\(\int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx\) [60]

Optimal result

Integrand size = 36, antiderivative size = 240 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {105 (4 A-7 B) c^5 x}{8 a^2}+\frac {35 (4 A-7 B) c^5 \cos ^3(e+f x)}{4 a^2 f}+\frac {105 (4 A-7 B) c^5 \cos (e+f x) \sin (e+f x)}{8 a^2 f}-\frac {a^5 (A-B) c^5 \cos ^{11}(e+f x)}{3 f (a+a \sin (e+f x))^7}+\frac {2 a^3 (4 A-7 B) c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^5}+\frac {6 a^4 (4 A-7 B) c^5 \cos ^7(e+f x)}{f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {21 (4 A-7 B) c^5 \cos ^5(e+f x)}{4 f \left (a^2+a^2 \sin (e+f x)\right )} \] Output:

105/8*(4*A-7*B)*c^5*x/a^2+35/4*(4*A-7*B)*c^5*cos(f*x+e)^3/a^2/f+105/8*(4*A 
-7*B)*c^5*cos(f*x+e)*sin(f*x+e)/a^2/f-1/3*a^5*(A-B)*c^5*cos(f*x+e)^11/f/(a 
+a*sin(f*x+e))^7+2/3*a^3*(4*A-7*B)*c^5*cos(f*x+e)^9/f/(a+a*sin(f*x+e))^5+6 
*a^4*(4*A-7*B)*c^5*cos(f*x+e)^7/f/(a^2+a^2*sin(f*x+e))^3+21/4*(4*A-7*B)*c^ 
5*cos(f*x+e)^5/f/(a^2+a^2*sin(f*x+e))
 

Mathematica [A] (verified)

Time = 12.57 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^5 \left (2048 (A-B) \sin \left (\frac {1}{2} (e+f x)\right )-1024 (A-B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-1024 (13 A-19 B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+1260 (4 A-7 B) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+24 (95 A-217 B) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-8 (A-7 B) \cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-24 (7 A-24 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (2 (e+f x))-3 B \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (4 (e+f x))\right )}{96 a^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (1+\sin (e+f x))^2} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(2048*(A - B 
)*Sin[(e + f*x)/2] - 1024*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 
1024*(13*A - 19*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^ 
2 + 1260*(4*A - 7*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 2 
4*(95*A - 217*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 8* 
(A - 7*B)*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 24*(7 
*A - 24*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)] - 3*B* 
(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[4*(e + f*x)]))/(96*a^2*f*(Cos[ 
(e + f*x)/2] - Sin[(e + f*x)/2])^10*(1 + Sin[e + f*x])^2)
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.87, 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, 3338, 3042, 3159, 3042, 3159, 3042, 3158, 3042, 3161, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3446

\(\displaystyle a^5 c^5 \int \frac {\cos ^{10}(e+f x) (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^7}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle a^5 c^5 \int \frac {\cos (e+f x)^{10} (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^7}dx\)

\(\Big \downarrow \) 3338

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3159

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3159

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3158

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3161

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3115

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

\(\Big \downarrow \) 24

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

Input:

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

Output:

a^5*c^5*(-1/3*((A - B)*Cos[e + f*x]^11)/(f*(a + a*Sin[e + f*x])^7) - ((4*A 
 - 7*B)*((-2*Cos[e + f*x]^9)/(a*f*(a + a*Sin[e + f*x])^5) - (9*((2*Cos[e + 
 f*x]^7)/(a*f*(a + a*Sin[e + f*x])^3) + (7*(Cos[e + f*x]^5/(4*f*(a^2 + a^2 
*Sin[e + f*x])) + (5*(Cos[e + f*x]^3/(3*a*f) + (x/2 + (Cos[e + f*x]*Sin[e 
+ f*x])/(2*f))/a))/(4*a)))/a^2))/a^2))/(3*a))
 

