3.686 \(\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx\)

Optimal. Leaf size=400 \[ \frac{(a d+b c) \left (a^2 d^2+8 a b c d+b^2 \left (c^2+6 d^2\right )\right ) \cos ^3(e+f x)}{3 f}-\frac{\left (3 a^2 b c \left (c^2+3 d^2\right )+a^3 \left (3 c^2 d+d^3\right )+3 a b^2 d \left (3 c^2+d^2\right )+b^3 c \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f}-\frac{3 b d \left (a^2 d^2+3 a b c d+b^2 c^2\right ) \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac{\left (18 a^2 b d \left (4 c^2+d^2\right )+24 a^3 c d^2+6 a b^2 c \left (4 c^2+9 d^2\right )+b^3 d \left (18 c^2+5 d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left (18 a^2 b d \left (4 c^2+d^2\right )+8 a^3 \left (2 c^3+3 c d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+b^3 d \left (18 c^2+5 d^2\right )\right )-\frac{3 b^2 d^2 (a d+b c) \cos ^5(e+f x)}{5 f}-\frac{b^3 d^3 \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac{5 b^3 d^3 \sin ^3(e+f x) \cos (e+f x)}{24 f} \]

[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 - ((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[e +
f*x])/f + ((b*c + a*d)*(8*a*b*c*d + a^2*d^2 + b^2*(c^2 + 6*d^2))*Cos[e + f*x]^3)/(3*f) - (3*b^2*d^2*(b*c + a*d
)*Cos[e + f*x]^5)/(5*f) - ((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[e + f*x]*Sin[e + f*x])/(16*f) - (5*b^3*d^3*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) - (3*b*d*(b^2*c
^2 + 3*a*b*c*d + a^2*d^2)*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f) - (b^3*d^3*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f)

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Rubi [A]  time = 0.947514, antiderivative size = 493, normalized size of antiderivative = 1.23, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2793, 3023, 2753, 2734} \[ -\frac{\left (90 a^2 b c d^2 \left (c^2+4 d^2\right )+40 a^3 d^3 \left (4 c^2+d^2\right )-6 a b^2 d \left (-52 c^2 d^2+3 c^4-16 d^4\right )+b^3 \left (17 c^3 d^2+2 c^5+96 c d^4\right )\right ) \cos (e+f x)}{60 d^2 f}+\frac{b \left (-90 a^2 d^2+18 a b c d+b^2 \left (-\left (2 c^2+25 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{120 d^2 f}-\frac{\left (90 a^2 b c d^2+40 a^3 d^3-a b^2 \left (18 c^2 d-96 d^3\right )+b^3 \left (2 c^3+21 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{120 d^2 f}-\frac{\left (90 a^2 b d^2 \left (2 c^2+3 d^2\right )+200 a^3 c d^3-6 a b^2 d \left (6 c^3-71 c d^2\right )+b^3 \left (36 c^2 d^2+4 c^4+75 d^4\right )\right ) \sin (e+f x) \cos (e+f x)}{240 d f}+\frac{1}{16} x \left (18 a^2 b d \left (4 c^2+d^2\right )+8 a^3 \left (2 c^3+3 c d^2\right )+6 a b^2 c \left (4 c^2+9 d^2\right )+b^3 d \left (18 c^2+5 d^2\right )\right )+\frac{b^2 (2 b c-13 a d) \cos (e+f x) (c+d \sin (e+f x))^4}{30 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^4}{6 d f} \]

Antiderivative was successfully verified.

[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 + 90*a^2*b*d^2*(2*c^2 + 3*d^2) -
6*a*b^2*d*(6*c^3 - 71*c*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 2793

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(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 3023

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]

Rule 2753

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)*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]

Rule 2734

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]

