\(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx\) [26]

Optimal result

Integrand size = 36, antiderivative size = 229 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx=\frac {9}{128} a^2 (8 A-3 B) c^5 x+\frac {3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac {9 a^2 (8 A-3 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac {3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac {a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac {a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac {3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f} \] Output:

9/128*a^2*(8*A-3*B)*c^5*x+3/80*a^2*(8*A-3*B)*c^5*cos(f*x+e)^5/f+9/128*a^2* 
(8*A-3*B)*c^5*cos(f*x+e)*sin(f*x+e)/f+3/64*a^2*(8*A-3*B)*c^5*cos(f*x+e)^3* 
sin(f*x+e)/f+1/56*a^2*(8*A-3*B)*c^3*cos(f*x+e)^5*(c-c*sin(f*x+e))^2/f-1/8* 
a^2*B*c^2*cos(f*x+e)^5*(c-c*sin(f*x+e))^3/f+3/112*a^2*(8*A-3*B)*cos(f*x+e) 
^5*(c^5-c^5*sin(f*x+e))/f
 

Mathematica [A] (verified)

Time = 5.53 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.96 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx=\frac {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5 (2520 (8 A-3 B) (e+f x)+560 (27 A-17 B) \cos (e+f x)+560 (13 A-7 B) \cos (3 (e+f x))+112 (11 A-B) \cos (5 (e+f x))-80 (A-3 B) \cos (7 (e+f x))+560 (19 A-3 B) \sin (2 (e+f x))-280 (2 A-7 B) \sin (4 (e+f x))-560 (A-B) \sin (6 (e+f x))-35 B \sin (8 (e+f x)))}{35840 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \] Input:

Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x]) 
^5,x]
 

Output:

((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5*(2520*(8*A - 3*B)*(e + f*x) 
 + 560*(27*A - 17*B)*Cos[e + f*x] + 560*(13*A - 7*B)*Cos[3*(e + f*x)] + 11 
2*(11*A - B)*Cos[5*(e + f*x)] - 80*(A - 3*B)*Cos[7*(e + f*x)] + 560*(19*A 
- 3*B)*Sin[2*(e + f*x)] - 280*(2*A - 7*B)*Sin[4*(e + f*x)] - 560*(A - B)*S 
in[6*(e + f*x)] - 35*B*Sin[8*(e + f*x)]))/(35840*f*(Cos[(e + f*x)/2] - Sin 
[(e + f*x)/2])^10*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 3446, 3042, 3339, 3042, 3157, 3042, 3157, 3042, 3148, 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.

\(\displaystyle \int (a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^5 (A+B \sin (e+f x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^5 (A+B \sin (e+f x))dx\)

\(\Big \downarrow \) 3446

\(\displaystyle a^2 c^2 \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^3dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 c^2 \int \cos (e+f x)^4 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3dx\)

\(\Big \downarrow \) 3339

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3157

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3148

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

Input:

Int[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5,x]
 

Output:

a^2*c^2*(-1/8*(B*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^3)/f + ((8*A - 3*B)*( 
(c*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(7*f) + (9*c*((Cos[e + f*x]^5*(c 
^2 - c^2*Sin[e + f*x]))/(6*f) + (7*c*((c*Cos[e + f*x]^5)/(5*f) + c*((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))/7))/8)
 

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. \(568\) vs. \(2(215)=430\).

Time = 0.11 (sec) , antiderivative size = 569, normalized size of antiderivative = 2.48

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

Input:

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.69 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx=-\frac {640 \, {\left (A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{7} - 3584 \, {\left (A - B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{5} - 315 \, {\left (8 \, A - 3 \, B\right )} a^{2} c^{5} f x + 35 \, {\left (16 \, B a^{2} c^{5} \cos \left (f x + e\right )^{7} + 8 \, {\left (8 \, A - 11 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{5} - 6 \, {\left (8 \, A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{3} - 9 \, {\left (8 \, A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4480 \, f} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x, algori 
thm="fricas")
 

Output:

-1/4480*(640*(A - 3*B)*a^2*c^5*cos(f*x + e)^7 - 3584*(A - B)*a^2*c^5*cos(f 
*x + e)^5 - 315*(8*A - 3*B)*a^2*c^5*f*x + 35*(16*B*a^2*c^5*cos(f*x + e)^7 
+ 8*(8*A - 11*B)*a^2*c^5*cos(f*x + e)^5 - 6*(8*A - 3*B)*a^2*c^5*cos(f*x + 
e)^3 - 9*(8*A - 3*B)*a^2*c^5*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. 1586 vs. \(2 (218) = 436\).