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 3158

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

rule 3159

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

rule 3161

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

rule 3338

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

Time = 88.20 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(\frac {2 c^{5} \left (\frac {\left (\frac {7 A}{2}-\frac {95 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (23 A -49 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (\frac {7 A}{2}-\frac {103 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (71 A -161 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-\frac {7 A}{2}+\frac {103 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (\frac {215 A}{3}-\frac {497 B}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {7 A}{2}+\frac {95 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {71 A}{3}-\frac {161 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4}}+\frac {105 \left (4 A -7 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}-\frac {-64 A +64 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-48 A +80 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {64 A -64 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{2}}\)	\(252\)
default	\(\frac {2 c^{5} \left (\frac {\left (\frac {7 A}{2}-\frac {95 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (23 A -49 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (\frac {7 A}{2}-\frac {103 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (71 A -161 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-\frac {7 A}{2}+\frac {103 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (\frac {215 A}{3}-\frac {497 B}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {7 A}{2}+\frac {95 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {71 A}{3}-\frac {161 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4}}+\frac {105 \left (4 A -7 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}-\frac {-64 A +64 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-48 A +80 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {64 A -64 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{2}}\)	\(252\)
parallelrisch	\(-\frac {75 \left (\frac {\left (-\frac {2543}{15} A +\frac {4514}{15} B -84 f x A +147 f x B \right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}+\frac {\left (-\frac {481}{45} A +\frac {712}{45} B +28 f x A -49 f x B \right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )}{5}+\frac {\left (147 f x B -84 f x A +\frac {2414}{15} B -\frac {1409}{15} A \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}+\frac {\left (-\frac {4433}{45} A +\frac {1528}{9} B -28 f x A +49 f x B \right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )}{5}+\left (-\frac {81 B}{40}+A \right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {\left (-2 A +\frac {139 B}{24}\right ) \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )}{25}+\frac {\left (\frac {47 B}{8}-A \right ) \cos \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )}{225}+\left (-A +\frac {81 B}{40}\right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {\left (-2 A +\frac {139 B}{24}\right ) \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )}{25}+\frac {\left (-\frac {47 B}{8}+A \right ) \sin \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )}{225}-\frac {B \left (\cos \left (\frac {11 f x}{2}+\frac {11 e}{2}\right )+\sin \left (\frac {11 f x}{2}+\frac {11 e}{2}\right )\right )}{600}\right ) c^{5}}{8 f \,a^{2} \left (\sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )\right )}\)	\(294\)
risch	\(\frac {105 c^{5} x A}{2 a^{2}}-\frac {735 c^{5} x B}{8 a^{2}}+\frac {7 i c^{5} {\mathrm e}^{2 i \left (f x +e \right )} A}{8 a^{2} f}-\frac {3 i c^{5} {\mathrm e}^{2 i \left (f x +e \right )} B}{a^{2} f}+\frac {95 c^{5} {\mathrm e}^{i \left (f x +e \right )} A}{8 a^{2} f}-\frac {217 c^{5} {\mathrm e}^{i \left (f x +e \right )} B}{8 a^{2} f}+\frac {95 c^{5} {\mathrm e}^{-i \left (f x +e \right )} A}{8 a^{2} f}-\frac {217 c^{5} {\mathrm e}^{-i \left (f x +e \right )} B}{8 a^{2} f}-\frac {7 i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )} A}{8 a^{2} f}+\frac {3 i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )} B}{a^{2} f}+\frac {256 i A \,c^{5} {\mathrm e}^{i \left (f x +e \right )}+160 A \,c^{5} {\mathrm e}^{2 i \left (f x +e \right )}-384 i B \,c^{5} {\mathrm e}^{i \left (f x +e \right )}-224 B \,c^{5} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {416 A \,c^{5}}{3}+\frac {608 B \,c^{5}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}-\frac {B \,c^{5} \sin \left (4 f x +4 e \right )}{32 a^{2} f}-\frac {c^{5} \cos \left (3 f x +3 e \right ) A}{12 a^{2} f}+\frac {7 c^{5} \cos \left (3 f x +3 e \right ) B}{12 a^{2} f}\)	\(354\)
norman	\(\text {Expression too large to display}\)	\(918\)

Input:

int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x,method=_RETUR 
NVERBOSE)
 

Output:

2/f*c^5/a^2*(((7/2*A-95/8*B)*tan(1/2*f*x+1/2*e)^7+(23*A-49*B)*tan(1/2*f*x+ 
1/2*e)^6+(7/2*A-103/8*B)*tan(1/2*f*x+1/2*e)^5+(71*A-161*B)*tan(1/2*f*x+1/2 
*e)^4+(-7/2*A+103/8*B)*tan(1/2*f*x+1/2*e)^3+(215/3*A-497/3*B)*tan(1/2*f*x+ 
1/2*e)^2+(-7/2*A+95/8*B)*tan(1/2*f*x+1/2*e)+71/3*A-161/3*B)/(1+tan(1/2*f*x 
+1/2*e)^2)^4+105/8*(4*A-7*B)*arctan(tan(1/2*f*x+1/2*e))-1/2*(-64*A+64*B)/( 
tan(1/2*f*x+1/2*e)+1)^2-(-48*A+80*B)/(tan(1/2*f*x+1/2*e)+1)-1/3*(64*A-64*B 
)/(tan(1/2*f*x+1/2*e)+1)^3)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=-\frac {6 \, B c^{5} \cos \left (f x + e\right )^{6} + 4 \, {\left (2 \, A - 11 \, B\right )} c^{5} \cos \left (f x + e\right )^{5} + {\left (76 \, A - 241 \, B\right )} c^{5} \cos \left (f x + e\right )^{4} - 2 \, {\left (212 \, A - 431 \, B\right )} c^{5} \cos \left (f x + e\right )^{3} + 630 \, {\left (4 \, A - 7 \, B\right )} c^{5} f x - 256 \, {\left (A - B\right )} c^{5} - {\left (315 \, {\left (4 \, A - 7 \, B\right )} c^{5} f x - {\left (2156 \, A - 3485 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (315 \, {\left (4 \, A - 7 \, B\right )} c^{5} f x + 2 \, {\left (1196 \, A - 2141 \, B\right )} c^{5}\right )} \cos \left (f x + e\right ) + {\left (6 \, B c^{5} \cos \left (f x + e\right )^{5} - 2 \, {\left (4 \, A - 25 \, B\right )} c^{5} \cos \left (f x + e\right )^{4} + {\left (68 \, A - 191 \, B\right )} c^{5} \cos \left (f x + e\right )^{3} + 630 \, {\left (4 \, A - 7 \, B\right )} c^{5} f x + 3 \, {\left (164 \, A - 351 \, B\right )} c^{5} \cos \left (f x + e\right )^{2} + 256 \, {\left (A - B\right )} c^{5} + {\left (315 \, {\left (4 \, A - 7 \, B\right )} c^{5} f x + 2 \, {\left (1324 \, A - 2269 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \] Input:

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

Output:

-1/24*(6*B*c^5*cos(f*x + e)^6 + 4*(2*A - 11*B)*c^5*cos(f*x + e)^5 + (76*A 
- 241*B)*c^5*cos(f*x + e)^4 - 2*(212*A - 431*B)*c^5*cos(f*x + e)^3 + 630*( 
4*A - 7*B)*c^5*f*x - 256*(A - B)*c^5 - (315*(4*A - 7*B)*c^5*f*x - (2156*A 
- 3485*B)*c^5)*cos(f*x + e)^2 + (315*(4*A - 7*B)*c^5*f*x + 2*(1196*A - 214 
1*B)*c^5)*cos(f*x + e) + (6*B*c^5*cos(f*x + e)^5 - 2*(4*A - 25*B)*c^5*cos( 
f*x + e)^4 + (68*A - 191*B)*c^5*cos(f*x + e)^3 + 630*(4*A - 7*B)*c^5*f*x + 
 3*(164*A - 351*B)*c^5*cos(f*x + e)^2 + 256*(A - B)*c^5 + (315*(4*A - 7*B) 
*c^5*f*x + 2*(1324*A - 2269*B)*c^5)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos 
(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f 
)*sin(f*x + e))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10608 vs. \(2 (224) = 448\).

