3.40 \(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx\)

Optimal. Leaf size=181 \[ \frac{a^3 c^4 (8 A-B) \cos ^7(e+f x)}{56 f}+\frac{a^3 c^4 (8 A-B) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 a^3 c^4 (8 A-B) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 a^3 c^4 (8 A-B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} a^3 c^4 x (8 A-B)-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f} \]

[Out]

(5*a^3*(8*A - B)*c^4*x)/128 + (a^3*(8*A - B)*c^4*Cos[e + f*x]^7)/(56*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]*Si
n[e + f*x])/(128*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (a^3*(8*A - B)*c^4*Cos[e + f
*x]^5*Sin[e + f*x])/(48*f) - (a^3*B*Cos[e + f*x]^7*(c^4 - c^4*Sin[e + f*x]))/(8*f)

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Rubi [A]  time = 0.233604, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139, Rules used = {2967, 2860, 2669, 2635, 8} \[ \frac{a^3 c^4 (8 A-B) \cos ^7(e+f x)}{56 f}+\frac{a^3 c^4 (8 A-B) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 a^3 c^4 (8 A-B) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 a^3 c^4 (8 A-B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} a^3 c^4 x (8 A-B)-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]

[Out]

(5*a^3*(8*A - B)*c^4*x)/128 + (a^3*(8*A - B)*c^4*Cos[e + f*x]^7)/(56*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]*Si
n[e + f*x])/(128*f) + (5*a^3*(8*A - B)*c^4*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (a^3*(8*A - B)*c^4*Cos[e + f
*x]^5*Sin[e + f*x])/(48*f) - (a^3*B*Cos[e + f*x]^7*(c^4 - c^4*Sin[e + f*x]))/(8*f)

Rule 2967

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

Rule 2860

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; Fre
eQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx\\ &=-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac{1}{8} \left (a^3 (8 A-B) c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac{1}{8} \left (a^3 (8 A-B) c^4\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac{a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac{a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac{1}{48} \left (5 a^3 (8 A-B) c^4\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac{5 a^3 (8 A-B) c^4 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac{1}{64} \left (5 a^3 (8 A-B) c^4\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac{5 a^3 (8 A-B) c^4 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{5 a^3 (8 A-B) c^4 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}+\frac{1}{128} \left (5 a^3 (8 A-B) c^4\right ) \int 1 \, dx\\ &=\frac{5}{128} a^3 (8 A-B) c^4 x+\frac{a^3 (8 A-B) c^4 \cos ^7(e+f x)}{56 f}+\frac{5 a^3 (8 A-B) c^4 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{5 a^3 (8 A-B) c^4 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{a^3 (8 A-B) c^4 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B \cos ^7(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{8 f}\\ \end{align*}

Mathematica [A]  time = 1.88535, size = 209, normalized size = 1.15 \[ \frac{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4 (840 (8 A-B) (e+f x)+336 (15 A-B) \sin (2 (e+f x))+168 (6 A+B) \sin (4 (e+f x))+112 (A+B) \sin (6 (e+f x))+1680 (A-B) \cos (e+f x)+1008 (A-B) \cos (3 (e+f x))+336 (A-B) \cos (5 (e+f x))+48 (A-B) \cos (7 (e+f x))+21 B \sin (8 (e+f x)))}{21504 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]

[Out]

((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4*(840*(8*A - B)*(e + f*x) + 1680*(A - B)*Cos[e + f*x] + 1008*(A
- B)*Cos[3*(e + f*x)] + 336*(A - B)*Cos[5*(e + f*x)] + 48*(A - B)*Cos[7*(e + f*x)] + 336*(15*A - B)*Sin[2*(e +
f*x)] + 168*(6*A + B)*Sin[4*(e + f*x)] + 112*(A + B)*Sin[6*(e + f*x)] + 21*B*Sin[8*(e + f*x)]))/(21504*f*(Cos
[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)

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Maple [B]  time = 0.032, size = 568, normalized size = 3.1 \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-c*sin(f*x+e))^4,x)

[Out]

