\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx\) [41]

Optimal result

Integrand size = 36, antiderivative size = 117 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {5}{16} a^3 A c^3 x-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 A c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 A c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f} \] Output:

5/16*a^3*A*c^3*x-1/7*a^3*B*c^3*cos(f*x+e)^7/f+5/16*a^3*A*c^3*cos(f*x+e)*si 
n(f*x+e)/f+5/24*a^3*A*c^3*cos(f*x+e)^3*sin(f*x+e)/f+1/6*a^3*A*c^3*cos(f*x+ 
e)^5*sin(f*x+e)/f
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.55 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {a^3 c^3 \left (-192 B \cos ^7(e+f x)+7 A (60 e+60 f x+45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x)))\right )}{1344 f} \] Input:

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

Output:

(a^3*c^3*(-192*B*Cos[e + f*x]^7 + 7*A*(60*e + 60*f*x + 45*Sin[2*(e + f*x)] 
 + 9*Sin[4*(e + f*x)] + Sin[6*(e + f*x)])))/(1344*f)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {3042, 3446, 3042, 3148, 3042, 3115, 3042, 3115, 3042, 3115, 24}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3446

\(\displaystyle a^3 c^3 \int \cos ^6(e+f x) (A+B \sin (e+f x))dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \int \cos (e+f x)^6 (A+B \sin (e+f x))dx\)

\(\Big \downarrow \) 3148

\(\displaystyle a^3 c^3 \left (A \int \cos ^6(e+f x)dx-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (A \int \sin \left (e+f x+\frac {\pi }{2}\right )^6dx-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^3 c^3 \left (A \left (\frac {5}{6} \int \cos ^4(e+f x)dx+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (A \left (\frac {5}{6} \int \sin \left (e+f x+\frac {\pi }{2}\right )^4dx+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^3 c^3 \left (A \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(e+f x)dx+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (A \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^3 c^3 \left (A \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}\right )+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle a^3 c^3 \left (A \left (\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac {5}{6} \left (\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac {3}{4} \left (\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}\right )\right )\right )-\frac {B \cos ^7(e+f x)}{7 f}\right )\)

Input:

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

Output:

a^3*c^3*(-1/7*(B*Cos[e + f*x]^7)/f + A*((Cos[e + f*x]^5*Sin[e + f*x])/(6*f 
) + (5*((Cos[e + f*x]^3*Sin[e + f*x])/(4*f) + (3*(x/2 + (Cos[e + f*x]*Sin[ 
e + f*x])/(2*f)))/4))/6))
 

Defintions of rubi rules used

rule 24

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3148

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] + 
 Simp[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 3446

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si 
mp[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] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] 
&& GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(262\) vs. \(2(107)=214\).

Time = 0.17 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.25

\[\frac {a^{3} A \,c^{3} \left (f x +e \right )-a^{3} B \,c^{3} \cos \left (f x +e \right )-3 a^{3} A \,c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{3} B \,c^{3} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )-a^{3} A \,c^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{5}+\frac {5 \sin \left (f x +e \right )^{3}}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+3 a^{3} A \,c^{3} \left (-\frac {\left (\sin \left (f x +e \right )^{3}+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {a^{3} B \,c^{3} \left (\frac {16}{5}+\sin \left (f x +e \right )^{6}+\frac {6 \sin \left (f x +e \right )^{4}}{5}+\frac {8 \sin \left (f x +e \right )^{2}}{5}\right ) \cos \left (f x +e \right )}{7}-\frac {3 a^{3} B \,c^{3} \left (\frac {8}{3}+\sin \left (f x +e \right )^{4}+\frac {4 \sin \left (f x +e \right )^{2}}{3}\right ) \cos \left (f x +e \right )}{5}}{f}\]

Input:

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

Output:

1/f*(a^3*A*c^3*(f*x+e)-a^3*B*c^3*cos(f*x+e)-3*a^3*A*c^3*(-1/2*sin(f*x+e)*c 
os(f*x+e)+1/2*f*x+1/2*e)+a^3*B*c^3*(2+sin(f*x+e)^2)*cos(f*x+e)-a^3*A*c^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^3*A*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)+1/7*a^3*B*c^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^3*B*c^3*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=-\frac {48 \, B a^{3} c^{3} \cos \left (f x + e\right )^{7} - 105 \, A a^{3} c^{3} f x - 7 \, {\left (8 \, A a^{3} c^{3} \cos \left (f x + e\right )^{5} + 10 \, A a^{3} c^{3} \cos \left (f x + e\right )^{3} + 15 \, A a^{3} c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{336 \, f} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3,x, algori 
thm="fricas")
 

Output:

-1/336*(48*B*a^3*c^3*cos(f*x + e)^7 - 105*A*a^3*c^3*f*x - 7*(8*A*a^3*c^3*c 
os(f*x + e)^5 + 10*A*a^3*c^3*cos(f*x + e)^3 + 15*A*a^3*c^3*cos(f*x + e))*s 
in(f*x + e))/f
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (116) = 232\).

Time = 0.55 (sec) , antiderivative size = 682, normalized size of antiderivative = 5.83 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((-5*A*a**3*c**3*x*sin(e + f*x)**6/16 - 15*A*a**3*c**3*x*sin(e + 
f*x)**4*cos(e + f*x)**2/16 + 9*A*a**3*c**3*x*sin(e + f*x)**4/8 - 15*A*a**3 
*c**3*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 9*A*a**3*c**3*x*sin(e + f*x)* 
*2*cos(e + f*x)**2/4 - 3*A*a**3*c**3*x*sin(e + f*x)**2/2 - 5*A*a**3*c**3*x 
*cos(e + f*x)**6/16 + 9*A*a**3*c**3*x*cos(e + f*x)**4/8 - 3*A*a**3*c**3*x* 
cos(e + f*x)**2/2 + A*a**3*c**3*x + 11*A*a**3*c**3*sin(e + f*x)**5*cos(e + 
 f*x)/(16*f) + 5*A*a**3*c**3*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 15*A* 
a**3*c**3*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 5*A*a**3*c**3*sin(e + f*x)* 
cos(e + f*x)**5/(16*f) - 9*A*a**3*c**3*sin(e + f*x)*cos(e + f*x)**3/(8*f) 
+ 3*A*a**3*c**3*sin(e + f*x)*cos(e + f*x)/(2*f) + B*a**3*c**3*sin(e + f*x) 
**6*cos(e + f*x)/f + 2*B*a**3*c**3*sin(e + f*x)**4*cos(e + f*x)**3/f - 3*B 
*a**3*c**3*sin(e + f*x)**4*cos(e + f*x)/f + 8*B*a**3*c**3*sin(e + f*x)**2* 
cos(e + f*x)**5/(5*f) - 4*B*a**3*c**3*sin(e + f*x)**2*cos(e + f*x)**3/f + 
3*B*a**3*c**3*sin(e + f*x)**2*cos(e + f*x)/f + 16*B*a**3*c**3*cos(e + f*x) 
**7/(35*f) - 8*B*a**3*c**3*cos(e + f*x)**5/(5*f) + 2*B*a**3*c**3*cos(e + f 
*x)**3/f - B*a**3*c**3*cos(e + f*x)/f, Ne(f, 0)), (x*(A + B*sin(e))*(a*sin 
(e) + a)**3*(-c*sin(e) + c)**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (107) = 214\).

Time = 0.04 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.26 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=-\frac {35 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{3} - 630 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{3} + 5040 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{3} - 6720 \, {\left (f x + e\right )} A a^{3} c^{3} + 192 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 1344 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 6720 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 6720 \, B a^{3} c^{3} \cos \left (f x + e\right )}{6720 \, f} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3,x, algori 
thm="maxima")
 

Output:

-1/6720*(35*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 4 
8*sin(2*f*x + 2*e))*A*a^3*c^3 - 630*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8* 
sin(2*f*x + 2*e))*A*a^3*c^3 + 5040*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3* 
c^3 - 6720*(f*x + e)*A*a^3*c^3 + 192*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 
 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*B*a^3*c^3 + 1344*(3*cos(f*x + e)^5 
 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c^3 + 6720*(cos(f*x + e)^3 - 
 3*cos(f*x + e))*B*a^3*c^3 + 6720*B*a^3*c^3*cos(f*x + e))/f
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.32 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {5}{16} \, A a^{3} c^{3} x - \frac {B a^{3} c^{3} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} - \frac {B a^{3} c^{3} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} - \frac {3 \, B a^{3} c^{3} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} - \frac {5 \, B a^{3} c^{3} \cos \left (f x + e\right )}{64 \, f} + \frac {A a^{3} c^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {3 \, A a^{3} c^{3} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {15 \, A a^{3} c^{3} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3,x, algori 
thm="giac")
 

