Integrand size = 36, antiderivative size = 243 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {21 (3 A-8 B) c^5 x}{2 a^3}-\frac {7 (3 A-8 B) c^5 \cos ^3(e+f x)}{a^3 f}-\frac {21 (3 A-8 B) c^5 \cos (e+f x) \sin (e+f x)}{2 a^3 f}-\frac {a^5 (A-B) c^5 \cos ^{11}(e+f x)}{5 f (a+a \sin (e+f x))^8}+\frac {2 a^3 (3 A-8 B) c^5 \cos ^9(e+f x)}{15 f (a+a \sin (e+f x))^6}-\frac {6 a^5 (3 A-8 B) c^5 \cos ^7(e+f x)}{5 f \left (a^2+a^2 \sin (e+f x)\right )^4}-\frac {42 a^5 (3 A-8 B) c^5 \cos ^5(e+f x)}{5 f \left (a^4+a^4 \sin (e+f x)\right )^2} \] Output:
-21/2*(3*A-8*B)*c^5*x/a^3-7*(3*A-8*B)*c^5*cos(f*x+e)^3/a^3/f-21/2*(3*A-8*B )*c^5*cos(f*x+e)*sin(f*x+e)/a^3/f-1/5*a^5*(A-B)*c^5*cos(f*x+e)^11/f/(a+a*s in(f*x+e))^8+2/15*a^3*(3*A-8*B)*c^5*cos(f*x+e)^9/f/(a+a*sin(f*x+e))^6-6/5* a^5*(3*A-8*B)*c^5*cos(f*x+e)^7/f/(a^2+a^2*sin(f*x+e))^4-42/5*a^5*(3*A-8*B) *c^5*cos(f*x+e)^5/f/(a^4+a^4*sin(f*x+e))^2
Time = 13.09 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^5 \left (768 (A-B) \sin \left (\frac {1}{2} (e+f x)\right )-384 (A-B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-128 (21 A-31 B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+64 (21 A-31 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+128 (54 A-119 B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-630 (3 A-8 B) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-15 (32 A-127 B) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-5 B \cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+15 (A-8 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sin (2 (e+f x))\right )}{60 a^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (1+\sin (e+f x))^3} \] Input:
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x ])^3,x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(768*(A - B) *Sin[(e + f*x)/2] - 384*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 12 8*(21*A - 31*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 64*(21*A - 31*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 128*(54*A - 11 9*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 630*(3*A - 8*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 15*(32*A - 127*B )*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 - 5*B*Cos[3*(e + f* x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5 + 15*(A - 8*B)*(Cos[(e + f*x)/ 2] + Sin[(e + f*x)/2])^5*Sin[2*(e + f*x)]))/(60*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(1 + Sin[e + f*x])^3)
Time = 1.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.85, 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, 3159, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^5 (A+B \sin (e+f x))}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^5 (A+B \sin (e+f x))}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^5 c^5 \int \frac {\cos ^{10}(e+f x) (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^8}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \int \frac {\cos (e+f x)^{10} (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^8}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \int \frac {\cos ^{10}(e+f x)}{(\sin (e+f x) a+a)^7}dx}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \int \frac {\cos (e+f x)^{10}}{(\sin (e+f x) a+a)^7}dx}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \int \frac {\cos ^8(e+f x)}{(\sin (e+f x) a+a)^5}dx}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \int \frac {\cos (e+f x)^8}{(\sin (e+f x) a+a)^5}dx}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \int \frac {\cos ^6(e+f x)}{(\sin (e+f x) a+a)^3}dx}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \int \frac {\cos (e+f x)^6}{(\sin (e+f x) a+a)^3}dx}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \int \frac {\cos ^4(e+f x)}{\sin (e+f x) a+a}dx}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \int \frac {\cos (e+f x)^4}{\sin (e+f x) a+a}dx}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\int \cos ^2(e+f x)dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^5 c^5 \left (-\frac {(3 A-8 B) \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\cos ^3(e+f x)}{3 a f}+\frac {\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}}{a}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )}{5 a}-\frac {(A-B) \cos ^{11}(e+f x)}{5 f (a \sin (e+f x)+a)^8}\right )\) |
