Integrand size = 36, antiderivative size = 180 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {35 (A-2 B) c^4 x}{2 a^2}+\frac {35 (A-2 B) c^4 \cos ^3(e+f x)}{3 a^2 f}+\frac {35 (A-2 B) c^4 \cos (e+f x) \sin (e+f x)}{2 a^2 f}-\frac {a^4 (A-B) c^4 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {2 a^2 (A-2 B) c^4 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac {14 (A-2 B) c^4 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2} \] Output:
35/2*(A-2*B)*c^4*x/a^2+35/3*(A-2*B)*c^4*cos(f*x+e)^3/a^2/f+35/2*(A-2*B)*c^ 4*cos(f*x+e)*sin(f*x+e)/a^2/f-1/3*a^4*(A-B)*c^4*cos(f*x+e)^9/f/(a+a*sin(f* x+e))^6+2*a^2*(A-2*B)*c^4*cos(f*x+e)^7/f/(a+a*sin(f*x+e))^4+14*(A-2*B)*c^4 *cos(f*x+e)^5/f/(a+a*sin(f*x+e))^2
Time = 11.87 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^4 \left (128 (A-B) \sin \left (\frac {1}{2} (e+f x)\right )-64 (A-B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-128 (5 A-8 B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+210 (A-2 B) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+3 (24 A-71 B) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+B \cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-3 (A-6 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (2 (e+f x))\right )}{12 a^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (1+\sin (e+f x))^2} \] Input:
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(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])^4*(128*(A - B) *Sin[(e + f*x)/2] - 64*(A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 128 *(5*A - 8*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 21 0*(A - 2*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 3*(24*A - 71*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + B*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 3*(A - 6*B)*(Cos[(e + f*x) /2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*a^2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(1 + Sin[e + f*x])^2)
Time = 0.94 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {3042, 3446, 3042, 3338, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^4 (A+B \sin (e+f x))}{(a \sin (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^4 (A+B \sin (e+f x))}{(a \sin (e+f x)+a)^2}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^4 c^4 \int \frac {\cos ^8(e+f x) (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^6}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \int \frac {\cos (e+f x)^8 (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^6}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \int \frac {\cos ^8(e+f x)}{(\sin (e+f x) a+a)^5}dx}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \int \frac {\cos (e+f x)^8}{(\sin (e+f x) a+a)^5}dx}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \int \frac {\cos ^6(e+f x)}{(\sin (e+f x) a+a)^3}dx}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \int \frac {\cos (e+f x)^6}{(\sin (e+f x) a+a)^3}dx}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \left (\frac {5 \int \frac {\cos ^4(e+f x)}{\sin (e+f x) a+a}dx}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \left (\frac {5 \int \frac {\cos (e+f x)^4}{\sin (e+f x) a+a}dx}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \left (\frac {5 \left (\frac {\int \cos ^2(e+f x)dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \left (\frac {5 \left (\frac {\int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \left (\frac {5 \left (\frac {\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^4 c^4 \left (-\frac {(A-2 B) \left (-\frac {7 \left (\frac {5 \left (\frac {\cos ^3(e+f x)}{3 a f}+\frac {\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}}{a}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}\right )\) |
Input:
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x])^2,x ]
Output:
a^4*c^4*(-1/3*((A - B)*Cos[e + f*x]^9)/(f*(a + a*Sin[e + f*x])^6) - ((A - 2*B)*((-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]*Sin[e + f*x])/(2*f))/a))/a^2))/a^2))/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 = 11.17 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(\frac {2 c^{4} \left (\frac {\left (\frac {A}{2}-3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (6 A -17 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (12 A -36 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {A}{2}+3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+6 A -\frac {53 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}+\frac {35 \left (A -2 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {-32 A +32 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-16 A +32 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {32 A -32 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{2}}\) | \(193\) |
