Integrand size = 36, antiderivative size = 190 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=-\frac {35 (4 A-5 B) c^4 x}{8 a}-\frac {35 (4 A-5 B) c^4 \cos ^3(e+f x)}{12 a f}-\frac {35 (4 A-5 B) c^4 \cos (e+f x) \sin (e+f x)}{8 a f}-\frac {a^4 (A-B) c^4 \cos ^9(e+f x)}{f (a+a \sin (e+f x))^5}-\frac {2 a^2 (4 A-5 B) c^4 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^3}-\frac {7 (4 A-5 B) c^4 \cos ^5(e+f x)}{4 f (a+a \sin (e+f x))} \] Output:
-35/8*(4*A-5*B)*c^4*x/a-35/12*(4*A-5*B)*c^4*cos(f*x+e)^3/a/f-35/8*(4*A-5*B )*c^4*cos(f*x+e)*sin(f*x+e)/a/f-a^4*(A-B)*c^4*cos(f*x+e)^9/f/(a+a*sin(f*x+ e))^5-2*a^2*(4*A-5*B)*c^4*cos(f*x+e)^7/f/(a+a*sin(f*x+e))^3-7/4*(4*A-5*B)* c^4*cos(f*x+e)^5/f/(a+a*sin(f*x+e))
Time = 14.23 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.44 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^4 \left (3072 (A-B) \sin \left (\frac {1}{2} (e+f x)\right )-420 (4 A-5 B) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-24 (47 A-75 B) \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+8 (A-5 B) \cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+24 (5 A-12 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (2 (e+f x))+3 B \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (4 (e+f x))\right )}{96 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 (1+\sin (e+f x))} \] Input:
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x ]),x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^4*(3072*(A - B )*Sin[(e + f*x)/2] - 420*(4*A - 5*B)*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 24*(47*A - 75*B)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f* x)/2]) + 8*(A - 5*B)*Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2] ) + 24*(5*A - 12*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[2*(e + f*x)] + 3*B*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[4*(e + f*x)]))/(96*a*f*(C os[(e + f*x)/2] - Sin[(e + f*x)/2])^8*(1 + Sin[e + f*x]))
Time = 0.95 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.90, 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, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^4 (A+B \sin (e+f x))}{a \sin (e+f x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^4 (A+B \sin (e+f x))}{a \sin (e+f x)+a}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^4 c^4 \int \frac {\cos ^8(e+f x) (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^5}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \int \frac {\cos (e+f x)^8 (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^5}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \int \frac {\cos ^8(e+f x)}{(\sin (e+f x) a+a)^4}dx}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \int \frac {\cos (e+f x)^8}{(\sin (e+f x) a+a)^4}dx}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \int \frac {\cos ^6(e+f x)}{(\sin (e+f x) a+a)^2}dx}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \int \frac {\cos (e+f x)^6}{(\sin (e+f x) a+a)^2}dx}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3158 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \left (\frac {5 \int \frac {\cos ^4(e+f x)}{\sin (e+f x) a+a}dx}{4 a}+\frac {\cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \left (\frac {5 \int \frac {\cos (e+f x)^4}{\sin (e+f x) a+a}dx}{4 a}+\frac {\cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \left (\frac {5 \left (\frac {\int \cos ^2(e+f x)dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{4 a}+\frac {\cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \left (\frac {5 \left (\frac {\int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{4 a}+\frac {\cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \left (\frac {5 \left (\frac {\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{4 a}+\frac {\cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}\right )}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^4 c^4 \left (-\frac {(4 A-5 B) \left (\frac {7 \left (\frac {\cos ^5(e+f x)}{4 f \left (a^2 \sin (e+f x)+a^2\right )}+\frac {5 \left (\frac {\cos ^3(e+f x)}{3 a f}+\frac {\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}}{a}\right )}{4 a}\right )}{a^2}+\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^3}\right )}{a}-\frac {(A-B) \cos ^9(e+f x)}{f (a \sin (e+f x)+a)^5}\right )\) |
