b^2*c*d^ 2*f*x - 1080*a^2*b*c^2*d*f*x)/(240*f)