3.258 \(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx\)

Optimal. Leaf size=604 \[ -\frac{a^3 \left (7 A d \left (107 c^3 d^2+472 c^2 d^3-18 c^4 d+2 c^5+456 c d^4+136 d^5\right )-3 B \left (51 c^4 d^2-189 c^3 d^3-920 c^2 d^4-14 c^5 d+2 c^6-952 c d^5-288 d^6\right )\right ) \cos (e+f x)}{420 d^3 f}-\frac{a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac{a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (-42 c^2 d+6 c^3+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}-\frac{a^3 \left (7 A d \left (-18 c^2 d+2 c^3+111 c d^2+136 d^3\right )-B \left (165 c^2 d^2-42 c^3 d+6 c^4-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{840 d^3 f}-\frac{a^3 \left (7 A d \left (216 c^2 d^2-36 c^3 d+4 c^4+626 c d^3+345 d^4\right )-3 B \left (104 c^3 d^2-392 c^2 d^3-28 c^4 d+4 c^5-1263 c d^4-735 d^5\right )\right ) \sin (e+f x) \cos (e+f x)}{1680 d^2 f}+\frac{1}{16} a^3 x \left (A \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right )+3 B \left (26 c^2 d+10 c^3+23 c d^2+7 d^3\right )\right )+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{42 d^2 f}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f} \]

[Out]

(a^3*(3*B*(10*c^3 + 26*c^2*d + 23*c*d^2 + 7*d^3) + A*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3))*x)/16 - (a^3*(7*
A*d*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 472*c^2*d^3 + 456*c*d^4 + 136*d^5) - 3*B*(2*c^6 - 14*c^5*d + 51*c^4*d^2
- 189*c^3*d^3 - 920*c^2*d^4 - 952*c*d^5 - 288*d^6))*Cos[e + f*x])/(420*d^3*f) - (a^3*(7*A*d*(4*c^4 - 36*c^3*d
+ 216*c^2*d^2 + 626*c*d^3 + 345*d^4) - 3*B*(4*c^5 - 28*c^4*d + 104*c^3*d^2 - 392*c^2*d^3 - 1263*c*d^4 - 735*d^
5))*Cos[e + f*x]*Sin[e + f*x])/(1680*d^2*f) - (a^3*(7*A*d*(2*c^3 - 18*c^2*d + 111*c*d^2 + 136*d^3) - B*(6*c^4
- 42*c^3*d + 165*c^2*d^2 - 651*c*d^3 - 864*d^4))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(840*d^3*f) - (a^3*(7*A*
d*(2*c^2 - 18*c*d + 115*d^2) - B*(6*c^3 - 42*c^2*d + 177*c*d^2 - 735*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3
)/(840*d^3*f) - (a^3*(6*B*c^2 - 14*A*c*d - 27*B*c*d + 91*A*d^2 + 87*B*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4
)/(210*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4)/(7*d*f) + ((3*B*(c - 3*d) - 7
*A*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(42*d^2*f)

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Rubi [A]  time = 1.48579, antiderivative size = 604, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2976, 2968, 3023, 2753, 2734} \[ -\frac{a^3 \left (7 A d \left (107 c^3 d^2+472 c^2 d^3-18 c^4 d+2 c^5+456 c d^4+136 d^5\right )-3 B \left (51 c^4 d^2-189 c^3 d^3-920 c^2 d^4-14 c^5 d+2 c^6-952 c d^5-288 d^6\right )\right ) \cos (e+f x)}{420 d^3 f}-\frac{a^3 \left (-14 A c d+91 A d^2+6 B c^2-27 B c d+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac{a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (-42 c^2 d+6 c^3+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}-\frac{a^3 \left (7 A d \left (-18 c^2 d+2 c^3+111 c d^2+136 d^3\right )-B \left (165 c^2 d^2-42 c^3 d+6 c^4-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{840 d^3 f}-\frac{a^3 \left (7 A d \left (216 c^2 d^2-36 c^3 d+4 c^4+626 c d^3+345 d^4\right )-3 B \left (104 c^3 d^2-392 c^2 