Input:
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4)/(a + a*Sin[e + f*x]),x]
Output:
a^4*c^4*(-(((A - B)*Cos[e + f*x]^9)/(f*(a + a*Sin[e + f*x])^5)) - ((4*A - 5*B)*((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)
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[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]
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 = 10.82 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {2 c^{4} \left (-\frac {\left (\frac {5 A}{2}-\frac {47 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (11 A -15 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (\frac {5 A}{2}-\frac {55 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (35 A -55 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-\frac {5 A}{2}+\frac {55 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (\frac {107 A}{3}-\frac {175 B}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {5 A}{2}+\frac {47 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {35 A}{3}-\frac {55 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4}}-\frac {35 \left (4 A -5 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}-\frac {16 A -16 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f a}\) | \(209\) |
| default | \(\frac {2 c^{4} \left (-\frac {\left (\frac {5 A}{2}-\frac {47 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (11 A -15 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (\frac {5 A}{2}-\frac {55 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (35 A -55 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-\frac {5 A}{2}+\frac {55 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (\frac {107 A}{3}-\frac {175 B}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-\frac {5 A}{2}+\frac {47 B}{8}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {35 A}{3}-\frac {55 B}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4}}-\frac {35 \left (4 A -5 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}-\frac {16 A -16 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f a}\) | \(209\) |
| parallelrisch | \(-\frac {7 \left (\left (30 f x A -\frac {75}{2} f x B +\frac {1189}{14} A -\frac {1433}{14} B \right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (30 f x A -\frac {75}{2} f x B +\frac {139}{14} A -\frac {215}{14} B \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+9 \left (A -\frac {3 B}{2}\right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (A -\frac {31 B}{14}\right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {\left (-A +\frac {37 B}{8}\right ) \cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )}{14}+9 \left (A -\frac {3 B}{2}\right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (-A +\frac {31 B}{14}\right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\frac {\left (-A +\frac {37 B}{8}\right ) \sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )}{14}+\frac {3 B \left (\cos \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )-\sin \left (\frac {9 f x}{2}+\frac {9 e}{2}\right )\right )}{112}\right ) c^{4}}{12 a f \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\sin \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}\) | \(212\) |
| risch | \(-\frac {35 c^{4} x A}{2 a}+\frac {175 c^{4} x B}{8 a}-\frac {47 c^{4} {\mathrm e}^{i \left (f x +e \right )} A}{8 a f}+\frac {75 c^{4} {\mathrm e}^{i \left (f x +e \right )} B}{8 a f}-\frac {47 c^{4} {\mathrm e}^{-i \left (f x +e \right )} A}{8 a f}+\frac {75 c^{4} {\mathrm e}^{-i \left (f x +e \right )} B}{8 a f}-\frac {32 c^{4} A}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {32 c^{4} B}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {B \,c^{4} \sin \left (4 f x +4 e \right )}{32 a f}+\frac {c^{4} \cos \left (3 f x +3 e \right ) A}{12 a f}-\frac {5 c^{4} \cos \left (3 f x +3 e \right ) B}{12 a f}+\frac {5 c^{4} \sin \left (2 f x +2 e \right ) A}{4 a f}-\frac {3 c^{4} \sin \left (2 f x +2 e \right ) B}{a f}\) | \(263\) |