Rubi steps

\begin{align*} \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3 \, dx &=-\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+90 a^2 b d^2 \left (2 c^2+3 d^2\right )-6 a b^2 d \left (6 c^3-71 c 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+90 a^2 b d^2 \left (2 c^2+3 d^2\right )-6 a b^2 d \left (6 c^3-71 c 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]  time = 1.15591, size = 552, normalized size = 1.38 \[ \frac{-360 \left (2 a^2 b c \left (4 c^2+9 d^2\right )+2 a^3 \left (4 c^2 d+d^3\right )+a b^2 d \left (18 c^2+5 d^2\right )+b^3 c \left (2 c^2+5 d^2\right )\right ) \cos (e+f x)+20 \left (36 a^2 b c d^2+4 a^3 d^3+3 a b^2 d \left (12 c^2+5 d^2\right )+b^3 \left (4 c^3+15 c d^2\right )\right ) \cos (3 (e+f x))-2160 a^2 b c^2 d \sin (2 (e+f x))+4320 a^2 b c^2 d e+4320 a^2 b c^2 d f x-720 a^2 b d^3 \sin (2 (e+f x))+90 a^2 b d^3 \sin (4 (e+f x))+1080 a^2 b d^3 e+1080 a^2 b d^3 f x+960 a^3 c^3 e+960 a^3 c^3 f x-720 a^3 c d^2 \sin (2 (e+f x))+1440 a^3 c d^2 e+1440 a^3 c d^2 f x-720 a b^2 c^3 \sin (2 (e+f x))+1440 a b^2 c^3 e+1440 a b^2 c^3 f x-2160 a b^2 c d^2 \sin (2 (e+f x))+270 a b^2 c d^2 \sin (4 (e+f x))+3240 a b^2 c d^2 e+3240 a b^2 c d^2 f x-36 a b^2 d^3 \cos (5 (e+f x))-720 b^3 c^2 d \sin (2 (e+f x))+90 b^3 c^2 d \sin (4 (e+f x))+1080 b^3 c^2 d e+1080 b^3 c^2 d f x-36 b^3 c d^2 \cos (5 (e+f x))-225 b^3 d^3 \sin (2 (e+f x))+45 b^3 d^3 \sin (4 (e+f x))-5 b^3 d^3 \sin (6 (e+f x))+300 b^3 d^3 e+300 b^3 d^3 f x}{960 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(960*a^3*c^3*e + 1440*a*b^2*c^3*e + 4320*a^2*b*c^2*d*e + 1080*b^3*c^2*d*e + 1440*a^3*c*d^2*e + 3240*a*b^2*c*d^
2*e + 1080*a^2*b*d^3*e + 300*b^3*d^3*e + 960*a^3*c^3*f*x + 1440*a*b^2*c^3*f*x + 4320*a^2*b*c^2*d*f*x + 1080*b^
3*c^2*d*f*x + 1440*a^3*c*d^2*f*x + 3240*a*b^2*c*d^2*f*x + 1080*a^2*b*d^3*f*x + 300*b^3*d^3*f*x - 360*(b^3*c*(2
*c^2 + 5*d^2) + a*b^2*d*(18*c^2 + 5*d^2) + 2*a^2*b*c*(4*c^2 + 9*d^2) + 2*a^3*(4*c^2*d + d^3))*Cos[e + f*x] + 2
0*(36*a^2*b*c*d^2 + 4*a^3*d^3 + 3*a*b^2*d*(12*c^2 + 5*d^2) + b^3*(4*c^3 + 15*c*d^2))*Cos[3*(e + f*x)] - 36*b^3
*c*d^2*Cos[5*(e + f*x)] - 36*a*b^2*d^3*Cos[5*(e + f*x)] - 720*a*b^2*c^3*Sin[2*(e + f*x)] - 2160*a^2*b*c^2*d*Si
n[2*(e + f*x)] - 720*b^3*c^2*d*Sin[2*(e + f*x)] - 720*a^3*c*d^2*Sin[2*(e + f*x)] - 2160*a*b^2*c*d^2*Sin[2*(e +
f*x)] - 720*a^2*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)] + 270*a
*b^2*c*d^2*Sin[4*(e + f*x)] + 90*a^2*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)

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Maple [A]  time = 0.041, size = 489, normalized size = 1.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(a^3*c^3*(f*x+e)-3*a^3*c^2*d*cos(f*x+e)+3*a^3*c*d^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-1/3*a^3*d^3
*(2+sin(f*x+e)^2)*cos(f*x+e)-3*a^2*b*c^3*cos(f*x+e)+9*a^2*b*c^2*d*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-3
*a^2*b*c*d^2*(2+sin(f*x+e)^2)*cos(f*x+e)+3*a^2*b*d^3*(-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*b^2*c^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-3*a*b^2*c^2*d*(2+sin(f*x+e)^2)*cos(f*x+e)+9*a*b^2*
c*d^2*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-3/5*a*b^2*d^3*(8/3+sin(f*x+e)^4+4/3*sin(f*
x+e)^2)*cos(f*x+e)-1/3*b^3*c^3*(2+sin(f*x+e)^2)*cos(f*x+e)+3*b^3*c^2*d*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos
(f*x+e)+3/8*f*x+3/8*e)-3/5*b^3*c*d^2*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)+b^3*d^3*(-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))

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Maxima [A]  time = 1.11274, size = 644, normalized size = 1.61 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.92437, size = 834, normalized size = 2.08 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 8.55227, size = 1217, normalized size = 3.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.67025, size = 562, normalized size = 1.4 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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