Time = 23.13 (sec) , antiderivative size = 10608, normalized size of antiderivative = 44.20 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((1260*A*c**5*f*x*tan(e/2 + f*x/2)**11/(24*a**2*f*tan(e/2 + f*x/2 
)**11 + 72*a**2*f*tan(e/2 + f*x/2)**10 + 168*a**2*f*tan(e/2 + f*x/2)**9 + 
312*a**2*f*tan(e/2 + f*x/2)**8 + 432*a**2*f*tan(e/2 + f*x/2)**7 + 528*a**2 
*f*tan(e/2 + f*x/2)**6 + 528*a**2*f*tan(e/2 + f*x/2)**5 + 432*a**2*f*tan(e 
/2 + f*x/2)**4 + 312*a**2*f*tan(e/2 + f*x/2)**3 + 168*a**2*f*tan(e/2 + f*x 
/2)**2 + 72*a**2*f*tan(e/2 + f*x/2) + 24*a**2*f) + 3780*A*c**5*f*x*tan(e/2 
 + f*x/2)**10/(24*a**2*f*tan(e/2 + f*x/2)**11 + 72*a**2*f*tan(e/2 + f*x/2) 
**10 + 168*a**2*f*tan(e/2 + f*x/2)**9 + 312*a**2*f*tan(e/2 + f*x/2)**8 + 4 
32*a**2*f*tan(e/2 + f*x/2)**7 + 528*a**2*f*tan(e/2 + f*x/2)**6 + 528*a**2* 
f*tan(e/2 + f*x/2)**5 + 432*a**2*f*tan(e/2 + f*x/2)**4 + 312*a**2*f*tan(e/ 
2 + f*x/2)**3 + 168*a**2*f*tan(e/2 + f*x/2)**2 + 72*a**2*f*tan(e/2 + f*x/2 
) + 24*a**2*f) + 8820*A*c**5*f*x*tan(e/2 + f*x/2)**9/(24*a**2*f*tan(e/2 + 
f*x/2)**11 + 72*a**2*f*tan(e/2 + f*x/2)**10 + 168*a**2*f*tan(e/2 + f*x/2)* 
*9 + 312*a**2*f*tan(e/2 + f*x/2)**8 + 432*a**2*f*tan(e/2 + f*x/2)**7 + 528 
*a**2*f*tan(e/2 + f*x/2)**6 + 528*a**2*f*tan(e/2 + f*x/2)**5 + 432*a**2*f* 
tan(e/2 + f*x/2)**4 + 312*a**2*f*tan(e/2 + f*x/2)**3 + 168*a**2*f*tan(e/2 
+ f*x/2)**2 + 72*a**2*f*tan(e/2 + f*x/2) + 24*a**2*f) + 16380*A*c**5*f*x*t 
an(e/2 + f*x/2)**8/(24*a**2*f*tan(e/2 + f*x/2)**11 + 72*a**2*f*tan(e/2 + f 
*x/2)**10 + 168*a**2*f*tan(e/2 + f*x/2)**9 + 312*a**2*f*tan(e/2 + f*x/2)** 
8 + 432*a**2*f*tan(e/2 + f*x/2)**7 + 528*a**2*f*tan(e/2 + f*x/2)**6 + 5...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2982 vs. \(2 (228) = 456\).

Time = 0.18 (sec) , antiderivative size = 2982, normalized size of antiderivative = 12.42 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/12*(B*c^5*((603*sin(f*x + e)/(cos(f*x + e) + 1) + 1297*sin(f*x + e)^2/( 
cos(f*x + e) + 1)^2 + 2228*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 2628*sin( 
f*x + e)^4/(cos(f*x + e) + 1)^4 + 3014*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 
 + 2618*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1980*sin(f*x + e)^7/(cos(f*x 
 + e) + 1)^7 + 1100*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 495*sin(f*x + e) 
^9/(cos(f*x + e) + 1)^9 + 165*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 256) 
/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 7*a^2*sin(f*x + e)^2/(cos( 
f*x + e) + 1)^2 + 13*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 18*a^2*sin( 
f*x + e)^4/(cos(f*x + e) + 1)^4 + 22*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1) 
^5 + 22*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 18*a^2*sin(f*x + e)^7/(c 
os(f*x + e) + 1)^7 + 13*a^2*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 7*a^2*si 
n(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3*a^2*sin(f*x + e)^10/(cos(f*x + e) + 
1)^10 + a^2*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 165*arctan(sin(f*x + 
e)/(cos(f*x + e) + 1))/a^2) - 20*A*c^5*((75*sin(f*x + e)/(cos(f*x + e) + 1 
) + 97*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 126*sin(f*x + e)^3/(cos(f*x + 
 e) + 1)^3 + 98*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5/(c 
os(f*x + e) + 1)^5 + 21*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 32)/(a^2 + 3 
*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 5*a^2*sin(f*x + e)^2/(cos(f*x + e) 
+ 1)^2 + 7*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 7*a^2*sin(f*x + e)^4/ 
(cos(f*x + e) + 1)^4 + 5*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^...
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {315 \, {\left (4 \, A c^{5} - 7 \, B c^{5}\right )} {\left (f x + e\right )}}{a^{2}} + \frac {256 \, {\left (9 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11 \, A c^{5} - 17 \, B c^{5}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {2 \, {\left (84 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 285 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 552 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1176 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 84 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 309 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1704 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3864 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 84 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 309 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1720 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3976 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 84 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 285 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 568 \, A c^{5} - 1288 \, B c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{4} a^{2}}}{24 \, f} \] Input:

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

Output:

1/24*(315*(4*A*c^5 - 7*B*c^5)*(f*x + e)/a^2 + 256*(9*A*c^5*tan(1/2*f*x + 1 
/2*e)^2 - 15*B*c^5*tan(1/2*f*x + 1/2*e)^2 + 24*A*c^5*tan(1/2*f*x + 1/2*e) 
- 36*B*c^5*tan(1/2*f*x + 1/2*e) + 11*A*c^5 - 17*B*c^5)/(a^2*(tan(1/2*f*x + 
 1/2*e) + 1)^3) + 2*(84*A*c^5*tan(1/2*f*x + 1/2*e)^7 - 285*B*c^5*tan(1/2*f 
*x + 1/2*e)^7 + 552*A*c^5*tan(1/2*f*x + 1/2*e)^6 - 1176*B*c^5*tan(1/2*f*x 
+ 1/2*e)^6 + 84*A*c^5*tan(1/2*f*x + 1/2*e)^5 - 309*B*c^5*tan(1/2*f*x + 1/2 
*e)^5 + 1704*A*c^5*tan(1/2*f*x + 1/2*e)^4 - 3864*B*c^5*tan(1/2*f*x + 1/2*e 
)^4 - 84*A*c^5*tan(1/2*f*x + 1/2*e)^3 + 309*B*c^5*tan(1/2*f*x + 1/2*e)^3 + 
 1720*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 3976*B*c^5*tan(1/2*f*x + 1/2*e)^2 - 8 
4*A*c^5*tan(1/2*f*x + 1/2*e) + 285*B*c^5*tan(1/2*f*x + 1/2*e) + 568*A*c^5 
- 1288*B*c^5)/((tan(1/2*f*x + 1/2*e)^2 + 1)^4*a^2))/f
 

Mupad [B] (verification not implemented)