| norman | \(\frac {-\frac {166 A \,c^{4}-206 B \,c^{4}}{3 a f}-\frac {35 \left (4 A -5 B \right ) c^{4} x}{8 a}-\frac {\left (108 A \,c^{4}-167 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{4 a f}-\frac {\left (148 A \,c^{4}-175 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{4 a f}-\frac {\left (204 A \,c^{4}-331 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{2 a f}-\frac {2 \left (206 A \,c^{4}-230 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a f}-\frac {2 \left (212 A \,c^{4}-340 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}-\frac {\left (220 A \,c^{4}-299 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{12 a f}-\frac {\left (384 A \,c^{4}-431 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{2 a f}-\frac {\left (508 A \,c^{4}-767 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{6 a f}-\frac {\left (2708 A \,c^{4}-3127 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{6 a f}-\frac {\left (2996 A \,c^{4}-3619 B \,c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{12 a f}-\frac {35 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{8 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{8 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{4 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{4 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{4 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{4 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{8 a}-\frac {175 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{8 a}-\frac {35 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{8 a}-\frac {35 \left (4 A -5 B \right ) c^{4} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{8 a}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\) | \(681\) |
Input:
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e)),x,method=_RETURNV ERBOSE)
Output:
2/f*c^4/a*(-((5/2*A-47/8*B)*tan(1/2*f*x+1/2*e)^7+(11*A-15*B)*tan(1/2*f*x+1 /2*e)^6+(5/2*A-55/8*B)*tan(1/2*f*x+1/2*e)^5+(35*A-55*B)*tan(1/2*f*x+1/2*e) ^4+(-5/2*A+55/8*B)*tan(1/2*f*x+1/2*e)^3+(107/3*A-175/3*B)*tan(1/2*f*x+1/2* e)^2+(-5/2*A+47/8*B)*tan(1/2*f*x+1/2*e)+35/3*A-55/3*B)/(1+tan(1/2*f*x+1/2* e)^2)^4-35/8*(4*A-5*B)*arctan(tan(1/2*f*x+1/2*e))-(16*A-16*B)/(tan(1/2*f*x +1/2*e)+1))
Time = 0.09 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=-\frac {6 \, B c^{4} \cos \left (f x + e\right )^{5} - 8 \, {\left (A - 5 \, B\right )} c^{4} \cos \left (f x + e\right )^{4} + {\left (52 \, A - 113 \, B\right )} c^{4} \cos \left (f x + e\right )^{3} + 105 \, {\left (4 \, A - 5 \, B\right )} c^{4} f x + 96 \, {\left (3 \, A - 5 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} + 384 \, {\left (A - B\right )} c^{4} + 3 \, {\left (35 \, {\left (4 \, A - 5 \, B\right )} c^{4} f x + {\left (204 \, A - 239 \, B\right )} c^{4}\right )} \cos \left (f x + e\right ) - {\left (6 \, B c^{4} \cos \left (f x + e\right )^{4} + 2 \, {\left (4 \, A - 17 \, B\right )} c^{4} \cos \left (f x + e\right )^{3} - 105 \, {\left (4 \, A - 5 \, B\right )} c^{4} f x + 3 \, {\left (20 \, A - 49 \, B\right )} c^{4} \cos \left (f x + e\right )^{2} - 3 \, {\left (76 \, A - 111 \, B\right )} c^{4} \cos \left (f x + e\right ) + 384 \, {\left (A - B\right )} c^{4}\right )} \sin \left (f x + e\right )}{24 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\right )}} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorith m="fricas")
Output:
-1/24*(6*B*c^4*cos(f*x + e)^5 - 8*(A - 5*B)*c^4*cos(f*x + e)^4 + (52*A - 1 13*B)*c^4*cos(f*x + e)^3 + 105*(4*A - 5*B)*c^4*f*x + 96*(3*A - 5*B)*c^4*co s(f*x + e)^2 + 384*(A - B)*c^4 + 3*(35*(4*A - 5*B)*c^4*f*x + (204*A - 239* B)*c^4)*cos(f*x + e) - (6*B*c^4*cos(f*x + e)^4 + 2*(4*A - 17*B)*c^4*cos(f* x + e)^3 - 105*(4*A - 5*B)*c^4*f*x + 3*(20*A - 49*B)*c^4*cos(f*x + e)^2 - 3*(76*A - 111*B)*c^4*cos(f*x + e) + 384*(A - B)*c^4)*sin(f*x + e))/(a*f*co s(f*x + e) + a*f*sin(f*x + e) + a*f)
Leaf count of result is larger than twice the leaf count of optimal. 6690 vs. \(2 (173) = 346\).
