\(\int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx\) [686]

Optimal result

Integrand size = 25, antiderivative size = 374 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {1}{16} \left (162 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+18 b^2 c \left (4 c^2+9 d^2\right )+216 \left (2 c^3+3 c d^2\right )\right ) x-\frac {\left (9 b^2 d \left (3 c^2+d^2\right )+27 b c \left (c^2+3 d^2\right )+b^3 c \left (c^2+3 d^2\right )+27 \left (3 c^2 d+d^3\right )\right ) \cos (e+f x)}{f}+\frac {(b c+3 d) \left (24 b c d+9 d^2+b^2 \left (c^2+6 d^2\right )\right ) \cos ^3(e+f x)}{3 f}-\frac {3 b^2 d^2 (b c+3 d) \cos ^5(e+f x)}{5 f}-\frac {\left (648 c d^2+162 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+18 b^2 c \left (4 c^2+9 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{16 f}-\frac {5 b^3 d^3 \cos (e+f x) \sin ^3(e+f x)}{24 f}-\frac {3 b d \left (b^2 c^2+9 b c d+9 d^2\right ) \cos (e+f x) \sin ^3(e+f x)}{4 f}-\frac {b^3 d^3 \cos (e+f x) \sin ^5(e+f x)}{6 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2872, 3102, 2832, 2813} \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {b \left (-90 a^2 d^2+18 a b c d-\left (b^2 \left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}-\frac {\left (200 a^3 c d^3+90 a^2 b d^2 \left (2 c^2+3 d^2\right )-6 a b^2 c d \left (6 c^2-71 d^2\right )+b^3 \left (4 c^4+36 c^2 d^2+75 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac {1}{16} x \left (8 a^3 \left (2 c^3+3 c d^2\right )+18 a^2 b d \left (4 c^2+d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+b^3 d \left (18 c^2+5 d^2\right )\right )-\frac {\left (40 a^3 d^3+90 a^2 b c d^2-a b^2 \left (18 c^2 d-96 d^3\right )+b^3 \left (2 c^3+21 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}-\frac {\left (40 a^3 d^3 \left (4 c^2+d^2\right )+90 a^2 b c d^2 \left (c^2+4 d^2\right )-6 a b^2 d \left (3 c^4-52 c^2 d^2-16 d^4\right )+b^3 \left (2 c^5+17 c^3 d^2+96 c d^4\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f} \]

[In]

Int[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]

[Out]

((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)/
16 - ((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])/(60*d^2*f) - ((200*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])/(240*d*f) - ((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)/(120*d^2*f) + (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)/(
120*d^2*f) + (b^2*(2*b*c - 13*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(30*d^2*f) - (b^2*Cos[e + f*x]*(a + b*
Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(6*d*f)

Rule 2813

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)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2832

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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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]

Rule 2872

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-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] + Dist[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))*Sin[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])))

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (b^3 c+6 a^3 d+4 a b^2 d-b \left (a b c-18 a^2 d-5 b^2 d\right ) \sin (e+f x)-b^2 (2 b c-13 a d) \sin ^2(e+f x)\right ) \, dx}{6 d} \\ & = \frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^3 \left (-3 d \left (b^3 c-10 a^3 d-24 a b^2 d\right )-b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{30 d^2} \\ & = \frac {b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (3 d \left (40 a^3 c d+78 a b^2 c d+90 a^2 b d^2-b^3 \left (2 c^2-25 d^2\right )\right )+3 \left (90 a^2 b c d^2+40 a^3 d^3+b^3 \left (2 c^3+21 c d^2\right )-a b^2 \left (18 c^2 d-96 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{120 d^2} \\ & = -\frac {\left (90 a^2 b c d^2+40 a^3 d^3+b^3 \left (2 c^3+21 c d^2\right )-a b^2 \left (18 c^2 d-96 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f}+\frac {\int (c+d \sin (e+f x)) \left (3 d \left (450 a^2 b c d^2+40 a^3 d \left (3 c^2+2 d^2\right )+6 a b^2 d \left (33 c^2+32 d^2\right )-b^3 \left (2 c^3-117 c d^2\right )\right )+3 \left (200 a^3 c d^3-6 a b^2 c d \left (6 c^2-71 d^2\right )+90 a^2 b d^2 \left (2 c^2+3 d^2\right )+b^3 \left (4 c^4+36 c^2 d^2+75 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{360 d^2} \\ & = \frac {1}{16} \left (18 a^2 b d \left (4 c^2+d^2\right )+b^3 d \left (18 c^2+5 d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+8 a^3 \left (2 c^3+3 c d^2\right )\right ) x-\frac {\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)}{60 d^2 f}-\frac {\left (200 a^3 c d^3-6 a b^2 c d \left (6 c^2-71 d^2\right )+90 a^2 b d^2 \left (2 c^2+3 d^2\right )+b^3 \left (4 c^4+36 c^2 d^2+75 d^4\right )\right ) \cos (e+f x) \sin (e+f x)}{240 d f}-\frac {\left (90 a^2 b c d^2+40 a^3 d^3+b^3 \left (2 c^3+21 c d^2\right )-a b^2 \left (18 c^2 d-96 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}+\frac {b \left (18 a b c d-90 a^2 d^2-b^2 \left (2 c^2+25 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}+\frac {b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac {b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.90 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.32 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {25920 c^3 e+4320 b^2 c^3 e+38880 b c^2 d e+1080 b^3 c^2 d e+38880 c d^2 e+9720 b^2 c d^2 e+9720 b d^3 e+300 b^3 d^3 e+25920 c^3 f x+4320 b^2 c^3 f x+38880 b c^2 d f x+1080 b^3 c^2 d f x+38880 c d^2 f x+9720 b^2 c d^2 f x+9720 b d^3 f x+300 b^3 d^3 f x-360 \left (b^3 \left (2 c^3+5 c d^2\right )+18 b \left (4 c^3+9 c d^2\right )+54 \left (4 c^2 d+d^3\right )+3 b^2 \left (18 c^2 d+5 d^3\right )\right ) \cos (e+f x)+20 \left (324 b c d^2+108 d^3+b^3 \left (4 c^3+15 c d^2\right )+9 b^2 \left (12 c^2 d+5 d^3\right )\right ) \cos (3 (e+f x))-36 b^3 c d^2 \cos (5 (e+f x))-108 b^2 d^3 \cos (5 (e+f x))-2160 b^2 c^3 \sin (2 (e+f x))-19440 b c^2 d \sin (2 (e+f x))-720 b^3 c^2 d \sin (2 (e+f x))-19440 c d^2 \sin (2 (e+f x))-6480 b^2 c d^2 \sin (2 (e+f x))-6480 b d^3 \sin (2 (e+f x))-225 b^3 d^3 \sin (2 (e+f x))+90 b^3 c^2 d \sin (4 (e+f x))+810 b^2 c d^2 \sin (4 (e+f x))+810 b d^3 \sin (4 (e+f x))+45 b^3 d^3 \sin (4 (e+f x))-5 b^3 d^3 \sin (6 (e+f x))}{960 f} \]