Time = 39.91 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.08 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(tan(e/2 + (f*x)/2)*(391*A*c^5 - (2729*B*c^5)/4) + (494*A*c^5)/3 - (866*B* 
c^5)/3 + tan(e/2 + (f*x)/2)^10*(103*A*c^5 - (735*B*c^5)/4) + tan(e/2 + (f* 
x)/2)^9*(323*A*c^5 - (2213*B*c^5)/4) + tan(e/2 + (f*x)/2)^7*(1332*A*c^5 - 
2253*B*c^5) + tan(e/2 + (f*x)/2)^4*(1632*A*c^5 - 2943*B*c^5) + tan(e/2 + ( 
f*x)/2)^8*((2002*A*c^5)/3 - (3637*B*c^5)/3) + tan(e/2 + (f*x)/2)^3*((4420* 
A*c^5)/3 - (7621*B*c^5)/3) + tan(e/2 + (f*x)/2)^2*((2489*A*c^5)/3 - (17609 
*B*c^5)/12) + tan(e/2 + (f*x)/2)^6*((4594*A*c^5)/3 - (16805*B*c^5)/6) + ta 
n(e/2 + (f*x)/2)^5*((6274*A*c^5)/3 - (21299*B*c^5)/6))/(f*(7*a^2*tan(e/2 + 
 (f*x)/2)^2 + 13*a^2*tan(e/2 + (f*x)/2)^3 + 18*a^2*tan(e/2 + (f*x)/2)^4 + 
22*a^2*tan(e/2 + (f*x)/2)^5 + 22*a^2*tan(e/2 + (f*x)/2)^6 + 18*a^2*tan(e/2 
 + (f*x)/2)^7 + 13*a^2*tan(e/2 + (f*x)/2)^8 + 7*a^2*tan(e/2 + (f*x)/2)^9 + 
 3*a^2*tan(e/2 + (f*x)/2)^10 + a^2*tan(e/2 + (f*x)/2)^11 + a^2 + 3*a^2*tan 
(e/2 + (f*x)/2))) + (105*c^5*atan((105*c^5*tan(e/2 + (f*x)/2)*(4*A - 7*B)) 
/(420*A*c^5 - 735*B*c^5))*(4*A - 7*B))/(4*a^2*f)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {c^{5} \left (824 a -1470 b +1260 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a f x -2205 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b f x +1470 \cos \left (f x +e \right ) b -6 \sin \left (f x +e \right )^{6} b -8 \sin \left (f x +e \right )^{5} a +2729 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b -6 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b -8 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a -1260 a f x +2205 b f x +1260 \cos \left (f x +e \right ) a f x -2205 \cos \left (f x +e \right ) b f x -2520 \sin \left (f x +e \right ) a f x +4410 \sin \left (f x +e \right ) b f x +774 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b -1564 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a -1564 a \sin \left (f x +e \right )-179 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -408 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +44 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +68 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -223 \sin \left (f x +e \right )^{4} b -476 \sin \left (f x +e \right )^{3} a +50 \sin \left (f x +e \right )^{5} b +76 \sin \left (f x +e \right )^{4} a -824 \cos \left (f x +e \right ) a +953 \sin \left (f x +e \right )^{3} b -3460 \sin \left (f x +e \right )^{2} a +5943 \sin \left (f x +e \right )^{2} b -1260 \sin \left (f x +e \right )^{2} a f x +2205 \sin \left (f x +e \right )^{2} b f x +2729 \sin \left (f x +e \right ) b \right )}{24 a^{2} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-\sin \left (f x +e \right )^{2}-2 \sin \left (f x +e \right )-1\right )} \] Input:

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

Output:

(c**5*( - 6*cos(e + f*x)*sin(e + f*x)**5*b - 8*cos(e + f*x)*sin(e + f*x)** 
4*a + 44*cos(e + f*x)*sin(e + f*x)**4*b + 68*cos(e + f*x)*sin(e + f*x)**3* 
a - 179*cos(e + f*x)*sin(e + f*x)**3*b - 408*cos(e + f*x)*sin(e + f*x)**2* 
a + 774*cos(e + f*x)*sin(e + f*x)**2*b + 1260*cos(e + f*x)*sin(e + f*x)*a* 
f*x - 1564*cos(e + f*x)*sin(e + f*x)*a - 2205*cos(e + f*x)*sin(e + f*x)*b* 
f*x + 2729*cos(e + f*x)*sin(e + f*x)*b + 1260*cos(e + f*x)*a*f*x - 824*cos 
(e + f*x)*a - 2205*cos(e + f*x)*b*f*x + 1470*cos(e + f*x)*b - 6*sin(e + f* 
x)**6*b - 8*sin(e + f*x)**5*a + 50*sin(e + f*x)**5*b + 76*sin(e + f*x)**4* 
a - 223*sin(e + f*x)**4*b - 476*sin(e + f*x)**3*a + 953*sin(e + f*x)**3*b 
- 1260*sin(e + f*x)**2*a*f*x - 3460*sin(e + f*x)**2*a + 2205*sin(e + f*x)* 
*2*b*f*x + 5943*sin(e + f*x)**2*b - 2520*sin(e + f*x)*a*f*x - 1564*sin(e + 
 f*x)*a + 4410*sin(e + f*x)*b*f*x + 2729*sin(e + f*x)*b - 1260*a*f*x + 824 
*a + 2205*b*f*x - 1470*b))/(24*a**2*f*(cos(e + f*x)*sin(e + f*x) + cos(e + 
 f*x) - sin(e + f*x)**2 - 2*sin(e + f*x) - 1))