| default | \(\frac {2 c^{4} \left (\frac {\left (\frac {A}{2}-3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (6 A -17 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (12 A -36 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {A}{2}+3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+6 A -\frac {53 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}+\frac {35 \left (A -2 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {-32 A +32 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-16 A +32 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {32 A -32 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{2}}\) | \(193\) |
| parallelrisch | \(-\frac {21 \left (\left (-\frac {277}{7} A +\frac {1712}{21} B -20 f x A +40 f x B \right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {\left (-\frac {191}{21} A +\frac {290}{21} B +20 f x A -40 f x B \right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )}{3}+\left (40 f x B -20 f x A +\frac {928}{21} B -\frac {481}{21} A \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (-\frac {167}{7} A +\frac {2930}{63} B -\frac {20}{3} f x A +\frac {40}{3} f x B \right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (-\frac {18 B}{7}+A \right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {\left (-A +5 B \right ) \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )}{21}+\left (-A +\frac {18 B}{7}\right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {\left (-A +5 B \right ) \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )}{21}+\frac {B \left (\cos \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )-\sin \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )\right )}{63}\right ) c^{4}}{8 f \,a^{2} \left (\sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+3 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )\right )}\) | \(259\) |
| risch | \(\frac {35 c^{4} x A}{2 a^{2}}-\frac {35 c^{4} x B}{a^{2}}+\frac {i c^{4} {\mathrm e}^{2 i \left (f x +e \right )} A}{8 a^{2} f}-\frac {3 i c^{4} {\mathrm e}^{2 i \left (f x +e \right )} B}{4 a^{2} f}+\frac {3 c^{4} {\mathrm e}^{i \left (f x +e \right )} A}{a^{2} f}-\frac {71 c^{4} {\mathrm e}^{i \left (f x +e \right )} B}{8 a^{2} f}+\frac {3 c^{4} {\mathrm e}^{-i \left (f x +e \right )} A}{a^{2} f}-\frac {71 c^{4} {\mathrm e}^{-i \left (f x +e \right )} B}{8 a^{2} f}-\frac {i c^{4} {\mathrm e}^{-2 i \left (f x +e \right )} A}{8 a^{2} f}+\frac {3 i c^{4} {\mathrm e}^{-2 i \left (f x +e \right )} B}{4 a^{2} f}+\frac {96 i A \,c^{4} {\mathrm e}^{i \left (f x +e \right )}+64 A \,c^{4} {\mathrm e}^{2 i \left (f x +e \right )}-160 i B \,c^{4} {\mathrm e}^{i \left (f x +e \right )}-96 B \,c^{4} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {160 A \,c^{4}}{3}+\frac {256 B \,c^{4}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}+\frac {B \,c^{4} \cos \left (3 f x +3 e \right )}{12 a^{2} f}\) | \(312\) |
| norman | \(\frac {\frac {35 c^{4} \left (A -2 B \right ) x}{2 a}+\frac {164 A \,c^{4}-330 B \,c^{4}}{3 a f}+\frac {\left (131 A \,c^{4}-260 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {\left (979 A \,c^{4}-2004 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{3 a f}+\frac {\left (1893 A \,c^{4}-3700 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}+\frac {\left (2387 A \,c^{4}-4998 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3 a f}+\frac {\left (3654 A \,c^{4}-7040 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}+\frac {2 \left (1513 A \,c^{4}-3232 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3 a f}+\frac {2 \left (1767 A \,c^{4}-3368 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3 a f}+\frac {\left (2086 A \,c^{4}-4510 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 a f}+\frac {\left (571 A \,c^{4}-1084 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{a f}+\frac {\left (731 A \,c^{4}-1580 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{3 a f}+\frac {\left (111 A \,c^{4}-212 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{a f}+\frac {\left (33 A \,c^{4}-70 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{a f}+\frac {700 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a}+\frac {1225 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{2 