d^3-28 c^4 d+4 c^5-1263 c d^4-735 d^5\right )\right ) \sin (e+f x) \cos (e+f x)}{1680 d^2 f}+\frac{1}{16} a^3 x \left (A \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right )+3 B \left (26 c^2 d+10 c^3+23 c d^2+7 d^3\right )\right )+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^4}{42 d^2 f}-\frac{a B \cos (e+f x) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^4}{7 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(3*B*(10*c^3 + 26*c^2*d + 23*c*d^2 + 7*d^3) + A*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3))*x)/16 - (a^3*(7*
A*d*(2*c^5 - 18*c^4*d + 107*c^3*d^2 + 472*c^2*d^3 + 456*c*d^4 + 136*d^5) - 3*B*(2*c^6 - 14*c^5*d + 51*c^4*d^2
- 189*c^3*d^3 - 920*c^2*d^4 - 952*c*d^5 - 288*d^6))*Cos[e + f*x])/(420*d^3*f) - (a^3*(7*A*d*(4*c^4 - 36*c^3*d
+ 216*c^2*d^2 + 626*c*d^3 + 345*d^4) - 3*B*(4*c^5 - 28*c^4*d + 104*c^3*d^2 - 392*c^2*d^3 - 1263*c*d^4 - 735*d^
5))*Cos[e + f*x]*Sin[e + f*x])/(1680*d^2*f) - (a^3*(7*A*d*(2*c^3 - 18*c^2*d + 111*c*d^2 + 136*d^3) - B*(6*c^4
- 42*c^3*d + 165*c^2*d^2 - 651*c*d^3 - 864*d^4))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(840*d^3*f) - (a^3*(7*A*
d*(2*c^2 - 18*c*d + 115*d^2) - B*(6*c^3 - 42*c^2*d + 177*c*d^2 - 735*d^3))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3
)/(840*d^3*f) - (a^3*(6*B*c^2 - 14*A*c*d - 27*B*c*d + 91*A*d^2 + 87*B*d^2)*Cos[e + f*x]*(c + d*Sin[e + f*x])^4
)/(210*d^3*f) - (a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^4)/(7*d*f) + ((3*B*(c - 3*d) - 7
*A*d)*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^4)/(42*d^2*f)

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx &=-\frac{a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac{\int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^3 (a (7 A d+2 B (c+2 d))-a (3 B c-7 A d-9 B d) \sin (e+f x)) \, dx}{7 d}\\ &=-\frac{a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{42 d^2 f}+\frac{\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^3 \left (a^2 \left (7 A d (c+10 d)-B \left (3 c^2-9 c d-60 d^2\right )\right )+a^2 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \sin (e+f x)\right ) \, dx}{42 d^2}\\ &=-\frac{a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{42 d^2 f}+\frac{\int (c+d \sin (e+f x))^3 \left (a^3 \left (7 A d (c+10 d)-B \left (3 c^2-9 c d-60 d^2\right )\right )+\left (a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right )+a^3 \left (7 A d (c+10 d)-B \left (3 c^2-9 c d-60 d^2\right )\right )\right ) \sin (e+f x)+a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \sin ^2(e+f x)\right ) \, dx}{42 d^2}\\ &=-\frac{a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac{a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{42 d^2 f}+\frac{\int (c+d \sin (e+f x))^3 \left (-3 a^3 d \left (7 A (c-34 d) d-3 B \left (c^2-7 c d+72 d^2\right )\right )+a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-3 B \left (2 c^3-14 c^2 d+59 c d^2-245 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{210 d^3}\\ &=-\frac{a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}-\frac{a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac{a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{42 d^2 f}+\frac{\int (c+d \sin (e+f x))^2 \left (-3 a^3 d \left (7 A d \left (2 c^2-118 c d-115 d^2\right )-B \left (6 c^3-42 c^2 d+687 c d^2+735 d^3\right )\right )+3 a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{840 d^3}\\ &=-\frac{a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{840 d^3 f}-\frac{a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}-\frac{a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac{a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{42 d^2 f}+\frac{\int (c+d \sin (e+f x)) \left (-3 a^3 d \left (7 A d \left (2 c^3-318 c^2 d-567 c d^2-272 d^3\right )-3 B \left (2 c^4-14 c^3 d+577 c^2 d^2+1169 c d^3+576 d^4\right )\right )+3 a^3 \left (7 A d \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right )-3 B \left (4 c^5-28 c^4 d+104 c^3 d^2-392 c^2 d^3-1263 c d^4-735 d^5\right )\right ) \sin (e+f x)\right ) \, dx}{2520 d^3}\\ &=\frac{1}{16} a^3 \left (3 B \left (10 c^3+26 c^2 d+23 c d^2+7 d^3\right )+A \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right )\right ) x-\frac{a^3 \left (7 A d \left (2 c^5-18 c^4 d+107 c^3 d^2+472 c^2 d^3+456 c d^4+136 d^5\right )-3 B \left (2 c^6-14 c^5 d+51 c^4 d^2-189 c^3 d^3-920 c^2 d^4-952 c d^5-288 d^6\right )\right ) \cos (e+f x)}{420 d^3 f}-\frac{a^3 \left (7 A d \left (4 c^4-36 c^3 d+216 c^2 d^2+626 c d^3+345 d^4\right )-3 B \left (4 c^5-28 c^4 d+104 c^3 d^2-392 c^2 d^3-1263 c d^4-735 d^5\right )\right ) \cos (e+f x) \sin (e+f x)}{1680 d^2 f}-\frac{a^3 \left (7 A d \left (2 c^3-18 c^2 d+111 c d^2+136 d^3\right )-B \left (6 c^4-42 c^3 d+165 c^2 d^2-651 c d^3-864 d^4\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{840 d^3 f}-\frac{a^3 \left (7 A d \left (2 c^2-18 c d+115 d^2\right )-B \left (6 c^3-42 c^2 d+177 c d^2-735 d^3\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{840 d^3 f}-\frac{a^3 \left (6 B c^2-14 A c d-27 B c d+91 A d^2+87 B d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^4}{210 d^3 f}-\frac{a B \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^4}{7 d f}+\frac{(3 B (c-3 d)-7 A d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^4}{42 d^2 f}\\ \end{align*}

Mathematica [A]  time = 4.76325, size = 528, normalized size = 0.87 \[ -\frac{a^3 \cos (e+f x) \left (420 \left (A \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right )+3 B \left (26 c^2 d+10 c^3+23 c d^2+7 d^3\right )\right ) \sin ^{-1}\left (\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right )+\sqrt{\cos ^2(e+f x)} \left (-\left (112 A \left (45 c^2 d+5 c^3+66 c d^2+26 d^3\right )+3 B \left (2464 c^2 d+560 c^3+2912 c d^2+1083 d^3\right )\right ) \cos (2 (e+f x))+18 d \left (14 A d (c+d)+B \left (14 c^2+42 c d+23 d^2\right )\right ) \cos (4 (e+f x))+20790 A c^2 d \sin (e+f x)-630 A c^2 d \sin (3 (e+f x))+35280 A c^2 d+5040 A c^3 \sin (e+f x)+12880 A c^3+22050 A c d^2 \sin (e+f x)-1890 A c d^2 \sin (3 (e+f x))+32676 A c d^2+7595 A d^3 \sin (e+f x)-980 A d^3 \sin (3 (e+f x))+35 A d^3 \sin (5 (e+f x))+10276 A d^3+22050 B c^2 d \sin (e+f x)-1890 B c^2 d \sin (3 (e+f x))+32676 B c^2 d+6930 B c^3 \sin (e+f x)-210 B c^3 \sin (3 (e+f x))+11760 B c^3+22785 B c d^2 \sin (e+f x)-2940 B c d^2 \sin (3 (e+f x))+105 B c d^2 \sin (5 (e+f x))+30828 B c d^2+7665 B d^3 \sin (e+f x)-1260 B d^3 \sin (3 (e+f x))+105 B d^3 \sin (5 (e+f x))-15 B d^3 \cos (6 (e+f x))+9762 B d^3\right )\right )}{3360 f \sqrt{\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*Cos[e + f*x]*(420*(3*B*(10*c^3 + 26*c^2*d + 23*c*d^2 + 7*d^3) + A*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3