1/f*(A*a^3*c^4*(f*x+e)+B*a^3*c^4*(-1/8*(sin(f*x+e)^7+7/6*sin(f*x+e)^5+35/24*sin(f*x+e)^3+35/16*sin(f*x+e))*cos
(f*x+e)+35/128*f*x+35/128*e)+1/7*B*a^3*c^4*(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*
B*a^3*c^4*(-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)-3/5*B*a^3*c^4*(8/3
+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)+A*a^3*c^4*cos(f*x+e)-B*a^3*c^4*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+
1/2*e)-1/7*A*a^3*c^4*(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)-A*a^3*c^4*(-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)+3/5*A*a^3*c^4*(8/3+sin(f*x+e)^4+4/3*sin(f*
x+e)^2)*cos(f*x+e)+3*A*a^3*c^4*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)+B*a^3*c^4*(2+sin(
f*x+e)^2)*cos(f*x+e)-3*A*a^3*c^4*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-A*a^3*c^4*(2+sin(f*x+e)^2)*cos(f*x
+e)+3*B*a^3*c^4*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8*e)-B*a^3*c^4*cos(f*x+e))

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Maxima [B]  time = 1.01738, size = 771, normalized size = 4.26 \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-c*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

1/107520*(3072*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*A*a^3*c^4 + 21504*
(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*c^4 + 107520*(cos(f*x + e)^3 - 3*cos(f*x + e))*
A*a^3*c^4 - 560*(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*c^4 +
10080*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c^4 - 80640*(2*f*x + 2*e - sin(2*f*x + 2*e
))*A*a^3*c^4 + 107520*(f*x + e)*A*a^3*c^4 - 3072*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 3
5*cos(f*x + e))*B*a^3*c^4 - 21504*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c^4 - 107520*
(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*c^4 + 35*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e
) + 168*sin(4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*B*a^3*c^4 - 1680*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*si
n(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*B*a^3*c^4 + 10080*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e)
)*B*a^3*c^4 - 26880*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^4 + 107520*A*a^3*c^4*cos(f*x + e) - 107520*B*a^3*
c^4*cos(f*x + e))/f

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Fricas [A]  time = 1.59575, size = 315, normalized size = 1.74 \begin{align*} \frac{384 \,{\left (A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{7} + 105 \,{\left (8 \, A - B\right )} a^{3} c^{4} f x + 7 \,{\left (48 \, B a^{3} c^{4} \cos \left (f x + e\right )^{7} + 8 \,{\left (8 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{5} + 10 \,{\left (8 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{3} + 15 \,{\left (8 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2688 \, 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-c*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

1/2688*(384*(A - B)*a^3*c^4*cos(f*x + e)^7 + 105*(8*A - B)*a^3*c^4*f*x + 7*(48*B*a^3*c^4*cos(f*x + e)^7 + 8*(8
*A - B)*a^3*c^4*cos(f*x + e)^5 + 10*(8*A - B)*a^3*c^4*cos(f*x + e)^3 + 15*(8*A - B)*a^3*c^4*cos(f*x + e))*sin(
f*x + e))/f

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Sympy [A]  time = 36.1194, size = 1579, normalized size = 8.72 \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-c*sin(f*x+e))**4,x)

[Out]