a}+\frac {875 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{2 a}+\frac {280 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{a}+\frac {140 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{a}+\frac {105 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{2 a}+\frac {35 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{2 a}+\frac {105 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {140 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{a}+\frac {280 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{a}+\frac {875 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{2 a}+\frac {1225 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{2 a}+\frac {700 c^{4} \left (A -2 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{5} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(770\) |
Input:
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x,method=_RETUR NVERBOSE)
Output:
2/f*c^4/a^2*(((1/2*A-3*B)*tan(1/2*f*x+1/2*e)^5+(6*A-17*B)*tan(1/2*f*x+1/2* e)^4+(12*A-36*B)*tan(1/2*f*x+1/2*e)^2+(-1/2*A+3*B)*tan(1/2*f*x+1/2*e)+6*A- 53/3*B)/(1+tan(1/2*f*x+1/2*e)^2)^3+35/2*(A-2*B)*arctan(tan(1/2*f*x+1/2*e)) -1/2*(-32*A+32*B)/(tan(1/2*f*x+1/2*e)+1)^2-(-16*A+32*B)/(tan(1/2*f*x+1/2*e )+1)-1/3*(32*A-32*B)/(tan(1/2*f*x+1/2*e)+1)^3)
Time = 0.09 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.79 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {2 \, B c^{4} \cos \left (f x + e\right )^{5} - {\left (3 \, A - 16 \, B\right )} c^{4} \cos \left (f x + e\right )^{4} + 2 \, {\left (15 \, A - 38 \, B\right )} c^{4} \cos \left (f x + e\right )^{3} - 210 \, {\left (A - 2 \, B\right )} c^{4} f x + 32 \, {\left (A - B\right )} c^{4} + {\left (105 \, {\left (A - 2 \, B\right )} c^{4} f x - {\left (193 \, A - 346 \, B\right )} c^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (105 \, {\left (A - 2 \, B\right )} c^{4} f x + 2 \, {\left (97 \, A - 202 \, B\right )} c^{4}\right )} \cos \left (f x + e\right ) - {\left (2 \, B c^{4} \cos \left (f x + e\right )^{4} + {\left (3 \, A - 14 \, B\right )} c^{4} \cos \left (f x + e\right )^{3} + 210 \, {\left (A - 2 \, B\right )} c^{4} f x + 3 \, {\left (11 \, A - 30 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 32 \, {\left (A - B\right )} c^{4} + {\left (105 \, {\left (A - 2 \, B\right )} c^{4} f x + 2 \, {\left (113 \, A - 218 \, B\right )} c^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algori thm="fricas")
Output:
1/6*(2*B*c^4*cos(f*x + e)^5 - (3*A - 16*B)*c^4*cos(f*x + e)^4 + 2*(15*A - 38*B)*c^4*cos(f*x + e)^3 - 210*(A - 2*B)*c^4*f*x + 32*(A - B)*c^4 + (105*( A - 2*B)*c^4*f*x - (193*A - 346*B)*c^4)*cos(f*x + e)^2 - (105*(A - 2*B)*c^ 4*f*x + 2*(97*A - 202*B)*c^4)*cos(f*x + e) - (2*B*c^4*cos(f*x + e)^4 + (3* A - 14*B)*c^4*cos(f*x + e)^3 + 210*(A - 2*B)*c^4*f*x + 3*(11*A - 30*B)*c^4 *cos(f*x + e)^2 + 32*(A - B)*c^4 + (105*(A - 2*B)*c^4*f*x + 2*(113*A - 218 *B)*c^4)*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))
Leaf count of result is larger than twice the leaf count of optimal. 7337 vs. \(2 (175) = 350\).
Time = 13.37 (sec) , antiderivative size = 7337, normalized size of antiderivative = 40.76 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))**4/(a+a*sin(f*x+e))**2,x)
Output:
Piecewise((105*A*c**4*f*x*tan(e/2 + f*x/2)**9/(6*a**2*f*tan(e/2 + f*x/2)** 9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a** 2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/ 2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2) **2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 315*A*c**4*f*x*tan(e/2 + f* x/2)**8/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36 *a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*ta n(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f* x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a **2*f) + 630*A*c**4*f*x*tan(e/2 + f*x/2)**7/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2* f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)** 2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 1050*A*c**4*f*x*tan(e/2 + f*x /2)**6/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36* a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan (e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x /2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a* *2*f) + 1260*A*c**4*f*x*tan(e/2 + f*x/2)**5/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a*...
Leaf count of result is larger than twice the leaf count of optimal. 2094 vs. \(2 (172) = 344\).