))*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(12880*A*c^3 + 11760*B*c^3 + 35280*A*c^2*d +
32676*B*c^2*d + 32676*A*c*d^2 + 30828*B*c*d^2 + 10276*A*d^3 + 9762*B*d^3 - (112*A*(5*c^3 + 45*c^2*d + 66*c*d^2
+ 26*d^3) + 3*B*(560*c^3 + 2464*c^2*d + 2912*c*d^2 + 1083*d^3))*Cos[2*(e + f*x)] + 18*d*(14*A*d*(c + d) + B*(
14*c^2 + 42*c*d + 23*d^2))*Cos[4*(e + f*x)] - 15*B*d^3*Cos[6*(e + f*x)] + 5040*A*c^3*Sin[e + f*x] + 6930*B*c^3
*Sin[e + f*x] + 20790*A*c^2*d*Sin[e + f*x] + 22050*B*c^2*d*Sin[e + f*x] + 22050*A*c*d^2*Sin[e + f*x] + 22785*B
*c*d^2*Sin[e + f*x] + 7595*A*d^3*Sin[e + f*x] + 7665*B*d^3*Sin[e + f*x] - 210*B*c^3*Sin[3*(e + f*x)] - 630*A*c
^2*d*Sin[3*(e + f*x)] - 1890*B*c^2*d*Sin[3*(e + f*x)] - 1890*A*c*d^2*Sin[3*(e + f*x)] - 2940*B*c*d^2*Sin[3*(e
+ f*x)] - 980*A*d^3*Sin[3*(e + f*x)] - 1260*B*d^3*Sin[3*(e + f*x)] + 105*B*c*d^2*Sin[5*(e + f*x)] + 35*A*d^3*S
in[5*(e + f*x)] + 105*B*d^3*Sin[5*(e + f*x)])))/(3360*f*Sqrt[Cos[e + f*x]^2])

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Maple [A]  time = 0.092, size = 1077, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(-B*a^3*c*d^2*(2+sin(f*x+e)^2)*cos(f*x+e)+9*B*a^3*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*A*a^3*c^2*d*cos(f*x+e)+3*A*a^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*B*a^3*c*d^2*(-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)+3*B*a^3*c^2
*d*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+9*B*a^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)-1/3*A*a^3*c^3*(2+sin(f*x+e)^2)*cos(f*x+e)+B*a^3*c^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*a^3*c^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+A*a^3*c^3*(f*x+e)+3*B*a^3*d^3*(-1/6*(si
n(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)-3*A*a^3*c^3*cos(f*x+e)-B*a^3*c^3*cos(
f*x+e)+B*a^3*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)+A*a^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)+3*A*a^3*d^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*c
os(f*x+e)+3/8*f*x+3/8*e)+3*B*a^3*c^3*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-9/5*B*a^3*c*d^2*(8/3+sin(f*x+e
)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-3*A*a^3*c*d^2*(2+sin(f*x+e)^2)*cos(f*x+e)-3*B*a^3*c^2*d*(2+sin(f*x+e)^2)*cos(
f*x+e)-3/5*A*a^3*c*d^2*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-3/5*B*a^3*c^2*d*(8/3+sin(f*x+e)^4+4/3*si
n(f*x+e)^2)*cos(f*x+e)-3*A*a^3*c^2*d*(2+sin(f*x+e)^2)*cos(f*x+e)+9*A*a^3*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)-1/7*B*a^3*d^3*(16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)-
3/5*A*a^3*d^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)+9*A*a^3*c^2*d*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x
+1/2*e)+3*A*a^3*c*d^2*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-B*a^3*c^3*(2+sin(f*x+e)^2)*cos(f*x+e)-3/5*B*a
^3*d^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-1/3*A*a^3*d^3*(2+sin(f*x+e)^2)*cos(f*x+e))