Input:
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5)/(a + a*Sin[e + f*x])^3,x ]
Output:
a^5*c^5*(-1/5*((A - B)*Cos[e + f*x]^11)/(f*(a + a*Sin[e + f*x])^8) - ((3*A - 8*B)*((-2*Cos[e + f*x]^9)/(3*a*f*(a + a*Sin[e + f*x])^6) - (3*((-2*Cos[ e + f*x]^7)/(a*f*(a + a*Sin[e + f*x])^4) - (7*((2*Cos[e + f*x]^5)/(a*f*(a + a*Sin[e + f*x])^2) + (5*(Cos[e + f*x]^3/(3*a*f) + (x/2 + (Cos[e + f*x]*S in[e + f*x])/(2*f))/a))/a^2))/a^2))/a^2))/(5*a))
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]
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]
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]
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]
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]))
Time = 87.96 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {2 c^{5} \left (-\frac {-256 A +256 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32 A -96 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32 A -80 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {96 A -32 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {128 A -128 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {\left (\frac {A}{2}-4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (8 A -31 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (16 A -64 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {A}{2}+4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8 A -\frac {95 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}-\frac {21 \left (3 A -8 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{3}}\) | \(240\) |
| default | \(\frac {2 c^{5} \left (-\frac {-256 A +256 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32 A -96 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32 A -80 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {96 A -32 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {128 A -128 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {\left (\frac {A}{2}-4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (8 A -31 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (16 A -64 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {A}{2}+4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+8 A -\frac {95 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}-\frac {21 \left (3 A -8 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{f \,a^{3}}\) | \(240\) |
| parallelrisch | \(-\frac {63 \left (\left (-\frac {1223}{63} A +\frac {19297}{378} B +\frac {80}{3} f x B -10 f x A \right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (-\frac {2543}{378} B +\frac {341}{126} A +5 f x A -\frac {40}{3} f x B \right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (-\frac {39439}{3780} B +\frac {2519}{630} A -\frac {8}{3} f x B +f x A \right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\frac {12367}{378} B -\frac {761}{63} A +\frac {80}{3} f x B -10 f x A \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\frac {13289}{378} B -\frac {1643}{126} A -5 f x A +\frac {40}{3} f x B \right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (-\frac {107}{126} A +\frac {1555}{756} B +f x A -\frac {8}{3} f x B \right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\frac {3 A}{28}-\frac {271 B}{756}\right ) \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+\left (-\frac {19 B}{756}+\frac {A}{252}\right ) \cos \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )+\left (\frac {3 A}{28}-\frac {271 B}{756}\right ) \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+\left (\frac {19 B}{756}-\frac {A}{252}\right ) \sin \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )+\frac {B \left (\cos \left (\frac {11 f x}{2}+\frac {11 e}{2}\right )+\sin \left (\frac {11 f x}{2}+\frac {11 e}{2}\right )\right )}{756}\right ) c^{5}}{2 f \,a^{3} \left (-5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\) | \(328\) |