Time = 7.35 (sec) , antiderivative size = 6690, normalized size of antiderivative = 35.21 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))**4/(a+a*sin(f*x+e)),x)
Output:
Piecewise((-420*A*c**4*f*x*tan(e/2 + f*x/2)**9/(24*a*f*tan(e/2 + f*x/2)**9 + 24*a*f*tan(e/2 + f*x/2)**8 + 96*a*f*tan(e/2 + f*x/2)**7 + 96*a*f*tan(e/ 2 + f*x/2)**6 + 144*a*f*tan(e/2 + f*x/2)**5 + 144*a*f*tan(e/2 + f*x/2)**4 + 96*a*f*tan(e/2 + f*x/2)**3 + 96*a*f*tan(e/2 + f*x/2)**2 + 24*a*f*tan(e/2 + f*x/2) + 24*a*f) - 420*A*c**4*f*x*tan(e/2 + f*x/2)**8/(24*a*f*tan(e/2 + f*x/2)**9 + 24*a*f*tan(e/2 + f*x/2)**8 + 96*a*f*tan(e/2 + f*x/2)**7 + 96* a*f*tan(e/2 + f*x/2)**6 + 144*a*f*tan(e/2 + f*x/2)**5 + 144*a*f*tan(e/2 + f*x/2)**4 + 96*a*f*tan(e/2 + f*x/2)**3 + 96*a*f*tan(e/2 + f*x/2)**2 + 24*a *f*tan(e/2 + f*x/2) + 24*a*f) - 1680*A*c**4*f*x*tan(e/2 + f*x/2)**7/(24*a* f*tan(e/2 + f*x/2)**9 + 24*a*f*tan(e/2 + f*x/2)**8 + 96*a*f*tan(e/2 + f*x/ 2)**7 + 96*a*f*tan(e/2 + f*x/2)**6 + 144*a*f*tan(e/2 + f*x/2)**5 + 144*a*f *tan(e/2 + f*x/2)**4 + 96*a*f*tan(e/2 + f*x/2)**3 + 96*a*f*tan(e/2 + f*x/2 )**2 + 24*a*f*tan(e/2 + f*x/2) + 24*a*f) - 1680*A*c**4*f*x*tan(e/2 + f*x/2 )**6/(24*a*f*tan(e/2 + f*x/2)**9 + 24*a*f*tan(e/2 + f*x/2)**8 + 96*a*f*tan (e/2 + f*x/2)**7 + 96*a*f*tan(e/2 + f*x/2)**6 + 144*a*f*tan(e/2 + f*x/2)** 5 + 144*a*f*tan(e/2 + f*x/2)**4 + 96*a*f*tan(e/2 + f*x/2)**3 + 96*a*f*tan( e/2 + f*x/2)**2 + 24*a*f*tan(e/2 + f*x/2) + 24*a*f) - 2520*A*c**4*f*x*tan( e/2 + f*x/2)**5/(24*a*f*tan(e/2 + f*x/2)**9 + 24*a*f*tan(e/2 + f*x/2)**8 + 96*a*f*tan(e/2 + f*x/2)**7 + 96*a*f*tan(e/2 + f*x/2)**6 + 144*a*f*tan(e/2 + f*x/2)**5 + 144*a*f*tan(e/2 + f*x/2)**4 + 96*a*f*tan(e/2 + f*x/2)**3...
Leaf count of result is larger than twice the leaf count of optimal. 1796 vs. \(2 (182) = 364\).
Time = 0.16 (sec) , antiderivative size = 1796, normalized size of antiderivative = 9.45 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorith m="maxima")
Output:
1/12*(B*c^4*((19*sin(f*x + e)/(cos(f*x + e) + 1) + 211*sin(f*x + e)^2/(cos (f*x + e) + 1)^2 + 91*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 219*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 165*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 165* sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 45*sin(f*x + e)^7/(cos(f*x + e) + 1) ^7 + 45*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 64)/(a + a*sin(f*x + e)/(cos (f*x + e) + 1) + 4*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4*a*sin(f*x + e )^3/(cos(f*x + e) + 1)^3 + 6*a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 6*a*s in(f*x + e)^5/(cos(f*x + e) + 1)^5 + 4*a*sin(f*x + e)^6/(cos(f*x + e) + 1) ^6 + 4*a*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + a*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 45*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a) - 4*A*c^4*((7*sin(f*x + e)/(cos(f*x + e) + 1) + 39*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 24*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 24*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 9*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 16)/(a + a*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*a*si n(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a*sin(f*x + e)^4/(cos(f*x + e) + 1)^ 4 + 3*a*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + a*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 9*arctan(sin(f*x + e) /(cos(f*x + e) + 1))/a) + 16*B*c^4*((7*sin(f*x + e)/(cos(f*x + e) + 1) + 3 9*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 24*sin(f*x + e)^3/(cos(f*x + e)...
