\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx\) [38]

Optimal result

Integrand size = 36, antiderivative size = 265 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx=\frac {11}{256} a^3 (10 A-3 B) c^6 x+\frac {11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos (e+f x) \sin (e+f x)}{256 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^3(e+f x) \sin (e+f x)}{384 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f} \] Output:

11/256*a^3*(10*A-3*B)*c^6*x+11/560*a^3*(10*A-3*B)*c^6*cos(f*x+e)^7/f+11/25 
6*a^3*(10*A-3*B)*c^6*cos(f*x+e)*sin(f*x+e)/f+11/384*a^3*(10*A-3*B)*c^6*cos 
(f*x+e)^3*sin(f*x+e)/f+11/480*a^3*(10*A-3*B)*c^6*cos(f*x+e)^5*sin(f*x+e)/f 
-1/10*a^3*B*cos(f*x+e)^7*(c^2-c^2*sin(f*x+e))^3/f+1/90*a^3*(10*A-3*B)*cos( 
f*x+e)^7*(c^3-c^3*sin(f*x+e))^2/f+11/720*a^3*(10*A-3*B)*cos(f*x+e)^7*(c^6- 
c^6*sin(f*x+e))/f
 

Mathematica [A] (verified)

Time = 12.47 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.96 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx=\frac {(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 (27720 (10 A-3 B) (e+f x)+5040 (33 A-19 B) \cos (e+f x)+3360 (29 A-15 B) \cos (3 (e+f x))+10080 (3 A-B) \cos (5 (e+f x))+360 (9 A+5 B) \cos (7 (e+f x))-280 (A-3 B) \cos (9 (e+f x))+1260 (144 A-25 B) \sin (2 (e+f x))+2520 (6 A+7 B) \sin (4 (e+f x))-210 (32 A-51 B) \sin (6 (e+f x))-315 (6 A-5 B) \sin (8 (e+f x))-126 B \sin (10 (e+f x)))}{645120 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{12} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6} \] Input:

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]) 
^6,x]
 

Output:

((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6*(27720*(10*A - 3*B)*(e + f* 
x) + 5040*(33*A - 19*B)*Cos[e + f*x] + 3360*(29*A - 15*B)*Cos[3*(e + f*x)] 
 + 10080*(3*A - B)*Cos[5*(e + f*x)] + 360*(9*A + 5*B)*Cos[7*(e + f*x)] - 2 
80*(A - 3*B)*Cos[9*(e + f*x)] + 1260*(144*A - 25*B)*Sin[2*(e + f*x)] + 252 
0*(6*A + 7*B)*Sin[4*(e + f*x)] - 210*(32*A - 51*B)*Sin[6*(e + f*x)] - 315* 
(6*A - 5*B)*Sin[8*(e + f*x)] - 126*B*Sin[10*(e + f*x)]))/(645120*f*(Cos[(e 
 + f*x)/2] - Sin[(e + f*x)/2])^12*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.81, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.472, Rules used = {3042, 3446, 3042, 3339, 3042, 3157, 3042, 3157, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3446

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3339

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \int \cos ^6(e+f x) (c-c \sin (e+f x))^3dx-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \int \cos (e+f x)^6 (c-c \sin (e+f x))^3dx-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3157

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \int \cos ^6(e+f x) (c-c \sin (e+f x))^2dx+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \int \cos (e+f x)^6 (c-c \sin (e+f x))^2dx+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3157

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \int \cos ^6(e+f x) (c-c \sin (e+f x))dx+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \int \cos (e+f x)^6 (c-c \sin (e+f x))dx+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3148

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \left (c \int \cos ^6(e+f x)dx+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \left (c \int \sin \left (e+f x+\frac {\pi }{2}\right )^6dx+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \int \cos ^4(e+f x)dx+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \int \sin \left (e+f x+\frac {\pi }{2}\right )^4dx+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(e+f x)dx+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {9}{8} c \left (c \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}\right )+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}\right )+\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}\right )+\frac {c \cos ^7(e+f x)}{7 f}\right )+\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

\(\Big \downarrow \) 24

\(\displaystyle a^3 c^3 \left (\frac {1}{10} (10 A-3 B) \left (\frac {11}{9} c \left (\frac {\cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{8 f}+\frac {9}{8} c \left (\frac {c \cos ^7(e+f x)}{7 f}+c \left (\frac {\sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac {5}{6} \left (\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac {3}{4} \left (\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}\right )\right )\right )\right )\right )+\frac {c \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}\right )-\frac {B \cos ^7(e+f x) (c-c \sin (e+f x))^3}{10 f}\right )\)

Input:

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^6,x]
 

Output:

a^3*c^3*(-1/10*(B*Cos[e + f*x]^7*(c - c*Sin[e + f*x])^3)/f + ((10*A - 3*B) 
*((c*Cos[e + f*x]^7*(c - c*Sin[e + f*x])^2)/(9*f) + (11*c*((Cos[e + f*x]^7 
*(c^2 - c^2*Sin[e + f*x]))/(8*f) + (9*c*((c*Cos[e + f*x]^7)/(7*f) + c*((Co 
s[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)))/8))/9))/10)
 

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 3157

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + 
f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p))   Int[(g* 
Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, 
g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers 
Q[2*m, 2*p]
 

rule 3339

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] + S 
imp[(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] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ 
a^2 - b^2, 0] && NeQ[m + p + 1, 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. \(650\) vs. \(2(249)=498\).

Time = 0.27 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.46

\[\frac {a^{3} A \,c^{6} \left (f x +e \right )-a^{3} B \,c^{6} \cos \left (f x +e \right )-\frac {8 a^{3} A \,c^{6} \left (2+\sin \left (f x +e \right )^{2}\right ) \cos \left (f x +e \right )}{3}+8 a^{3} B \,c^{6} \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 )+3 a^{3} A \,c^{6} \cos \left (f x +e \right )-3 a^{3} B \,c^{6} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{3} A \,c^{6} \left (\frac {128}{35}+\sin \left (f x +e \right )^{8}+\frac {8 \sin \left (f x +e \right )^{6}}{7}+\frac {48 \sin \left (f x +e \right )^{4}}{35}+\frac {64 \sin \left (f x +e \right )^{2}}{35}\right ) \cos \left (f x +e \right )}{9}-3 a^{3} A \,c^{6} \left (-\frac {\left (\sin \left (f x +e \right )^{7}+\frac {7 \sin \left (f x +e \right )^{5}}{6}+\frac {35 \sin \left (f x +e \right )^{3}}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )+8 a^{3} A \,c^{6} \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 )+\frac {6 a^{3} A \,c^{6} \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}-6 a^{3} A \,c^{6} \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 )+a^{3} B \,c^{6} \left (-\frac {\left (\sin \left (f x +e \right )^{9}+\frac {9 \sin \left (f x +e \right )^{7}}{8}+\frac {21 \sin \left (f x +e \right )^{5}}{16}+\frac {105 \sin \left (f x +e \right )^{3}}{64}+\frac {315 \sin \left (f x +e \right )}{128}\right ) \cos \left (f x +e \right )}{10}+\frac {63 f x}{256}+\frac {63 e}{256}\right )+\frac {a^{3} B \,c^{6} \left (\frac {128}{35}+\sin \left (f x +e \right )^{8}+\frac {8 \sin \left (f x +e \right )^{6}}{7}+\frac {48 \sin \left (f x +e \right )^{4}}{35}+\frac {64 \sin \left (f x +e \right )^{2}}{35}\right ) \cos \left (f x +e \right )}{3}-\frac {8 a^{3} B \,c^{6} \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}-6 a^{3} B \,c^{6} \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 )+\frac {6 a^{3} B \,c^{6} \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))^6,x)
 

Output:

1/f*(a^3*A*c^6*(f*x+e)-a^3*B*c^6*cos(f*x+e)-8/3*a^3*A*c^6*(2+sin(f*x+e)^2) 
*cos(f*x+e)+8*a^3*B*c^6*(-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*A*c^6*cos(f*x+e)-3*a^3*B*c^6*(-1/2*sin(f*x+e)*cos(f*x+e) 
+1/2*f*x+1/2*e)-1/9*a^3*A*c^6*(128/35+sin(f*x+e)^8+8/7*sin(f*x+e)^6+48/35* 
sin(f*x+e)^4+64/35*sin(f*x+e)^2)*cos(f*x+e)-3*a^3*A*c^6*(-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)+8*a^3*A*c^6*(-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)+6/5*a^3*A*c^6*(8/3+sin(f*x+e)^4+4/3*sin( 
f*x+e)^2)*cos(f*x+e)-6*a^3*A*c^6*(-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^3*B*c^6*(-1/10*(sin(f*x+e)^9+9/8*sin(f*x+e)^7+21/16 
*sin(f*x+e)^5+105/64*sin(f*x+e)^3+315/128*sin(f*x+e))*cos(f*x+e)+63/256*f* 
x+63/256*e)+1/3*a^3*B*c^6*(128/35+sin(f*x+e)^8+8/7*sin(f*x+e)^6+48/35*sin( 
f*x+e)^4+64/35*sin(f*x+e)^2)*cos(f*x+e)-8/7*a^3*B*c^6*(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)-6*a^3*B*c^6*(-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)+6/5*a^3*B 
*c^6*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.68 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx=-\frac {8960 \, {\left (A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{9} - 46080 \, {\left (A - B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{7} - 3465 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} f x + 21 \, {\left (384 \, B a^{3} c^{6} \cos \left (f x + e\right )^{9} + 48 \, {\left (30 \, A - 41 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{7} - 88 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{5} - 110 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{3} - 165 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{80640 \, f} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x, algori 
thm="fricas")
 