| risch | \(-\frac {63 c^{5} x A}{2 a^{3}}+\frac {84 c^{5} x B}{a^{3}}-\frac {B \,c^{5} {\mathrm e}^{3 i \left (f x +e \right )}}{24 a^{3} f}-\frac {i c^{5} {\mathrm e}^{2 i \left (f x +e \right )} A}{8 a^{3} f}+\frac {i c^{5} {\mathrm e}^{2 i \left (f x +e \right )} B}{a^{3} f}-\frac {4 c^{5} {\mathrm e}^{i \left (f x +e \right )} A}{a^{3} f}+\frac {127 c^{5} {\mathrm e}^{i \left (f x +e \right )} B}{8 a^{3} f}-\frac {4 c^{5} {\mathrm e}^{-i \left (f x +e \right )} A}{a^{3} f}+\frac {127 c^{5} {\mathrm e}^{-i \left (f x +e \right )} B}{8 a^{3} f}+\frac {i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )} A}{8 a^{3} f}-\frac {i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )} B}{a^{3} f}-\frac {B \,c^{5} {\mathrm e}^{-3 i \left (f x +e \right )}}{24 a^{3} f}-\frac {32 \left (-315 A \,c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+225 i A \,c^{5} {\mathrm e}^{3 i \left (f x +e \right )}-195 i A \,c^{5} {\mathrm e}^{i \left (f x +e \right )}+75 A \,c^{5} {\mathrm e}^{4 i \left (f x +e \right )}+695 B \,c^{5} {\mathrm e}^{2 i \left (f x +e \right )}-495 i B \,c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+445 i B \,c^{5} {\mathrm e}^{i \left (f x +e \right )}-150 B \,c^{5} {\mathrm e}^{4 i \left (f x +e \right )}+54 A \,c^{5}-119 B \,c^{5}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(395\) |
| norman | \(\text {Expression too large to display}\) | \(1038\) |
Input:
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x,method=_RETUR NVERBOSE)
Output:
2/f*c^5/a^3*(-1/4*(-256*A+256*B)/(tan(1/2*f*x+1/2*e)+1)^4-1/2*(32*A-96*B)/ (tan(1/2*f*x+1/2*e)+1)^2-(32*A-80*B)/(tan(1/2*f*x+1/2*e)+1)-1/3*(96*A-32*B )/(tan(1/2*f*x+1/2*e)+1)^3-1/5*(128*A-128*B)/(tan(1/2*f*x+1/2*e)+1)^5-((1/ 2*A-4*B)*tan(1/2*f*x+1/2*e)^5+(8*A-31*B)*tan(1/2*f*x+1/2*e)^4+(16*A-64*B)* tan(1/2*f*x+1/2*e)^2+(-1/2*A+4*B)*tan(1/2*f*x+1/2*e)+8*A-95/3*B)/(1+tan(1/ 2*f*x+1/2*e)^2)^3-21/2*(3*A-8*B)*arctan(tan(1/2*f*x+1/2*e)))
Time = 0.10 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.77 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {10 \, B c^{5} \cos \left (f x + e\right )^{6} + 15 \, {\left (A - 6 \, B\right )} c^{5} \cos \left (f x + e\right )^{5} + 10 \, {\left (21 \, A - 74 \, B\right )} c^{5} \cos \left (f x + e\right )^{4} - 1260 \, {\left (3 \, A - 8 \, B\right )} c^{5} f x - 192 \, {\left (A - B\right )} c^{5} + {\left (315 \, {\left (3 \, A - 8 \, B\right )} c^{5} f x + {\left (2373 \, A - 6128 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (945 \, {\left (3 \, A - 8 \, B\right )} c^{5} f x - 2 \, {\left (753 \, A - 2248 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (105 \, {\left (3 \, A - 8 \, B\right )} c^{5} f x + 2 \, {\left (323 \, A - 848 \, B\right )} c^{5}\right )} \cos \left (f x + e\right ) + {\left (10 \, B c^{5} \cos \left (f x + e\right )^{5} - 5 \, {\left (3 \, A - 20 \, B\right )} c^{5} \cos \left (f x + e\right )^{4} + 5 \, {\left (39 \, A - 128 \, B\right )} c^{5} \cos \left (f x + e\right )^{3} - 1260 \, {\left (3 \, A - 8 \, B\right )} c^{5} f x + 192 \, {\left (A - B\right )} c^{5} + {\left (315 \, {\left (3 \, A - 8 \, B\right )} c^{5} f x - 2 \, {\left (1089 \, A - 2744 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )^{2} - 6 \, {\left (105 \, {\left (3 \, A - 8 \, B\right )} c^{5} f x + 2 \, {\left (307 \, A - 832 \, B\right )} c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{30 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algori thm="fricas")
Output:
-1/30*(10*B*c^5*cos(f*x + e)^6 + 15*(A - 6*B)*c^5*cos(f*x + e)^5 + 10*(21* A - 74*B)*c^5*cos(f*x + e)^4 - 1260*(3*A - 8*B)*c^5*f*x - 192*(A - B)*c^5 + (315*(3*A - 8*B)*c^5*f*x + (2373*A - 6128*B)*c^5)*cos(f*x + e)^3 + (945* (3*A - 8*B)*c^5*f*x - 2*(753*A - 2248*B)*c^5)*cos(f*x + e)^2 - 6*(105*(3*A - 8*B)*c^5*f*x + 2*(323*A - 848*B)*c^5)*cos(f*x + e) + (10*B*c^5*cos(f*x + e)^5 - 5*(3*A - 20*B)*c^5*cos(f*x + e)^4 + 5*(39*A - 128*B)*c^5*cos(f*x + e)^3 - 1260*(3*A - 8*B)*c^5*f*x + 192*(A - B)*c^5 + (315*(3*A - 8*B)*c^5 *f*x - 2*(1089*A - 2744*B)*c^5)*cos(f*x + e)^2 - 6*(105*(3*A - 8*B)*c^5*f* x + 2*(307*A - 832*B)*c^5)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e) ^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos( f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 10608 vs. \(2 (228) = 456\).