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Maxima [A]  time = 1.08227, size = 1426, normalized size = 2.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6720*(2240*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c^3 + 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3*c^3 + 6
720*(f*x + e)*A*a^3*c^3 + 6720*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*c^3 + 210*(12*f*x + 12*e + sin(4*f*x +
4*e) - 8*sin(2*f*x + 2*e))*B*a^3*c^3 + 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^3 + 20160*(cos(f*x + e)^3
- 3*cos(f*x + e))*A*a^3*c^2*d + 630*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c^2*d + 151
20*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3*c^2*d - 1344*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e)
)*B*a^3*c^2*d + 20160*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*c^2*d + 1890*(12*f*x + 12*e + sin(4*f*x + 4*e) -
8*sin(2*f*x + 2*e))*B*a^3*c^2*d + 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^2*d - 1344*(3*cos(f*x + e)^5
- 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*c*d^2 + 20160*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c*d^2 + 189
0*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c*d^2 + 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*
A*a^3*c*d^2 - 4032*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c*d^2 + 6720*(cos(f*x + e)^3
- 3*cos(f*x + e))*B*a^3*c*d^2 + 105*(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*a^3*c*d^2 + 1890*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*c*d^2 - 1344*(3*cos(
f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*d^3 + 2240*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*d^3
+ 35*(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))*A*a^3*d^3 + 630*(12*f*
x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*d^3 + 192*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*c
os(f*x + e)^3 - 35*cos(f*x + e))*B*a^3*d^3 - 1344*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a
^3*d^3 + 105*(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*a^3*d^3 + 210
*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*d^3 - 20160*A*a^3*c^3*cos(f*x + e) - 6720*B*a^3
*c^3*cos(f*x + e) - 20160*A*a^3*c^2*d*cos(f*x + e))/f

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Fricas [A]  time = 2.69028, size = 1019, normalized size = 1.69 \begin{align*} \frac{240 \, B a^{3} d^{3} \cos \left (f x + e\right )^{7} - 1008 \,{\left (B a^{3} c^{2} d +{\left (A + 3 \, B\right )} a^{3} c d^{2} +{\left (A + 2 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} + 560 \,{\left ({\left (A + 3 \, B\right )} a^{3} c^{3} + 3 \,{\left (3 \, A + 5 \, B\right )} a^{3} c^{2} d + 3 \,{\left (5 \, A + 7 \, B\right )} a^{3} c d^{2} +{\left (7 \, A + 9 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 105 \,{\left (10 \,{\left (4 \, A + 3 \, B\right )} a^{3} c^{3} + 6 \,{\left (15 \, A + 13 \, B\right )} a^{3} c^{2} d + 3 \,{\left (26 \, A + 23 \, B\right )} a^{3} c d^{2} +{\left (23 \, A + 21 \, B\right )} a^{3} d^{3}\right )} f x - 6720 \,{\left ({\left (A + B\right )} a^{3} c^{3} + 3 \,{\left (A + B\right )} a^{3} c^{2} d + 3 \,{\left (A + B\right )} a^{3} c d^{2} +{\left (A + B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right ) - 35 \,{\left (8 \,{\left (3 \, B a^{3} c d^{2} +{\left (A + 3 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} - 2 \,{\left (6 \, B a^{3} c^{3} + 18 \,{\left (A + 3 \, B\right )} a^{3} c^{2} d + 3 \,{\left (18 \, A + 31 \, B\right )} a^{3} c d^{2} +{\left (31 \, A + 45 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (2 \,{\left (12 \, A + 17 \, B\right )} a^{3} c^{3} + 6 \,{\left (17 \, A + 19 \, B\right )} a^{3} c^{2} d + 3 \,{\left (38 \, A + 41 \, B\right )} a^{3} c d^{2} +{\left (41 \, A + 43 \, B\right )} a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{1680 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(240*B*a^3*d^3*cos(f*x + e)^7 - 1008*(B*a^3*c^2*d + (A + 3*B)*a^3*c*d^2 + (A + 2*B)*a^3*d^3)*cos(f*x +
e)^5 + 560*((A + 3*B)*a^3*c^3 + 3*(3*A + 5*B)*a^3*c^2*d + 3*(5*A + 7*B)*a^3*c*d^2 + (7*A + 9*B)*a^3*d^3)*cos(f
*x + e)^3 + 105*(10*(4*A + 3*B)*a^3*c^3 + 6*(15*A + 13*B)*a^3*c^2*d + 3*(26*A + 23*B)*a^3*c*d^2 + (23*A + 21*B
)*a^3*d^3)*f*x - 6720*((A + B)*a^3*c^3 + 3*(A + B)*a^3*c^2*d + 3*(A + B)*a^3*c*d^2 + (A + B)*a^3*d^3)*cos(f*x
+ e) - 35*(8*(3*B*a^3*c*d^2 + (A + 3*B)*a^3*d^3)*cos(f*x + e)^5 - 2*(6*B*a^3*c^3 + 18*(A + 3*B)*a^3*c^2*d + 3*
(18*A + 31*B)*a^3*c*d^2 + (31*A + 45*B)*a^3*d^3)*cos(f*x + e)^3 + 3*(2*(12*A + 17*B)*a^3*c^3 + 6*(17*A + 19*B)
*a^3*c^2*d + 3*(38*A + 41*B)*a^3*c*d^2 + (41*A + 43*B)*a^3*d^3)*cos(f*x + e))*sin(f*x + e))/f

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Sympy [A]  time = 22.4938, size = 2878, normalized size = 4.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((3*A*a**3*c**3*x*sin(e + f*x)**2/2 + 3*A*a**3*c**3*x*cos(e + f*x)**2/2 + A*a**3*c**3*x - A*a**3*c**3
*sin(e + f*x)**2*cos(e + f*x)/f - 3*A*a**3*c**3*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a**3*c**3*cos(e + f*x)**
3/(3*f) - 3*A*a**3*c**3*cos(e + f*x)/f + 9*A*a**3*c**2*d*x*sin(e + f*x)**4/8 + 9*A*a**3*c**2*d*x*sin(e + f*x)*
*2*cos(e + f*x)**2/4 + 9*A*a**3*c**2*d*x*sin(e + f*x)**2/2 + 9*A*a**3*c**2*d*x*cos(e + f*x)**4/8 + 9*A*a**3*c*
*2*d*x*cos(e + f*x)**2/2 - 15*A*a**3*c**2*d*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 9*A*a**3*c**2*d*sin(e + f*x)*
*2*cos(e + f*x)/f - 9*A*a**3*c**2*d*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 9*A*a**3*c**2*d*sin(e + f*x)*cos(e +
f*x)/(2*f) - 6*A*a**3*c**2*d*cos(e + f*x)**3/f - 3*A*a**3*c**2*d*cos(e + f*x)/f + 27*A*a**3*c*d**2*x*sin(e + f
*x)**4/8 + 27*A*a**3*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*A*a**3*c*d**2*x*sin(e + f*x)**2/2 + 27*A*a
**3*c*d**2*x*cos(e + f*x)**4/8 + 3*A*a**3*c*d**2*x*cos(e + f*x)**2/2 - 3*A*a**3*c*d**2*sin(e + f*x)**4*cos(e +
f*x)/f - 45*A*a**3*c*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*A*a**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)**
3/f - 9*A*a**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 27*A*a**3*c*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3
*A*a**3*c*d**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 8*A*a**3*c*d**2*cos(e + f*x)**5/(5*f) - 6*A*a**3*c*d**2*cos(e
+ f*x)**3/f + 5*A*a**3*d**3*x*sin(e + f*x)**6/16 + 15*A*a**3*d**3*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*A*
a**3*d**3*x*sin(e + f*x)**4/8 + 15*A*a**3*d**3*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*A*a**3*d**3*x*sin(e +