Output:

5/16*A*a^3*c^3*x - 1/448*B*a^3*c^3*cos(7*f*x + 7*e)/f - 1/64*B*a^3*c^3*cos 
(5*f*x + 5*e)/f - 3/64*B*a^3*c^3*cos(3*f*x + 3*e)/f - 5/64*B*a^3*c^3*cos(f 
*x + e)/f + 1/192*A*a^3*c^3*sin(6*f*x + 6*e)/f + 3/64*A*a^3*c^3*sin(4*f*x 
+ 4*e)/f + 15/64*A*a^3*c^3*sin(2*f*x + 2*e)/f
 

Mupad [B] (verification not implemented)

Time = 37.43 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.78 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {5\,A\,a^3\,c^3\,x}{16}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (\frac {a^3\,c^3\,\left (672\,B-735\,A\,\left (e+f\,x\right )\right )}{336}+\frac {35\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^3\,c^3\,\left (2016\,B-2205\,A\,\left (e+f\,x\right )\right )}{336}+\frac {105\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {a^3\,c^3\,\left (3360\,B-3675\,A\,\left (e+f\,x\right )\right )}{336}+\frac {175\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}\right )+\frac {a^3\,c^3\,\left (96\,B-105\,A\,\left (e+f\,x\right )\right )}{336}+\frac {5\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}-\frac {7\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6}-\frac {85\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{24}+\frac {85\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}+\frac {7\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{6}+\frac {11\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{8}-\frac {11\,A\,a^3\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^7} \] Input:

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

Output:

(5*A*a^3*c^3*x)/16 - (tan(e/2 + (f*x)/2)^12*((a^3*c^3*(672*B - 735*A*(e + 
f*x)))/336 + (35*A*a^3*c^3*(e + f*x))/16) + tan(e/2 + (f*x)/2)^4*((a^3*c^3 
*(2016*B - 2205*A*(e + f*x)))/336 + (105*A*a^3*c^3*(e + f*x))/16) + tan(e/ 
2 + (f*x)/2)^8*((a^3*c^3*(3360*B - 3675*A*(e + f*x)))/336 + (175*A*a^3*c^3 
*(e + f*x))/16) + (a^3*c^3*(96*B - 105*A*(e + f*x)))/336 + (5*A*a^3*c^3*(e 
 + f*x))/16 - (7*A*a^3*c^3*tan(e/2 + (f*x)/2)^3)/6 - (85*A*a^3*c^3*tan(e/2 
 + (f*x)/2)^5)/24 + (85*A*a^3*c^3*tan(e/2 + (f*x)/2)^9)/24 + (7*A*a^3*c^3* 
tan(e/2 + (f*x)/2)^11)/6 + (11*A*a^3*c^3*tan(e/2 + (f*x)/2)^13)/8 - (11*A* 
a^3*c^3*tan(e/2 + (f*x)/2))/8)/(f*(tan(e/2 + (f*x)/2)^2 + 1)^7)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.10 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx=\frac {a^{3} c^{3} \left (48 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} b +56 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} a -144 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b -182 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a +144 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +231 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a -48 \cos \left (f x +e \right ) b +105 a f x +48 b \right )}{336 f} \] Input:

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

Output:

(a**3*c**3*(48*cos(e + f*x)*sin(e + f*x)**6*b + 56*cos(e + f*x)*sin(e + f* 
x)**5*a - 144*cos(e + f*x)*sin(e + f*x)**4*b - 182*cos(e + f*x)*sin(e + f* 
x)**3*a + 144*cos(e + f*x)*sin(e + f*x)**2*b + 231*cos(e + f*x)*sin(e + f* 
x)*a - 48*cos(e + f*x)*b + 105*a*f*x + 48*b))/(336*f)