Time = 0.90 (sec) , antiderivative size = 1586, normalized size of antiderivative = 6.93 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**5,x)
 

Output:

Piecewise((15*A*a**2*c**5*x*sin(e + f*x)**6/16 + 45*A*a**2*c**5*x*sin(e + 
f*x)**4*cos(e + f*x)**2/16 - 15*A*a**2*c**5*x*sin(e + f*x)**4/8 + 45*A*a** 
2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/16 - 15*A*a**2*c**5*x*sin(e + f*x 
)**2*cos(e + f*x)**2/4 + A*a**2*c**5*x*sin(e + f*x)**2/2 + 15*A*a**2*c**5* 
x*cos(e + f*x)**6/16 - 15*A*a**2*c**5*x*cos(e + f*x)**4/8 + A*a**2*c**5*x* 
cos(e + f*x)**2/2 + A*a**2*c**5*x + A*a**2*c**5*sin(e + f*x)**6*cos(e + f* 
x)/f - 33*A*a**2*c**5*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 2*A*a**2*c**5* 
sin(e + f*x)**4*cos(e + f*x)**3/f + A*a**2*c**5*sin(e + f*x)**4*cos(e + f* 
x)/f - 5*A*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) + 25*A*a**2*c** 
5*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 8*A*a**2*c**5*sin(e + f*x)**2*cos(e 
 + f*x)**5/(5*f) + 4*A*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 5 
*A*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)/f - 15*A*a**2*c**5*sin(e + f*x)* 
cos(e + f*x)**5/(16*f) + 15*A*a**2*c**5*sin(e + f*x)*cos(e + f*x)**3/(8*f) 
 - A*a**2*c**5*sin(e + f*x)*cos(e + f*x)/(2*f) + 16*A*a**2*c**5*cos(e + f* 
x)**7/(35*f) + 8*A*a**2*c**5*cos(e + f*x)**5/(15*f) - 10*A*a**2*c**5*cos(e 
 + f*x)**3/(3*f) + 3*A*a**2*c**5*cos(e + f*x)/f - 35*B*a**2*c**5*x*sin(e + 
 f*x)**8/128 - 35*B*a**2*c**5*x*sin(e + f*x)**6*cos(e + f*x)**2/32 - 5*B*a 
**2*c**5*x*sin(e + f*x)**6/16 - 105*B*a**2*c**5*x*sin(e + f*x)**4*cos(e + 
f*x)**4/64 - 15*B*a**2*c**5*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 15*B*a* 
*2*c**5*x*sin(e + f*x)**4/8 - 35*B*a**2*c**5*x*sin(e + f*x)**2*cos(e + ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (218) = 436\).

Time = 0.05 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.49 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx =\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x, algori 
thm="maxima")
 

Output:

-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^2*c^5 - 7168*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 
+ 15*cos(f*x + e))*A*a^2*c^5 - 179200*(cos(f*x + e)^3 - 3*cos(f*x + e))*A* 
a^2*c^5 - 1680*(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^2*c^5 + 16800*(12*f*x + 12*e + sin(4*f*x + 4*e) 
 - 8*sin(2*f*x + 2*e))*A*a^2*c^5 - 26880*(2*f*x + 2*e - sin(2*f*x + 2*e))* 
A*a^2*c^5 - 107520*(f*x + e)*A*a^2*c^5 - 9216*(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^2*c^5 - 35840*(3*cos( 
f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^2*c^5 - 35840*(cos(f 
*x + e)^3 - 3*cos(f*x + e))*B*a^2*c^5 + 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^2*c^5 + 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))*B*a^2*c^5 - 16800*(12*f*x + 12*e + sin(4*f*x 
+ 4*e) - 8*sin(2*f*x + 2*e))*B*a^2*c^5 + 80640*(2*f*x + 2*e - sin(2*f*x + 
2*e))*B*a^2*c^5 - 322560*A*a^2*c^5*cos(f*x + e) + 107520*B*a^2*c^5*cos(f*x 
 + e))/f
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.18 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx=-\frac {B a^{2} c^{5} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac {9}{128} \, {\left (8 \, A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} x - \frac {{\left (A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {{\left (11 \, A a^{2} c^{5} - B a^{2} c^{5}\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {{\left (13 \, A a^{2} c^{5} - 7 \, B a^{2} c^{5}\right )} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {{\left (27 \, A a^{2} c^{5} - 17 \, B a^{2} c^{5}\right )} \cos \left (f x + e\right )}{64 \, f} - \frac {{\left (A a^{2} c^{5} - B a^{2} c^{5}\right )} \sin \left (6 \, f x + 6 \, e\right )}{64 \, f} - \frac {{\left (2 \, A a^{2} c^{5} - 7 \, B a^{2} c^{5}\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {{\left (19 \, A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x, algori 
thm="giac")
 

Output:

-1/1024*B*a^2*c^5*sin(8*f*x + 8*e)/f + 9/128*(8*A*a^2*c^5 - 3*B*a^2*c^5)*x 
 - 1/448*(A*a^2*c^5 - 3*B*a^2*c^5)*cos(7*f*x + 7*e)/f + 1/320*(11*A*a^2*c^ 
5 - B*a^2*c^5)*cos(5*f*x + 5*e)/f + 1/64*(13*A*a^2*c^5 - 7*B*a^2*c^5)*cos( 
3*f*x + 3*e)/f + 1/64*(27*A*a^2*c^5 - 17*B*a^2*c^5)*cos(f*x + e)/f - 1/64* 
(A*a^2*c^5 - B*a^2*c^5)*sin(6*f*x + 6*e)/f - 1/128*(2*A*a^2*c^5 - 7*B*a^2* 
c^5)*sin(4*f*x + 4*e)/f + 1/64*(19*A*a^2*c^5 - 3*B*a^2*c^5)*sin(2*f*x + 2* 
e)/f
 