Time = 0.16 (sec) , antiderivative size = 2094, normalized size of antiderivative = 11.63 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algori thm="maxima")
Output:
1/3*(A*c^4*((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/(cos(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^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e) /(cos(f*x + e) + 1))/a^2) - 4*B*c^4*((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/(cos( 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/(co s(f*x + e) + 1)^4 + 5*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^2*sin( f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - 2*B*c^4*((57*sin(f*x + e)/(cos(f*x + e) + 1) + 99*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 155*sin (f*x + e)^3/(cos(f*x + e) + 1)^3 + 153*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 135*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 85*sin(f*x + e)^6/(cos(f*x...
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (172) = 344\).
Time = 0.25 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.94 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {105 \, {\left (A c^{4} - 2 \, B c^{4}\right )} {\left (f x + e\right )}}{a^{2}} + \frac {2 \, {\left (99 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 210 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 333 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 636 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 533 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1160 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 1047 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 1980 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 921 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1980 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1107 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2140 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 651 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1344 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 393 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 780 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 164 \, A c^{4} - 330 \, B c^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3} a^{2}}}{6 \, f} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algori thm="giac")
Output:
1/6*(105*(A*c^4 - 2*B*c^4)*(f*x + e)/a^2 + 2*(99*A*c^4*tan(1/2*f*x + 1/2*e )^8 - 210*B*c^4*tan(1/2*f*x + 1/2*e)^8 + 333*A*c^4*tan(1/2*f*x + 1/2*e)^7 - 636*B*c^4*tan(1/2*f*x + 1/2*e)^7 + 533*A*c^4*tan(1/2*f*x + 1/2*e)^6 - 11 60*B*c^4*tan(1/2*f*x + 1/2*e)^6 + 1047*A*c^4*tan(1/2*f*x + 1/2*e)^5 - 1980 *B*c^4*tan(1/2*f*x + 1/2*e)^5 + 921*A*c^4*tan(1/2*f*x + 1/2*e)^4 - 1980*B* c^4*tan(1/2*f*x + 1/2*e)^4 + 1107*A*c^4*tan(1/2*f*x + 1/2*e)^3 - 2140*B*c^ 4*tan(1/2*f*x + 1/2*e)^3 + 651*A*c^4*tan(1/2*f*x + 1/2*e)^2 - 1344*B*c^4*t an(1/2*f*x + 1/2*e)^2 + 393*A*c^4*tan(1/2*f*x + 1/2*e) - 780*B*c^4*tan(1/2 *f*x + 1/2*e) + 164*A*c^4 - 330*B*c^4)/((tan(1/2*f*x + 1/2*e)^3 + tan(1/2* f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e) + 1)^3*a^2))/f
Time = 39.07 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.30 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (131\,A\,c^4-260\,B\,c^4\right )+\frac {164\,A\,c^4}{3}-110\,B\,c^4+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (33\,A\,c^4-70\,B\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (111\,A\,c^4-212\,B\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (217\,A\,c^4-448\,B\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (307\,A\,c^4-660\,B\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (349\,A\,c^4-660\,B\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {533\,A\,c^4}{3}-\frac {1160\,B\,c^4}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (369\,A\,c^4-\frac {2140\,B\,c^4}{3}\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+6\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+10\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+12\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+12\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+10\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+6\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^2\right )}+\frac {35\,c^4\,\mathrm {atan}\left (\frac {35\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A-2\,B\right )}{35\,A\,c^4-70\,B\,c^4}\right )\,\left (A-2\,B\right )}{a^2\,f} \] Input:
int(((A + B*sin(e + f*x))*(c - c*sin(e + f*x))^4)/(a + a*sin(e + f*x))^2,x )
Output:
(tan(e/2 + (f*x)/2)*(131*A*c^4 - 260*B*c^4) + (164*A*c^4)/3 - 110*B*c^4 + tan(e/2 + (f*x)/2)^8*(33*A*c^4 - 70*B*c^4) + tan(e/2 + (f*x)/2)^7*(111*A*c ^4 - 212*B*c^4) + tan(e/2 + (f*x)/2)^2*(217*A*c^4 - 448*B*c^4) + tan(e/2 + (f*x)/2)^4*(307*A*c^4 - 660*B*c^4) + tan(e/2 + (f*x)/2)^5*(349*A*c^4 - 66 0*B*c^4) + tan(e/2 + (f*x)/2)^6*((533*A*c^4)/3 - (1160*B*c^4)/3) + tan(e/2 + (f*x)/2)^3*(369*A*c^4 - (2140*B*c^4)/3))/(f*(6*a^2*tan(e/2 + (f*x)/2)^2 + 10*a^2*tan(e/2 + (f*x)/2)^3 + 12*a^2*tan(e/2 + (f*x)/2)^4 + 12*a^2*tan( e/2 + (f*x)/2)^5 + 10*a^2*tan(e/2 + (f*x)/2)^6 + 6*a^2*tan(e/2 + (f*x)/2)^ 7 + 3*a^2*tan(e/2 + (f*x)/2)^8 + a^2*tan(e/2 + (f*x)/2)^9 + a^2 + 3*a^2*ta n(e/2 + (f*x)/2))) + (35*c^4*atan((35*c^4*tan(e/2 + (f*x)/2)*(A - 2*B))/(3 5*A*c^4 - 70*B*c^4))*(A - 2*B))/(a^2*f)
Time = 0.18 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.23 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx=\frac {c^{4} \left (66 a -140 b +105 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a f x -210 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b f x +140 \cos \left (f x +e \right ) b +260 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b -105 a f x +210 b f x +105 \cos \left (f x +e \right ) a f x -210 \cos \left (f x +e \right ) b f x -210 \sin \left (f x +e \right ) a f x +420 \sin \left (f x +e \right ) b f x +72 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b -131 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a -131 a \sin \left (f x +e \right )-14 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -30 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +3 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -16 \sin \left (f x +e \right )^{4} b -33 \sin \left (f x +e \right )^{3} a +2 \sin \left (f x +e \right )^{5} b +3 \sin \left (f x +e \right )^{4} a -66 \cos \left (f x +e \right ) a +86 \sin \left (f x +e \right )^{3} b -297 \sin \left (f x +e \right )^{2} a +568 \sin \left (f x +e \right )^{2} b -105 \sin \left (f x +e \right )^{2} a f x +210 \sin \left (f x +e \right )^{2} b f x +260 \sin \left (f x +e \right ) b \right )}{6 a^{2} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-\sin \left (f x +e \right )^{2}-2 \sin \left (f x +e \right )-1\right )} \] Input:
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x)
Output:
(c**4*(2*cos(e + f*x)*sin(e + f*x)**4*b + 3*cos(e + f*x)*sin(e + f*x)**3*a - 14*cos(e + f*x)*sin(e + f*x)**3*b - 30*cos(e + f*x)*sin(e + f*x)**2*a + 72*cos(e + f*x)*sin(e + f*x)**2*b + 105*cos(e + f*x)*sin(e + f*x)*a*f*x - 131*cos(e + f*x)*sin(e + f*x)*a - 210*cos(e + f*x)*sin(e + f*x)*b*f*x + 2 60*cos(e + f*x)*sin(e + f*x)*b + 105*cos(e + f*x)*a*f*x - 66*cos(e + f*x)* a - 210*cos(e + f*x)*b*f*x + 140*cos(e + f*x)*b + 2*sin(e + f*x)**5*b + 3* sin(e + f*x)**4*a - 16*sin(e + f*x)**4*b - 33*sin(e + f*x)**3*a + 86*sin(e + f*x)**3*b - 105*sin(e + f*x)**2*a*f*x - 297*sin(e + f*x)**2*a + 210*sin (e + f*x)**2*b*f*x + 568*sin(e + f*x)**2*b - 210*sin(e + f*x)*a*f*x - 131* sin(e + f*x)*a + 420*sin(e + f*x)*b*f*x + 260*sin(e + f*x)*b - 105*a*f*x + 66*a + 210*b*f*x - 140*b))/(6*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))