f*x)**2*cos(e + f*x)**2/4 + 5*A*a**3*d**3*x*cos(e + f*x)**6/16 + 9*A*a**3*d**3*x*cos(e + f*x)**4/8 - 11*A*a**3
*d**3*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 3*A*a**3*d**3*sin(e + f*x)**4*cos(e + f*x)/f - 5*A*a**3*d**3*sin(e
+ f*x)**3*cos(e + f*x)**3/(6*f) - 15*A*a**3*d**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*A*a**3*d**3*sin(e + f
*x)**2*cos(e + f*x)**3/f - A*a**3*d**3*sin(e + f*x)**2*cos(e + f*x)/f - 5*A*a**3*d**3*sin(e + f*x)*cos(e + f*x
)**5/(16*f) - 9*A*a**3*d**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 8*A*a**3*d**3*cos(e + f*x)**5/(5*f) - 2*A*a**
3*d**3*cos(e + f*x)**3/(3*f) + 3*B*a**3*c**3*x*sin(e + f*x)**4/8 + 3*B*a**3*c**3*x*sin(e + f*x)**2*cos(e + f*x
)**2/4 + 3*B*a**3*c**3*x*sin(e + f*x)**2/2 + 3*B*a**3*c**3*x*cos(e + f*x)**4/8 + 3*B*a**3*c**3*x*cos(e + f*x)*
*2/2 - 5*B*a**3*c**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 3*B*a**3*c**3*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*a
**3*c**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*B*a**3*c**3*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*B*a**3*c**3*co
s(e + f*x)**3/f - B*a**3*c**3*cos(e + f*x)/f + 27*B*a**3*c**2*d*x*sin(e + f*x)**4/8 + 27*B*a**3*c**2*d*x*sin(e
+ f*x)**2*cos(e + f*x)**2/4 + 3*B*a**3*c**2*d*x*sin(e + f*x)**2/2 + 27*B*a**3*c**2*d*x*cos(e + f*x)**4/8 + 3*
B*a**3*c**2*d*x*cos(e + f*x)**2/2 - 3*B*a**3*c**2*d*sin(e + f*x)**4*cos(e + f*x)/f - 45*B*a**3*c**2*d*sin(e +
f*x)**3*cos(e + f*x)/(8*f) - 4*B*a**3*c**2*d*sin(e + f*x)**2*cos(e + f*x)**3/f - 9*B*a**3*c**2*d*sin(e + f*x)*
*2*cos(e + f*x)/f - 27*B*a**3*c**2*d*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*B*a**3*c**2*d*sin(e + f*x)*cos(e +
f*x)/(2*f) - 8*B*a**3*c**2*d*cos(e + f*x)**5/(5*f) - 6*B*a**3*c**2*d*cos(e + f*x)**3/f + 15*B*a**3*c*d**2*x*s
in(e + f*x)**6/16 + 45*B*a**3*c*d**2*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 27*B*a**3*c*d**2*x*sin(e + f*x)**4
/8 + 45*B*a**3*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 27*B*a**3*c*d**2*x*sin(e + f*x)**2*cos(e + f*x)**
2/4 + 15*B*a**3*c*d**2*x*cos(e + f*x)**6/16 + 27*B*a**3*c*d**2*x*cos(e + f*x)**4/8 - 33*B*a**3*c*d**2*sin(e +
f*x)**5*cos(e + f*x)/(16*f) - 9*B*a**3*c*d**2*sin(e + f*x)**4*cos(e + f*x)/f - 5*B*a**3*c*d**2*sin(e + f*x)**3
*cos(e + f*x)**3/(2*f) - 45*B*a**3*c*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 12*B*a**3*c*d**2*sin(e + f*x)**
2*cos(e + f*x)**3/f - 3*B*a**3*c*d**2*sin(e + f*x)**2*cos(e + f*x)/f - 15*B*a**3*c*d**2*sin(e + f*x)*cos(e + f
*x)**5/(16*f) - 27*B*a**3*c*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 24*B*a**3*c*d**2*cos(e + f*x)**5/(5*f) -
2*B*a**3*c*d**2*cos(e + f*x)**3/f + 15*B*a**3*d**3*x*sin(e + f*x)**6/16 + 45*B*a**3*d**3*x*sin(e + f*x)**4*co
s(e + f*x)**2/16 + 3*B*a**3*d**3*x*sin(e + f*x)**4/8 + 45*B*a**3*d**3*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 3
*B*a**3*d**3*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 15*B*a**3*d**3*x*cos(e + f*x)**6/16 + 3*B*a**3*d**3*x*cos(e
+ f*x)**4/8 - B*a**3*d**3*sin(e + f*x)**6*cos(e + f*x)/f - 33*B*a**3*d**3*sin(e + f*x)**5*cos(e + f*x)/(16*f)
- 2*B*a**3*d**3*sin(e + f*x)**4*cos(e + f*x)**3/f - 3*B*a**3*d**3*sin(e + f*x)**4*cos(e + f*x)/f - 5*B*a**3*d