Piecewise((-5*A*a**3*c**4*x*sin(e + f*x)**6/16 - 15*A*a**3*c**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*A*a**
3*c**4*x*sin(e + f*x)**4/8 - 15*A*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*A*a**3*c**4*x*sin(e + f*x
)**2*cos(e + f*x)**2/4 - 3*A*a**3*c**4*x*sin(e + f*x)**2/2 - 5*A*a**3*c**4*x*cos(e + f*x)**6/16 + 9*A*a**3*c**
4*x*cos(e + f*x)**4/8 - 3*A*a**3*c**4*x*cos(e + f*x)**2/2 + A*a**3*c**4*x - A*a**3*c**4*sin(e + f*x)**6*cos(e
+ f*x)/f + 11*A*a**3*c**4*sin(e + f*x)**5*cos(e + f*x)/(16*f) - 2*A*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)**3/
f + 3*A*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)/f + 5*A*a**3*c**4*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 15*A*
a**3*c**4*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 8*A*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 4*A*a**3*
c**4*sin(e + f*x)**2*cos(e + f*x)**3/f - 3*A*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)/f + 5*A*a**3*c**4*sin(e +
f*x)*cos(e + f*x)**5/(16*f) - 9*A*a**3*c**4*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*A*a**3*c**4*sin(e + f*x)*co
s(e + f*x)/(2*f) - 16*A*a**3*c**4*cos(e + f*x)**7/(35*f) + 8*A*a**3*c**4*cos(e + f*x)**5/(5*f) - 2*A*a**3*c**4
*cos(e + f*x)**3/f + A*a**3*c**4*cos(e + f*x)/f + 35*B*a**3*c**4*x*sin(e + f*x)**8/128 + 35*B*a**3*c**4*x*sin(
e + f*x)**6*cos(e + f*x)**2/32 - 15*B*a**3*c**4*x*sin(e + f*x)**6/16 + 105*B*a**3*c**4*x*sin(e + f*x)**4*cos(e
+ f*x)**4/64 - 45*B*a**3*c**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 9*B*a**3*c**4*x*sin(e + f*x)**4/8 + 35*B
*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**6/32 - 45*B*a**3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*B*a*
*3*c**4*x*sin(e + f*x)**2*cos(e + f*x)**2/4 - B*a**3*c**4*x*sin(e + f*x)**2/2 + 35*B*a**3*c**4*x*cos(e + f*x)*
*8/128 - 15*B*a**3*c**4*x*cos(e + f*x)**6/16 + 9*B*a**3*c**4*x*cos(e + f*x)**4/8 - B*a**3*c**4*x*cos(e + f*x)*
*2/2 - 93*B*a**3*c**4*sin(e + f*x)**7*cos(e + f*x)/(128*f) + B*a**3*c**4*sin(e + f*x)**6*cos(e + f*x)/f - 511*
B*a**3*c**4*sin(e + f*x)**5*cos(e + f*x)**3/(384*f) + 33*B*a**3*c**4*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 2*B
*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)**3/f - 3*B*a**3*c**4*sin(e + f*x)**4*cos(e + f*x)/f - 385*B*a**3*c**4*
sin(e + f*x)**3*cos(e + f*x)**5/(384*f) + 5*B*a**3*c**4*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) - 15*B*a**3*c**4
*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 8*B*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) - 4*B*a**3*c**4*sin(
e + f*x)**2*cos(e + f*x)**3/f + 3*B*a**3*c**4*sin(e + f*x)**2*cos(e + f*x)/f - 35*B*a**3*c**4*sin(e + f*x)*cos
(e + f*x)**7/(128*f) + 15*B*a**3*c**4*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 9*B*a**3*c**4*sin(e + f*x)*cos(e +
f*x)**3/(8*f) + B*a**3*c**4*sin(e + f*x)*cos(e + f*x)/(2*f) + 16*B*a**3*c**4*cos(e + f*x)**7/(35*f) - 8*B*a**
3*c**4*cos(e + f*x)**5/(5*f) + 2*B*a**3*c**4*cos(e + f*x)**3/f - B*a**3*c**4*cos(e + f*x)/f, Ne(f, 0)), (x*(A
+ B*sin(e))*(a*sin(e) + a)**3*(-c*sin(e) + c)**4, True))

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Giac [A]  time = 1.29101, size = 369, normalized size = 2.04 \begin{align*} \frac{B a^{3} c^{4} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac{5}{128} \,{\left (8 \, A a^{3} c^{4} - B a^{3} c^{4}\right )} x + \frac{{\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac{{\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac{3 \,{\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac{5 \,{\left (A a^{3} c^{4} - B a^{3} c^{4}\right )} \cos \left (f x + e\right )}{64 \, f} + \frac{{\left (A a^{3} c^{4} + B a^{3} c^{4}\right )} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac{{\left (6 \, A a^{3} c^{4} + B a^{3} c^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac{{\left (15 \, A a^{3} c^{4} - B a^{3} c^{4}\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-c*sin(f*x+e))^4,x, algorithm="giac")

[Out]

1/1024*B*a^3*c^4*sin(8*f*x + 8*e)/f + 5/128*(8*A*a^3*c^4 - B*a^3*c^4)*x + 1/448*(A*a^3*c^4 - B*a^3*c^4)*cos(7*
f*x + 7*e)/f + 1/64*(A*a^3*c^4 - B*a^3*c^4)*cos(5*f*x + 5*e)/f + 3/64*(A*a^3*c^4 - B*a^3*c^4)*cos(3*f*x + 3*e)
/f + 5/64*(A*a^3*c^4 - B*a^3*c^4)*cos(f*x + e)/f + 1/192*(A*a^3*c^4 + B*a^3*c^4)*sin(6*f*x + 6*e)/f + 1/128*(6
*A*a^3*c^4 + B*a^3*c^4)*sin(4*f*x + 4*e)/f + 1/64*(15*A*a^3*c^4 - B*a^3*c^4)*sin(2*f*x + 2*e)/f