Time = 0.24 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.72 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=-\frac {\frac {105 \, {\left (4 \, A c^{4} - 5 \, B c^{4}\right )} {\left (f x + e\right )}}{a} + \frac {768 \, {\left (A c^{4} - B c^{4}\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {2 \, {\left (60 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 141 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 264 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 360 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 60 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 165 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 840 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1320 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 60 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 165 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 856 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1400 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 60 \, A c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 141 \, B c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 280 \, A c^{4} - 440 \, B c^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{4} a}}{24 \, f} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorith m="giac")
Output:
-1/24*(105*(4*A*c^4 - 5*B*c^4)*(f*x + e)/a + 768*(A*c^4 - B*c^4)/(a*(tan(1 /2*f*x + 1/2*e) + 1)) + 2*(60*A*c^4*tan(1/2*f*x + 1/2*e)^7 - 141*B*c^4*tan (1/2*f*x + 1/2*e)^7 + 264*A*c^4*tan(1/2*f*x + 1/2*e)^6 - 360*B*c^4*tan(1/2 *f*x + 1/2*e)^6 + 60*A*c^4*tan(1/2*f*x + 1/2*e)^5 - 165*B*c^4*tan(1/2*f*x + 1/2*e)^5 + 840*A*c^4*tan(1/2*f*x + 1/2*e)^4 - 1320*B*c^4*tan(1/2*f*x + 1 /2*e)^4 - 60*A*c^4*tan(1/2*f*x + 1/2*e)^3 + 165*B*c^4*tan(1/2*f*x + 1/2*e) ^3 + 856*A*c^4*tan(1/2*f*x + 1/2*e)^2 - 1400*B*c^4*tan(1/2*f*x + 1/2*e)^2 - 60*A*c^4*tan(1/2*f*x + 1/2*e) + 141*B*c^4*tan(1/2*f*x + 1/2*e) + 280*A*c ^4 - 440*B*c^4)/((tan(1/2*f*x + 1/2*e)^2 + 1)^4*a))/f
Time = 39.09 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.09 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {55\,A\,c^4}{3}-\frac {299\,B\,c^4}{12}\right )+\frac {166\,A\,c^4}{3}-\frac {206\,B\,c^4}{3}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (27\,A\,c^4-\frac {167\,B\,c^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (37\,A\,c^4-\frac {175\,B\,c^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (75\,A\,c^4-\frac {495\,B\,c^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (155\,A\,c^4-\frac {687\,B\,c^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (257\,A\,c^4-\frac {1153\,B\,c^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {199\,A\,c^4}{3}-\frac {1235\,B\,c^4}{12}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {583\,A\,c^4}{3}-\frac {2795\,B\,c^4}{12}\right )}{f\,\left (a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+4\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+6\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+4\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\right )}-\frac {35\,c^4\,\mathrm {atan}\left (\frac {35\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,A-5\,B\right )}{140\,A\,c^4-175\,B\,c^4}\right )\,\left (4\,A-5\,B\right )}{4\,a\,f} \] Input:
int(((A + B*sin(e + f*x))*(c - c*sin(e + f*x))^4)/(a + a*sin(e + f*x)),x)
Output:
- (tan(e/2 + (f*x)/2)*((55*A*c^4)/3 - (299*B*c^4)/12) + (166*A*c^4)/3 - (2 06*B*c^4)/3 + tan(e/2 + (f*x)/2)^7*(27*A*c^4 - (167*B*c^4)/4) + tan(e/2 + (f*x)/2)^8*(37*A*c^4 - (175*B*c^4)/4) + tan(e/2 + (f*x)/2)^5*(75*A*c^4 - ( 495*B*c^4)/4) + tan(e/2 + (f*x)/2)^6*(155*A*c^4 - (687*B*c^4)/4) + tan(e/2 + (f*x)/2)^4*(257*A*c^4 - (1153*B*c^4)/4) + tan(e/2 + (f*x)/2)^3*((199*A* c^4)/3 - (1235*B*c^4)/12) + tan(e/2 + (f*x)/2)^2*((583*A*c^4)/3 - (2795*B* c^4)/12))/(f*(a + a*tan(e/2 + (f*x)/2) + 4*a*tan(e/2 + (f*x)/2)^2 + 4*a*ta n(e/2 + (f*x)/2)^3 + 6*a*tan(e/2 + (f*x)/2)^4 + 6*a*tan(e/2 + (f*x)/2)^5 + 4*a*tan(e/2 + (f*x)/2)^6 + 4*a*tan(e/2 + (f*x)/2)^7 + a*tan(e/2 + (f*x)/2 )^8 + a*tan(e/2 + (f*x)/2)^9)) - (35*c^4*atan((35*c^4*tan(e/2 + (f*x)/2)*( 4*A - 5*B))/(140*A*c^4 - 175*B*c^4))*(4*A - 5*B))/(4*a*f)
Time = 0.19 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.67 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^4}{a+a \sin (e+f x)} \, dx=\frac {c^{4} \left (-888 a +1050 b -1050 \cos \left (f x +e \right ) b -299 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +420 a f x -525 b f x -420 \cos \left (f x +e \right ) a f x +525 \cos \left (f x +e \right ) b f x +420 \sin \left (f x +e \right ) a f x -525 \sin \left (f x +e \right ) b f x +101 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +220 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +220 a \sin \left (f x +e \right )-34 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -52 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +6 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +8 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -40 \sin \left (f x +e \right )^{4} b -60 \sin \left (f x +e \right )^{3} a +6 \sin \left (f x +e \right )^{5} b +8 \sin \left (f x +e \right )^{4} a +888 \cos \left (f x +e \right ) a +135 \sin \left (f x +e \right )^{3} b +272 \sin \left (f x +e \right )^{2} a -400 \sin \left (f x +e \right )^{2} b -299 \sin \left (f x +e \right ) b \right )}{24 a f \left (\cos \left (f x +e \right )-\sin \left (f x +e \right )-1\right )} \] Input:
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4/(a+a*sin(f*x+e)),x)
Output:
(c**4*(6*cos(e + f*x)*sin(e + f*x)**4*b + 8*cos(e + f*x)*sin(e + f*x)**3*a - 34*cos(e + f*x)*sin(e + f*x)**3*b - 52*cos(e + f*x)*sin(e + f*x)**2*a + 101*cos(e + f*x)*sin(e + f*x)**2*b + 220*cos(e + f*x)*sin(e + f*x)*a - 29 9*cos(e + f*x)*sin(e + f*x)*b - 420*cos(e + f*x)*a*f*x + 888*cos(e + f*x)* a + 525*cos(e + f*x)*b*f*x - 1050*cos(e + f*x)*b + 6*sin(e + f*x)**5*b + 8 *sin(e + f*x)**4*a - 40*sin(e + f*x)**4*b - 60*sin(e + f*x)**3*a + 135*sin (e + f*x)**3*b + 272*sin(e + f*x)**2*a - 400*sin(e + f*x)**2*b + 420*sin(e + f*x)*a*f*x + 220*sin(e + f*x)*a - 525*sin(e + f*x)*b*f*x - 299*sin(e + f*x)*b + 420*a*f*x - 888*a - 525*b*f*x + 1050*b))/(24*a*f*(cos(e + f*x) - sin(e + f*x) - 1))