Time = 39.74 (sec) , antiderivative size = 10608, normalized size of antiderivative = 43.65 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))**5/(a+a*sin(f*x+e))**3,x)
Output:
Piecewise((-945*A*c**5*f*x*tan(e/2 + f*x/2)**11/(30*a**3*f*tan(e/2 + f*x/2 )**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a **3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f* tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 4725*A*c**5*f*x*t an(e/2 + f*x/2)**10/(30*a**3*f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2) **8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2)**5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a* *3*f*tan(e/2 + f*x/2)**3 + 390*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan (e/2 + f*x/2) + 30*a**3*f) - 12285*A*c**5*f*x*tan(e/2 + f*x/2)**9/(30*a**3 *f*tan(e/2 + f*x/2)**11 + 150*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan (e/2 + f*x/2)**9 + 750*a**3*f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*tan(e/2 + f*x/2)**6 + 1380*a**3*f*tan(e/2 + f*x/2) **5 + 1140*a**3*f*tan(e/2 + f*x/2)**4 + 750*a**3*f*tan(e/2 + f*x/2)**3 + 3 90*a**3*f*tan(e/2 + f*x/2)**2 + 150*a**3*f*tan(e/2 + f*x/2) + 30*a**3*f) - 23625*A*c**5*f*x*tan(e/2 + f*x/2)**8/(30*a**3*f*tan(e/2 + f*x/2)**11 + 15 0*a**3*f*tan(e/2 + f*x/2)**10 + 390*a**3*f*tan(e/2 + f*x/2)**9 + 750*a**3* f*tan(e/2 + f*x/2)**8 + 1140*a**3*f*tan(e/2 + f*x/2)**7 + 1380*a**3*f*t...
Leaf count of result is larger than twice the leaf count of optimal. 3282 vs. \(2 (231) = 462\).
Time = 0.21 (sec) , antiderivative size = 3282, normalized size of antiderivative = 13.51 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algori thm="maxima")
Output:
1/15*(B*c^5*((2375*sin(f*x + e)/(cos(f*x + e) + 1) + 5347*sin(f*x + e)^2/( cos(f*x + e) + 1)^2 + 9230*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 12622*sin (f*x + e)^4/(cos(f*x + e) + 1)^4 + 13340*sin(f*x + e)^5/(cos(f*x + e) + 1) ^5 + 11684*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 8050*sin(f*x + e)^7/(cos( f*x + e) + 1)^7 + 4370*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1725*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 345*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 544)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 13*a^3*sin(f*x + e)^2/ (cos(f*x + e) + 1)^2 + 25*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 38*a^3 *sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 46*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 46*a^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 38*a^3*sin(f*x + e) ^7/(cos(f*x + e) + 1)^7 + 25*a^3*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 13* a^3*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 5*a^3*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + a^3*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 345*arctan(sin( f*x + e)/(cos(f*x + e) + 1))/a^3) - A*c^5*((1325*sin(f*x + e)/(cos(f*x + e ) + 1) + 2673*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3805*sin(f*x + e)^3/(c os(f*x + e) + 1)^3 + 4329*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 3575*sin(f *x + e)^5/(cos(f*x + e) + 1)^5 + 2275*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 975*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 195*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 304)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^3*sin (f*x + e)^2/(cos(f*x + e) + 1)^2 + 20*a^3*sin(f*x + e)^3/(cos(f*x + e) ...