**3*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) - 5*B*a**3*d**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 8*B*a**3*d**3*s
in(e + f*x)**2*cos(e + f*x)**5/(5*f) - 4*B*a**3*d**3*sin(e + f*x)**2*cos(e + f*x)**3/f - 15*B*a**3*d**3*sin(e
+ f*x)*cos(e + f*x)**5/(16*f) - 3*B*a**3*d**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 16*B*a**3*d**3*cos(e + f*x)
**7/(35*f) - 8*B*a**3*d**3*cos(e + f*x)**5/(5*f), Ne(f, 0)), (x*(A + B*sin(e))*(c + d*sin(e))**3*(a*sin(e) + a
)**3, True))

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Giac [A]  time = 1.34768, size = 764, normalized size = 1.26 \begin{align*} \frac{B a^{3} d^{3} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac{1}{16} \,{\left (40 \, A a^{3} c^{3} + 30 \, B a^{3} c^{3} + 90 \, A a^{3} c^{2} d + 78 \, B a^{3} c^{2} d + 78 \, A a^{3} c d^{2} + 69 \, B a^{3} c d^{2} + 23 \, A a^{3} d^{3} + 21 \, B a^{3} d^{3}\right )} x - \frac{{\left (12 \, B a^{3} c^{2} d + 12 \, A a^{3} c d^{2} + 36 \, B a^{3} c d^{2} + 12 \, A a^{3} d^{3} + 19 \, B a^{3} d^{3}\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac{{\left (16 \, A a^{3} c^{3} + 48 \, B a^{3} c^{3} + 144 \, A a^{3} c^{2} d + 204 \, B a^{3} c^{2} d + 204 \, A a^{3} c d^{2} + 228 \, B a^{3} c d^{2} + 76 \, A a^{3} d^{3} + 81 \, B a^{3} d^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} - \frac{{\left (240 \, A a^{3} c^{3} + 208 \, B a^{3} c^{3} + 624 \, A a^{3} c^{2} d + 552 \, B a^{3} c^{2} d + 552 \, A a^{3} c d^{2} + 504 \, B a^{3} c d^{2} + 168 \, A a^{3} d^{3} + 155 \, B a^{3} d^{3}\right )} \cos \left (f x + e\right )}{64 \, f} - \frac{{\left (3 \, B a^{3} c d^{2} + A a^{3} d^{3} + 3 \, B a^{3} d^{3}\right )} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{{\left (2 \, B a^{3} c^{3} + 6 \, A a^{3} c^{2} d + 18 \, B a^{3} c^{2} d + 18 \, A a^{3} c d^{2} + 27 \, B a^{3} c d^{2} + 9 \, A a^{3} d^{3} + 11 \, B a^{3} d^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac{{\left (48 \, A a^{3} c^{3} + 64 \, B a^{3} c^{3} + 192 \, A a^{3} c^{2} d + 192 \, B a^{3} c^{2} d + 192 \, A a^{3} c d^{2} + 189 \, B a^{3} c d^{2} + 63 \, A a^{3} d^{3} + 61 \, B a^{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+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/448*B*a^3*d^3*cos(7*f*x + 7*e)/f + 1/16*(40*A*a^3*c^3 + 30*B*a^3*c^3 + 90*A*a^3*c^2*d + 78*B*a^3*c^2*d + 78*
A*a^3*c*d^2 + 69*B*a^3*c*d^2 + 23*A*a^3*d^3 + 21*B*a^3*d^3)*x - 1/320*(12*B*a^3*c^2*d + 12*A*a^3*c*d^2 + 36*B*
a^3*c*d^2 + 12*A*a^3*d^3 + 19*B*a^3*d^3)*cos(5*f*x + 5*e)/f + 1/192*(16*A*a^3*c^3 + 48*B*a^3*c^3 + 144*A*a^3*c
^2*d + 204*B*a^3*c^2*d + 204*A*a^3*c*d^2 + 228*B*a^3*c*d^2 + 76*A*a^3*d^3 + 81*B*a^3*d^3)*cos(3*f*x + 3*e)/f -
1/64*(240*A*a^3*c^3 + 208*B*a^3*c^3 + 624*A*a^3*c^2*d + 552*B*a^3*c^2*d + 552*A*a^3*c*d^2 + 504*B*a^3*c*d^2 +
168*A*a^3*d^3 + 155*B*a^3*d^3)*cos(f*x + e)/f - 1/192*(3*B*a^3*c*d^2 + A*a^3*d^3 + 3*B*a^3*d^3)*sin(6*f*x + 6
*e)/f + 1/64*(2*B*a^3*c^3 + 6*A*a^3*c^2*d + 18*B*a^3*c^2*d + 18*A*a^3*c*d^2 + 27*B*a^3*c*d^2 + 9*A*a^3*d^3 + 1
1*B*a^3*d^3)*sin(4*f*x + 4*e)/f - 1/64*(48*A*a^3*c^3 + 64*B*a^3*c^3 + 192*A*a^3*c^2*d + 192*B*a^3*c^2*d + 192*
A*a^3*c*d^2 + 189*B*a^3*c*d^2 + 63*A*a^3*d^3 + 61*B*a^3*d^3)*sin(2*f*x + 2*e)/f