[In]

Integrate[(3 + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3,x]

[Out]

(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)

Maple [A] (verified)

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	\(\text {Expression too large to display}\)	\(1540\)

[In]

int((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {144 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} \cos \left (f x + e\right )^{5} - 80 \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 3 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d^{2} + {\left (a^{3} + 6 \, a b^{2}\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (8 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{3} + 18 \, {\left (4 \, a^{2} b + b^{3}\right )} c^{2} d + 6 \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c d^{2} + {\left (18 \, a^{2} b + 5 \, b^{3}\right )} d^{3}\right )} f x + 240 \, {\left ({\left (3 \, a^{2} b + b^{3}\right )} c^{3} + 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d + 3 \, {\left (3 \, a^{2} b + b^{3}\right )} c d^{2} + {\left (a^{3} + 3 \, a b^{2}\right )} d^{3}\right )} \cos \left (f x + e\right ) + 5 \, {\left (8 \, b^{3} d^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (18 \, b^{3} c^{2} d + 54 \, a b^{2} c d^{2} + {\left (18 \, a^{2} b + 13 \, b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (24 \, a b^{2} c^{3} + 6 \, {\left (12 \, a^{2} b + 5 \, b^{3}\right )} c^{2} d + 6 \, {\left (4 \, a^{3} + 15 \, a b^{2}\right )} c d^{2} + {\left (30 \, a^{2} b + 11 \, b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]

[In]

integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

-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

Sympy [B] (verification not implemented)

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 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sin(f*x+e))**3*(c+d*sin(f*x+e))**3,x)

[Out]

Piecewise((a**3*c**3*x - 3*a**3*c**2*d*cos(e + f*x)/f + 3*a**3*c*d**2*x*sin(e + f*x)**2/2 + 3*a**3*c*d**2*x*co
s(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*s
in(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*cos(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**3*c**2*d*x*cos(e + f*x)**4/8 - 15*b**3*c**2*d*sin(e + f*x)**3*c
os(e + f*x)/(8*f) - 9*b**3*c**2*d*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*b**3*c*d**2*sin(e + f*x)**4*cos(e + f
*x)/f - 4*b**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)**3/f - 8*b**3*c*d**2*cos(e + f*x)**5/(5*f) + 5*b**3*d**3*x*
sin(e + f*x)**6/16 + 15*b**3*d**3*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 15*b**3*d**3*x*sin(e + f*x)**2*cos(e
+ f*x)**4/16 + 5*b**3*d**3*x*cos(e + f*x)**6/16 - 11*b**3*d**3*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 5*b**3*d*
*3*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 5*b**3*d**3*sin(e + f*x)*cos(e + f*x)**5/(16*f), Ne(f, 0)), (x*(a +
b*sin(e))**3*(c + d*sin(e))**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.28 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=\frac {960 \, {\left (f x + e\right )} a^{3} c^{3} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} c^{3} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{3} c^{3} + 2160 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b c^{2} d + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a b^{2} c^{2} d + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{3} c^{2} d + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c d^{2} + 2880 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} b c d^{2} + 270 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a b^{2} c d^{2} - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} b^{3} c d^{2} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} d^{3} + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} b d^{3} - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a b^{2} d^{3} + 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{3} d^{3} - 2880 \, a^{2} b c^{3} \cos \left (f x + e\right ) - 2880 \, a^{3} c^{2} d \cos \left (f x + e\right )}{960 \, f} \]