Time = 0.29 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=-\frac {\frac {315 \, {\left (3 \, A c^{5} - 8 \, B c^{5}\right )} {\left (f x + e\right )}}{a^{3}} + \frac {10 \, {\left (3 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 48 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 186 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 96 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 384 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 48 \, A c^{5} - 190 \, B c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} a^{3}} + \frac {64 \, {\left (30 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 75 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 135 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 345 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 255 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 595 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 165 \, A c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 395 \, B c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 39 \, A c^{5} - 94 \, B c^{5}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{30 \, f} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algori thm="giac")
Output:
-1/30*(315*(3*A*c^5 - 8*B*c^5)*(f*x + e)/a^3 + 10*(3*A*c^5*tan(1/2*f*x + 1 /2*e)^5 - 24*B*c^5*tan(1/2*f*x + 1/2*e)^5 + 48*A*c^5*tan(1/2*f*x + 1/2*e)^ 4 - 186*B*c^5*tan(1/2*f*x + 1/2*e)^4 + 96*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 3 84*B*c^5*tan(1/2*f*x + 1/2*e)^2 - 3*A*c^5*tan(1/2*f*x + 1/2*e) + 24*B*c^5* tan(1/2*f*x + 1/2*e) + 48*A*c^5 - 190*B*c^5)/((tan(1/2*f*x + 1/2*e)^2 + 1) ^3*a^3) + 64*(30*A*c^5*tan(1/2*f*x + 1/2*e)^4 - 75*B*c^5*tan(1/2*f*x + 1/2 *e)^4 + 135*A*c^5*tan(1/2*f*x + 1/2*e)^3 - 345*B*c^5*tan(1/2*f*x + 1/2*e)^ 3 + 255*A*c^5*tan(1/2*f*x + 1/2*e)^2 - 595*B*c^5*tan(1/2*f*x + 1/2*e)^2 + 165*A*c^5*tan(1/2*f*x + 1/2*e) - 395*B*c^5*tan(1/2*f*x + 1/2*e) + 39*A*c^5 - 94*B*c^5)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f
Time = 39.66 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.06 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:
int(((A + B*sin(e + f*x))*(c - c*sin(e + f*x))^5)/(a + a*sin(e + f*x))^3,x )
Output:
- (tan(e/2 + (f*x)/2)*(431*A*c^5 - (3454*B*c^5)/3) + (496*A*c^5)/5 - (3958 *B*c^5)/15 + tan(e/2 + (f*x)/2)^10*(65*A*c^5 - 168*B*c^5) + tan(e/2 + (f*x )/2)^9*(309*A*c^5 - 838*B*c^5) + tan(e/2 + (f*x)/2)^8*(826*A*c^5 - (6418*B *c^5)/3) + tan(e/2 + (f*x)/2)^7*(1418*A*c^5 - (11636*B*c^5)/3) + tan(e/2 + (f*x)/2)^3*(1654*A*c^5 - (13372*B*c^5)/3) + tan(e/2 + (f*x)/2)^5*(2332*A* c^5 - (19072*B*c^5)/3) + tan(e/2 + (f*x)/2)^2*((4903*A*c^5)/5 - (38884*B*c ^5)/15) + tan(e/2 + (f*x)/2)^6*((11156*A*c^5)/5 - (86708*B*c^5)/15) + tan( e/2 + (f*x)/2)^4*((11758*A*c^5)/5 - (92224*B*c^5)/15))/(f*(13*a^3*tan(e/2 + (f*x)/2)^2 + 25*a^3*tan(e/2 + (f*x)/2)^3 + 38*a^3*tan(e/2 + (f*x)/2)^4 + 46*a^3*tan(e/2 + (f*x)/2)^5 + 46*a^3*tan(e/2 + (f*x)/2)^6 + 38*a^3*tan(e/ 2 + (f*x)/2)^7 + 25*a^3*tan(e/2 + (f*x)/2)^8 + 13*a^3*tan(e/2 + (f*x)/2)^9 + 5*a^3*tan(e/2 + (f*x)/2)^10 + a^3*tan(e/2 + (f*x)/2)^11 + a^3 + 5*a^3*t an(e/2 + (f*x)/2))) - (21*c^5*atan((21*c^5*tan(e/2 + (f*x)/2)*(3*A - 8*B)) /(63*A*c^5 - 168*B*c^5))*(3*A - 8*B))/(a^3*f)