Mupad [B] (verification not implemented)

Time = 37.82 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.89 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx =\text {Too large to display} \] Input:

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^5,x)
                                                                                    
                                                                                    
 

Output:

(tan(e/2 + (f*x)/2)^14*(6*A*a^2*c^5 - 2*B*a^2*c^5) + tan(e/2 + (f*x)/2)^10 
*(30*A*a^2*c^5 - 10*B*a^2*c^5) + tan(e/2 + (f*x)/2)^12*(22*A*a^2*c^5 - 18* 
B*a^2*c^5) + tan(e/2 + (f*x)/2)^8*(46*A*a^2*c^5 - 26*B*a^2*c^5) + tan(e/2 
+ (f*x)/2)^4*((74*A*a^2*c^5)/5 - (14*B*a^2*c^5)/5) - tan(e/2 + (f*x)/2)^15 
*((7*A*a^2*c^5)/8 + (27*B*a^2*c^5)/64) + tan(e/2 + (f*x)/2)^2*((158*A*a^2* 
c^5)/35 - (138*B*a^2*c^5)/35) + tan(e/2 + (f*x)/2)^6*((218*A*a^2*c^5)/5 - 
(158*B*a^2*c^5)/5) + tan(e/2 + (f*x)/2)^3*((75*A*a^2*c^5)/8 - (305*B*a^2*c 
^5)/64) - tan(e/2 + (f*x)/2)^13*((75*A*a^2*c^5)/8 - (305*B*a^2*c^5)/64) + 
tan(e/2 + (f*x)/2)^5*((55*A*a^2*c^5)/8 - (437*B*a^2*c^5)/64) - tan(e/2 + ( 
f*x)/2)^11*((55*A*a^2*c^5)/8 - (437*B*a^2*c^5)/64) - tan(e/2 + (f*x)/2)^7* 
((13*A*a^2*c^5)/8 - (919*B*a^2*c^5)/64) + tan(e/2 + (f*x)/2)^9*((13*A*a^2* 
c^5)/8 - (919*B*a^2*c^5)/64) + tan(e/2 + (f*x)/2)*((7*A*a^2*c^5)/8 + (27*B 
*a^2*c^5)/64) + (46*A*a^2*c^5)/35 - (26*B*a^2*c^5)/35)/(f*(8*tan(e/2 + (f* 
x)/2)^2 + 28*tan(e/2 + (f*x)/2)^4 + 56*tan(e/2 + (f*x)/2)^6 + 70*tan(e/2 + 
 (f*x)/2)^8 + 56*tan(e/2 + (f*x)/2)^10 + 28*tan(e/2 + (f*x)/2)^12 + 8*tan( 
e/2 + (f*x)/2)^14 + tan(e/2 + (f*x)/2)^16 + 1)) + (9*a^2*c^5*atan((9*a^2*c 
^5*tan(e/2 + (f*x)/2)*(8*A - 3*B))/(64*((9*A*a^2*c^5)/8 - (27*B*a^2*c^5)/6 
4)))*(8*A - 3*B))/(64*f)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.15 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx=\frac {a^{2} c^{5} \left (560 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7} b +640 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} a -1920 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} b -2240 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} a +1400 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b +1664 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a +2176 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +2800 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -3850 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -5248 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +1408 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +1960 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +945 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +2944 \cos \left (f x +e \right ) a -1664 \cos \left (f x +e \right ) b +2520 a f x -2944 a -945 b f x +1664 b \right )}{4480 f} \] Input:

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x)
 

Output:

(a**2*c**5*(560*cos(e + f*x)*sin(e + f*x)**7*b + 640*cos(e + f*x)*sin(e + 
f*x)**6*a - 1920*cos(e + f*x)*sin(e + f*x)**6*b - 2240*cos(e + f*x)*sin(e 
+ f*x)**5*a + 1400*cos(e + f*x)*sin(e + f*x)**5*b + 1664*cos(e + f*x)*sin( 
e + f*x)**4*a + 2176*cos(e + f*x)*sin(e + f*x)**4*b + 2800*cos(e + f*x)*si 
n(e + f*x)**3*a - 3850*cos(e + f*x)*sin(e + f*x)**3*b - 5248*cos(e + f*x)* 
sin(e + f*x)**2*a + 1408*cos(e + f*x)*sin(e + f*x)**2*b + 1960*cos(e + f*x 
)*sin(e + f*x)*a + 945*cos(e + f*x)*sin(e + f*x)*b + 2944*cos(e + f*x)*a - 
 1664*cos(e + f*x)*b + 2520*a*f*x - 2944*a - 945*b*f*x + 1664*b))/(4480*f)