Output:

-1/80640*(8960*(A - 3*B)*a^3*c^6*cos(f*x + e)^9 - 46080*(A - B)*a^3*c^6*co 
s(f*x + e)^7 - 3465*(10*A - 3*B)*a^3*c^6*f*x + 21*(384*B*a^3*c^6*cos(f*x + 
 e)^9 + 48*(30*A - 41*B)*a^3*c^6*cos(f*x + e)^7 - 88*(10*A - 3*B)*a^3*c^6* 
cos(f*x + e)^5 - 110*(10*A - 3*B)*a^3*c^6*cos(f*x + e)^3 - 165*(10*A - 3*B 
)*a^3*c^6*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. 1948 vs. \(2 (252) = 504\).

Time = 1.59 (sec) , antiderivative size = 1948, normalized size of antiderivative = 7.35 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**6,x)
 

Output:

Piecewise((-105*A*a**3*c**6*x*sin(e + f*x)**8/128 - 105*A*a**3*c**6*x*sin( 
e + f*x)**6*cos(e + f*x)**2/32 + 5*A*a**3*c**6*x*sin(e + f*x)**6/2 - 315*A 
*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*A*a**3*c**6*x*sin(e + 
 f*x)**4*cos(e + f*x)**2/2 - 9*A*a**3*c**6*x*sin(e + f*x)**4/4 - 105*A*a** 
3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**6/32 + 15*A*a**3*c**6*x*sin(e + f*x 
)**2*cos(e + f*x)**4/2 - 9*A*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**2/2 
 - 105*A*a**3*c**6*x*cos(e + f*x)**8/128 + 5*A*a**3*c**6*x*cos(e + f*x)**6 
/2 - 9*A*a**3*c**6*x*cos(e + f*x)**4/4 + A*a**3*c**6*x - A*a**3*c**6*sin(e 
 + f*x)**8*cos(e + f*x)/f + 279*A*a**3*c**6*sin(e + f*x)**7*cos(e + f*x)/( 
128*f) - 8*A*a**3*c**6*sin(e + f*x)**6*cos(e + f*x)**3/(3*f) + 511*A*a**3* 
c**6*sin(e + f*x)**5*cos(e + f*x)**3/(128*f) - 11*A*a**3*c**6*sin(e + f*x) 
**5*cos(e + f*x)/(2*f) - 16*A*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)**5/(5 
*f) + 6*A*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)/f + 385*A*a**3*c**6*sin(e 
 + f*x)**3*cos(e + f*x)**5/(128*f) - 20*A*a**3*c**6*sin(e + f*x)**3*cos(e 
+ f*x)**3/(3*f) + 15*A*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 64*A 
*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**7/(35*f) + 8*A*a**3*c**6*sin(e + 
f*x)**2*cos(e + f*x)**3/f - 8*A*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)/f + 
 105*A*a**3*c**6*sin(e + f*x)*cos(e + f*x)**7/(128*f) - 5*A*a**3*c**6*sin( 
e + f*x)*cos(e + f*x)**5/(2*f) + 9*A*a**3*c**6*sin(e + f*x)*cos(e + f*x)** 
3/(4*f) - 128*A*a**3*c**6*cos(e + f*x)**9/(315*f) + 16*A*a**3*c**6*cos(...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (252) = 504\).

Time = 0.05 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.49 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x, algori 
thm="maxima")
 

Output:

-1/645120*(2048*(35*cos(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e) 
^5 - 420*cos(f*x + e)^3 + 315*cos(f*x + e))*A*a^3*c^6 - 258048*(3*cos(f*x 
+ e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*c^6 - 1720320*(cos(f*x 
 + e)^3 - 3*cos(f*x + e))*A*a^3*c^6 + 630*(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))*A*a^3*c^6 - 26880*(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^6 + 120960*(12*f*x + 12*e + sin(4*f* 
x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*c^6 - 645120*(f*x + e)*A*a^3*c^6 - 61 
44*(35*cos(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e)^5 - 420*cos( 
f*x + e)^3 + 315*cos(f*x + e))*B*a^3*c^6 - 147456*(5*cos(f*x + e)^7 - 21*c 
os(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*B*a^3*c^6 - 258048*(3 
*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c^6 + 63*(32* 
sin(2*f*x + 2*e)^5 - 640*sin(2*f*x + 2*e)^3 - 2520*f*x - 2520*e - 25*sin(8 
*f*x + 8*e) - 600*sin(4*f*x + 4*e) + 2560*sin(2*f*x + 2*e))*B*a^3*c^6 + 20 
160*(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^6 - 161280*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2 
*f*x + 2*e))*B*a^3*c^6 + 483840*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^6 
 - 1935360*A*a^3*c^6*cos(f*x + e) + 645120*B*a^3*c^6*cos(f*x + e))/f
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.27 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx=-\frac {B a^{3} c^{6} \sin \left (10 \, f x + 10 \, e\right )}{5120 \, f} + \frac {11}{256} \, {\left (10 \, A a^{3} c^{6} - 3 \, B a^{3} c^{6}\right )} x - \frac {{\left (A a^{3} c^{6} - 3 \, B a^{3} c^{6}\right )} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac {{\left (9 \, A a^{3} c^{6} + 5 \, B a^{3} c^{6}\right )} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac {{\left (3 \, A a^{3} c^{6} - B a^{3} c^{6}\right )} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {{\left (29 \, A a^{3} c^{6} - 15 \, B a^{3} c^{6}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac {{\left (33 \, A a^{3} c^{6} - 19 \, B a^{3} c^{6}\right )} \cos \left (f x + e\right )}{128 \, f} - \frac {{\left (6 \, A a^{3} c^{6} - 5 \, B a^{3} c^{6}\right )} \sin \left (8 \, f x + 8 \, e\right )}{2048 \, f} - \frac {{\left (32 \, A a^{3} c^{6} - 51 \, B a^{3} c^{6}\right )} \sin \left (6 \, f x + 6 \, e\right )}{3072 \, f} + \frac {{\left (6 \, A a^{3} c^{6} + 7 \, B a^{3} c^{6}\right )} \sin \left (4 \, f x + 4 \, e\right )}{256 \, f} + \frac {{\left (144 \, A a^{3} c^{6} - 25 \, B a^{3} c^{6}\right )} \sin \left (2 \, f x + 2 \, e\right )}{512 \, f} \] Input:

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x, algori 
thm="giac")
 

Output:

-1/5120*B*a^3*c^6*sin(10*f*x + 10*e)/f + 11/256*(10*A*a^3*c^6 - 3*B*a^3*c^ 
6)*x - 1/2304*(A*a^3*c^6 - 3*B*a^3*c^6)*cos(9*f*x + 9*e)/f + 1/1792*(9*A*a 
^3*c^6 + 5*B*a^3*c^6)*cos(7*f*x + 7*e)/f + 1/64*(3*A*a^3*c^6 - B*a^3*c^6)* 
cos(5*f*x + 5*e)/f + 1/192*(29*A*a^3*c^6 - 15*B*a^3*c^6)*cos(3*f*x + 3*e)/ 
f + 1/128*(33*A*a^3*c^6 - 19*B*a^3*c^6)*cos(f*x + e)/f - 1/2048*(6*A*a^3*c 
^6 - 5*B*a^3*c^6)*sin(8*f*x + 8*e)/f - 1/3072*(32*A*a^3*c^6 - 51*B*a^3*c^6 
)*sin(6*f*x + 6*e)/f + 1/256*(6*A*a^3*c^6 + 7*B*a^3*c^6)*sin(4*f*x + 4*e)/ 
f + 1/512*(144*A*a^3*c^6 - 25*B*a^3*c^6)*sin(2*f*x + 2*e)/f
 

Mupad [B] (verification not implemented)

Time = 38.07 (sec) , antiderivative size = 812, normalized size of antiderivative = 3.06 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx=\text {Too large to display} \] Input:

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^6,x)
                                                                                    
                                                                                    
 

Output:

(tan(e/2 + (f*x)/2)^18*(6*A*a^3*c^6 - 2*B*a^3*c^6) + tan(e/2 + (f*x)/2)^16 
*(22*A*a^3*c^6 - 18*B*a^3*c^6) + tan(e/2 + (f*x)/2)^8*(84*A*a^3*c^6 - 28*B 
*a^3*c^6) + tan(e/2 + (f*x)/2)^14*((136*A*a^3*c^6)/3 - 8*B*a^3*c^6) + tan( 
e/2 + (f*x)/2)^4*((136*A*a^3*c^6)/7 - (24*B*a^3*c^6)/7) + tan(e/2 + (f*x)/ 
2)^10*(116*A*a^3*c^6 - 60*B*a^3*c^6) - tan(e/2 + (f*x)/2)^19*((73*A*a^3*c^ 
6)/64 + (33*B*a^3*c^6)/128) + tan(e/2 + (f*x)/2)^2*((202*A*a^3*c^6)/63 - ( 
58*B*a^3*c^6)/21) + tan(e/2 + (f*x)/2)^12*((328*A*a^3*c^6)/3 - 72*B*a^3*c^ 
6) + tan(e/2 + (f*x)/2)^7*((341*A*a^3*c^6)/16 + (333*B*a^3*c^6)/32) - tan( 
e/2 + (f*x)/2)^13*((341*A*a^3*c^6)/16 + (333*B*a^3*c^6)/32) + tan(e/2 + (f 
*x)/2)^6*((456*A*a^3*c^6)/7 - (344*B*a^3*c^6)/7) + tan(e/2 + (f*x)/2)^5*(( 
449*A*a^3*c^6)/48 - (577*B*a^3*c^6)/160) - tan(e/2 + (f*x)/2)^15*((449*A*a 
^3*c^6)/48 - (577*B*a^3*c^6)/160) + tan(e/2 + (f*x)/2)^3*((2117*A*a^3*c^6) 
/192 - (705*B*a^3*c^6)/128) - tan(e/2 + (f*x)/2)^17*((2117*A*a^3*c^6)/192 
- (705*B*a^3*c^6)/128) + tan(e/2 + (f*x)/2)^9*((699*A*a^3*c^6)/32 - (2749* 
B*a^3*c^6)/64) - tan(e/2 + (f*x)/2)^11*((699*A*a^3*c^6)/32 - (2749*B*a^3*c 
^6)/64) + tan(e/2 + (f*x)/2)*((73*A*a^3*c^6)/64 + (33*B*a^3*c^6)/128) + (5 
8*A*a^3*c^6)/63 - (10*B*a^3*c^6)/21)/(f*(10*tan(e/2 + (f*x)/2)^2 + 45*tan( 
e/2 + (f*x)/2)^4 + 120*tan(e/2 + (f*x)/2)^6 + 210*tan(e/2 + (f*x)/2)^8 + 2 
52*tan(e/2 + (f*x)/2)^10 + 210*tan(e/2 + (f*x)/2)^12 + 120*tan(e/2 + (f*x) 
/2)^14 + 45*tan(e/2 + (f*x)/2)^16 + 10*tan(e/2 + (f*x)/2)^18 + tan(e/2 ...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.25 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx=\frac {a^{3} c^{6} \left (-8064 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{9} b -8960 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{8} a +26880 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{8} b +30240 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7} a -9072 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7} b -10240 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} a -61440 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} b -72240 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} a +70056 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b +84480 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a +23040 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +30660 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -73710 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -102400 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +30720 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +45990 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +10395 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +37120 \cos \left (f x +e \right ) a -19200 \cos \left (f x +e \right ) b +34650 a f x -37120 a -10395 b f x +19200 b \right )}{80640 f} \] Input:

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^6,x)
 

Output:

(a**3*c**6*( - 8064*cos(e + f*x)*sin(e + f*x)**9*b - 8960*cos(e + f*x)*sin 
(e + f*x)**8*a + 26880*cos(e + f*x)*sin(e + f*x)**8*b + 30240*cos(e + f*x) 
*sin(e + f*x)**7*a - 9072*cos(e + f*x)*sin(e + f*x)**7*b - 10240*cos(e + f 
*x)*sin(e + f*x)**6*a - 61440*cos(e + f*x)*sin(e + f*x)**6*b - 72240*cos(e 
 + f*x)*sin(e + f*x)**5*a + 70056*cos(e + f*x)*sin(e + f*x)**5*b + 84480*c 
os(e + f*x)*sin(e + f*x)**4*a + 23040*cos(e + f*x)*sin(e + f*x)**4*b + 306 
60*cos(e + f*x)*sin(e + f*x)**3*a - 73710*cos(e + f*x)*sin(e + f*x)**3*b - 
 102400*cos(e + f*x)*sin(e + f*x)**2*a + 30720*cos(e + f*x)*sin(e + f*x)** 
2*b + 45990*cos(e + f*x)*sin(e + f*x)*a + 10395*cos(e + f*x)*sin(e + f*x)* 
b + 37120*cos(e + f*x)*a - 19200*cos(e + f*x)*b + 34650*a*f*x - 37120*a - 
10395*b*f*x + 19200*b))/(80640*f)