[In]

integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.10 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {b^{3} d^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (16 \, a^{3} c^{3} + 24 \, a b^{2} c^{3} + 72 \, a^{2} b c^{2} d + 18 \, b^{3} c^{2} d + 24 \, a^{3} c d^{2} + 54 \, a b^{2} c d^{2} + 18 \, a^{2} b d^{3} + 5 \, b^{3} d^{3}\right )} x - \frac {3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (4 \, b^{3} c^{3} + 36 \, a b^{2} c^{2} d + 36 \, a^{2} b c d^{2} + 15 \, b^{3} c d^{2} + 4 \, a^{3} d^{3} + 15 \, a b^{2} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {3 \, {\left (8 \, a^{2} b c^{3} + 2 \, b^{3} c^{3} + 8 \, a^{3} c^{2} d + 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} + 5 \, b^{3} c d^{2} + 2 \, a^{3} d^{3} + 5 \, a b^{2} d^{3}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {3 \, {\left (2 \, b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3} + b^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {3 \, {\left (16 \, a b^{2} c^{3} + 48 \, a^{2} b c^{2} d + 16 \, b^{3} c^{2} d + 16 \, a^{3} c d^{2} + 48 \, a b^{2} c d^{2} + 16 \, a^{2} b d^{3} + 5 \, b^{3} d^{3}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

-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*(1
6*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

Mupad [B] (verification not implemented)

Time = 9.52 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.53 \[ \int (3+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx=-\frac {180\,a^3\,d^3\,\cos \left (e+f\,x\right )+180\,b^3\,c^3\,\cos \left (e+f\,x\right )-20\,a^3\,d^3\,\cos \left (3\,e+3\,f\,x\right )-20\,b^3\,c^3\,\cos \left (3\,e+3\,f\,x\right )+\frac {225\,b^3\,d^3\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {45\,b^3\,d^3\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {5\,b^3\,d^3\,\sin \left (6\,e+6\,f\,x\right )}{4}-75\,a\,b^2\,d^3\,\cos \left (3\,e+3\,f\,x\right )+9\,a\,b^2\,d^3\,\cos \left (5\,e+5\,f\,x\right )-75\,b^3\,c\,d^2\,\cos \left (3\,e+3\,f\,x\right )+9\,b^3\,c\,d^2\,\cos \left (5\,e+5\,f\,x\right )+180\,a\,b^2\,c^3\,\sin \left (2\,e+2\,f\,x\right )+180\,a^2\,b\,d^3\,\sin \left (2\,e+2\,f\,x\right )-\frac {45\,a^2\,b\,d^3\,\sin \left (4\,e+4\,f\,x\right )}{2}+180\,a^3\,c\,d^2\,\sin \left (2\,e+2\,f\,x\right )+180\,b^3\,c^2\,d\,\sin \left (2\,e+2\,f\,x\right )-\frac {45\,b^3\,c^2\,d\,\sin \left (4\,e+4\,f\,x\right )}{2}+720\,a^2\,b\,c^3\,\cos \left (e+f\,x\right )+450\,a\,b^2\,d^3\,\cos \left (e+f\,x\right )+720\,a^3\,c^2\,d\,\cos \left (e+f\,x\right )+450\,b^3\,c\,d^2\,\cos \left (e+f\,x\right )-240\,a^3\,c^3\,f\,x-75\,b^3\,d^3\,f\,x+1620\,a\,b^2\,c^2\,d\,\cos \left (e+f\,x\right )+1620\,a^2\,b\,c\,d^2\,\cos \left (e+f\,x\right )-360\,a\,b^2\,c^3\,f\,x-270\,a^2\,b\,d^3\,f\,x-360\,a^3\,c\,d^2\,f\,x-270\,b^3\,c^2\,d\,f\,x-180\,a\,b^2\,c^2\,d\,\cos \left (3\,e+3\,f\,x\right )-180\,a^2\,b\,c\,d^2\,\cos \left (3\,e+3\,f\,x\right )+540\,a\,b^2\,c\,d^2\,\sin \left (2\,e+2\,f\,x\right )+540\,a^2\,b\,c^2\,d\,\sin \left (2\,e+2\,f\,x\right )-\frac {135\,a\,b^2\,c\,d^2\,\sin \left (4\,e+4\,f\,x\right )}{2}-810\,a\,b^2\,c\,d^2\,f\,x-1080\,a^2\,b\,c^2\,d\,f\,x}{240\,f} \]

[In]

int((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^3,x)

[Out]

-(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*cos(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 + 72
0*a^2*b*c^3*cos(e + f*x) + 450*a*b^2*d^3*cos(e + f*x) + 720*a^3*c^2*d*cos(e + f*x) + 450*b^3*c*d^2*cos(e + f*x
) - 240*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) - 18
0*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)