Time = 0.19 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.25 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx=\frac {c^{5} \left (-390 a +1008 b -1890 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a f x +5040 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b f x -1008 \cos \left (f x +e \right ) b -10 \sin \left (f x +e \right )^{6} b -15 \sin \left (f x +e \right )^{5} a -3454 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b -10 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b -15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a +945 a f x -2520 b f x -945 \cos \left (f x +e \right ) a f x +2520 \cos \left (f x +e \right ) b f x +2835 \sin \left (f x +e \right ) a f x -7560 \sin \left (f x +e \right ) b f x -3358 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +1293 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +1293 a \sin \left (f x +e \right )-620 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b +1305 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +90 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +195 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -710 \sin \left (f x +e \right )^{4} b +3306 \sin \left (f x +e \right )^{3} a +100 \sin \left (f x +e \right )^{5} b +210 \sin \left (f x +e \right )^{4} a +390 \cos \left (f x +e \right ) a -8638 \sin \left (f x +e \right )^{3} b +4380 \sin \left (f x +e \right )^{2} a -11896 \sin \left (f x +e \right )^{2} b -945 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a f x +2520 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b f x +945 \sin \left (f x +e \right )^{3} a f x -2520 \sin \left (f x +e \right )^{3} b f x +2835 \sin \left (f x +e \right )^{2} a f x -7560 \sin \left (f x +e \right )^{2} b f x -3454 \sin \left (f x +e \right ) b \right )}{30 a^{3} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+2 \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-\sin \left (f x +e \right )^{3}-3 \sin \left (f x +e \right )^{2}-3 \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))^3,x)
Output:
(c**5*( - 10*cos(e + f*x)*sin(e + f*x)**5*b - 15*cos(e + f*x)*sin(e + f*x) **4*a + 90*cos(e + f*x)*sin(e + f*x)**4*b + 195*cos(e + f*x)*sin(e + f*x)* *3*a - 620*cos(e + f*x)*sin(e + f*x)**3*b - 945*cos(e + f*x)*sin(e + f*x)* *2*a*f*x + 1305*cos(e + f*x)*sin(e + f*x)**2*a + 2520*cos(e + f*x)*sin(e + f*x)**2*b*f*x - 3358*cos(e + f*x)*sin(e + f*x)**2*b - 1890*cos(e + f*x)*s in(e + f*x)*a*f*x + 1293*cos(e + f*x)*sin(e + f*x)*a + 5040*cos(e + f*x)*s in(e + f*x)*b*f*x - 3454*cos(e + f*x)*sin(e + f*x)*b - 945*cos(e + f*x)*a* f*x + 390*cos(e + f*x)*a + 2520*cos(e + f*x)*b*f*x - 1008*cos(e + f*x)*b - 10*sin(e + f*x)**6*b - 15*sin(e + f*x)**5*a + 100*sin(e + f*x)**5*b + 210 *sin(e + f*x)**4*a - 710*sin(e + f*x)**4*b + 945*sin(e + f*x)**3*a*f*x + 3 306*sin(e + f*x)**3*a - 2520*sin(e + f*x)**3*b*f*x - 8638*sin(e + f*x)**3* b + 2835*sin(e + f*x)**2*a*f*x + 4380*sin(e + f*x)**2*a - 7560*sin(e + f*x )**2*b*f*x - 11896*sin(e + f*x)**2*b + 2835*sin(e + f*x)*a*f*x + 1293*sin( e + f*x)*a - 7560*sin(e + f*x)*b*f*x - 3454*sin(e + f*x)*b + 945*a*f*x - 3 90*a - 2520*b*f*x + 1008*b))/(30*a**3*f*(cos(e + f*x)*sin(e + f*x)**2 + 2* cos(e + f*x)*sin(e + f*x) + cos(e + f*x) - sin(e + f*x)**3 - 3*sin(e + f*x )**2 - 3*sin(e + f*x) - 1))