Integrand size = 26, antiderivative size = 118 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx=\frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {4 \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {4 \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{35 a c^4 f} \]
1/7*sec(f*x+e)/a/c/f/(c-c*sin(f*x+e))^3+4/35*sec(f*x+e)/a/f/(c^2-c^2*sin(f *x+e))^2+4/35*sec(f*x+e)/a/f/(c^4-c^4*sin(f*x+e))+8/35*tan(f*x+e)/a/c^4/f
Time = 1.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (9170 \cos (e+f x)+1792 \cos (2 (e+f x))-3930 \cos (3 (e+f x))-128 \cos (4 (e+f x))+1792 \sin (e+f x)-9170 \sin (2 (e+f x))-768 \sin (3 (e+f x))+655 \sin (4 (e+f x)))}{4480 a c^4 f (-1+\sin (e+f x))^4 (1+\sin (e+f x))} \]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ])*(9170*Cos[e + f*x] + 1792*Cos[2*(e + f*x)] - 3930*Cos[3*(e + f*x)] - 12 8*Cos[4*(e + f*x)] + 1792*Sin[e + f*x] - 9170*Sin[2*(e + f*x)] - 768*Sin[3 *(e + f*x)] + 655*Sin[4*(e + f*x)]))/(4480*a*c^4*f*(-1 + Sin[e + f*x])^4*( 1 + Sin[e + f*x]))
Time = 0.66 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3042, 3215, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 4254, 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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a) (c-c \sin (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a) (c-c \sin (e+f x))^4}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \frac {\sec ^2(e+f x)}{(c-c \sin (e+f x))^3}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\cos (e+f x)^2 (c-c \sin (e+f x))^3}dx}{a c}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {4 \int \frac {\sec ^2(e+f x)}{(c-c \sin (e+f x))^2}dx}{7 c}+\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4 \int \frac {1}{\cos (e+f x)^2 (c-c \sin (e+f x))^2}dx}{7 c}+\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}}{a c}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {4 \left (\frac {3 \int \frac {\sec ^2(e+f x)}{c-c \sin (e+f x)}dx}{5 c}+\frac {\sec (e+f x)}{5 f (c-c \sin (e+f x))^2}\right )}{7 c}+\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4 \left (\frac {3 \int \frac {1}{\cos (e+f x)^2 (c-c \sin (e+f x))}dx}{5 c}+\frac {\sec (e+f x)}{5 f (c-c \sin (e+f x))^2}\right )}{7 c}+\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}}{a c}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {4 \left (\frac {3 \left (\frac {2 \int \sec ^2(e+f x)dx}{3 c}+\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}\right )}{5 c}+\frac {\sec (e+f x)}{5 f (c-c \sin (e+f x))^2}\right )}{7 c}+\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {4 \left (\frac {3 \left (\frac {2 \int \csc \left (e+f x+\frac {\pi }{2}\right )^2dx}{3 c}+\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}\right )}{5 c}+\frac {\sec (e+f x)}{5 f (c-c \sin (e+f x))^2}\right )}{7 c}+\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}}{a c}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {4 \left (\frac {3 \left (\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}-\frac {2 \int 1d(-\tan (e+f x))}{3 c f}\right )}{5 c}+\frac {\sec (e+f x)}{5 f (c-c \sin (e+f x))^2}\right )}{7 c}+\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}}{a c}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\sec (e+f x)}{7 f (c-c \sin (e+f x))^3}+\frac {4 \left (\frac {\sec (e+f x)}{5 f (c-c \sin (e+f x))^2}+\frac {3 \left (\frac {2 \tan (e+f x)}{3 c f}+\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}\right )}{5 c}\right )}{7 c}}{a c}\) |
(Sec[e + f*x]/(7*f*(c - c*Sin[e + f*x])^3) + (4*(Sec[e + f*x]/(5*f*(c - c* Sin[e + f*x])^2) + (3*(Sec[e + f*x]/(3*f*(c - c*Sin[e + f*x])) + (2*Tan[e + f*x])/(3*c*f)))/(5*c)))/(7*c))/(a*c)
3.3.68.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains complex when optimal does not.
Time = 1.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(-\frac {16 \left (-6 \,{\mathrm e}^{i \left (f x +e \right )}+i+14 \,{\mathrm e}^{3 i \left (f x +e \right )}-14 i {\mathrm e}^{2 i \left (f x +e \right )}\right )}{35 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a \,c^{4} f}\) | \(77\) |
| parallelrisch | \(\frac {-\frac {26}{35}-10 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+6 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+6 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {2 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {86 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}-\frac {22 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}}{f a \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(129\) |
| derivativedivides | \(\frac {-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {38}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {9}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {15}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {17}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a \,c^{4} f}\) | \(133\) |
| default | \(\frac {-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {38}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {9}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {15}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {17}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a \,c^{4} f}\) | \(133\) |
| norman | \(\frac {\frac {6 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {26}{35 a c f}-\frac {2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {6 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {10 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {2 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}-\frac {22 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}+\frac {86 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 a c f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(195\) |
-16/35*(-6*exp(I*(f*x+e))+I+14*exp(3*I*(f*x+e))-14*I*exp(2*I*(f*x+e)))/(ex p(I*(f*x+e))-I)^7/(exp(I*(f*x+e))+I)/a/c^4/f
Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx=\frac {8 \, \cos \left (f x + e\right )^{4} - 36 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (6 \, \cos \left (f x + e\right )^{2} - 5\right )} \sin \left (f x + e\right ) + 15}{35 \, {\left (3 \, a c^{4} f \cos \left (f x + e\right )^{3} - 4 \, a c^{4} f \cos \left (f x + e\right ) - {\left (a c^{4} f \cos \left (f x + e\right )^{3} - 4 \, a c^{4} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
1/35*(8*cos(f*x + e)^4 - 36*cos(f*x + e)^2 + 4*(6*cos(f*x + e)^2 - 5)*sin( f*x + e) + 15)/(3*a*c^4*f*cos(f*x + e)^3 - 4*a*c^4*f*cos(f*x + e) - (a*c^4 *f*cos(f*x + e)^3 - 4*a*c^4*f*cos(f*x + e))*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 1307 vs. \(2 (97) = 194\).
Time = 4.89 (sec) , antiderivative size = 1307, normalized size of antiderivative = 11.08 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]
Piecewise((-70*tan(e/2 + f*x/2)**7/(35*a*c**4*f*tan(e/2 + f*x/2)**8 - 210* a*c**4*f*tan(e/2 + f*x/2)**7 + 490*a*c**4*f*tan(e/2 + f*x/2)**6 - 490*a*c* *4*f*tan(e/2 + f*x/2)**5 + 490*a*c**4*f*tan(e/2 + f*x/2)**3 - 490*a*c**4*f *tan(e/2 + f*x/2)**2 + 210*a*c**4*f*tan(e/2 + f*x/2) - 35*a*c**4*f) + 210* tan(e/2 + f*x/2)**6/(35*a*c**4*f*tan(e/2 + f*x/2)**8 - 210*a*c**4*f*tan(e/ 2 + f*x/2)**7 + 490*a*c**4*f*tan(e/2 + f*x/2)**6 - 490*a*c**4*f*tan(e/2 + f*x/2)**5 + 490*a*c**4*f*tan(e/2 + f*x/2)**3 - 490*a*c**4*f*tan(e/2 + f*x/ 2)**2 + 210*a*c**4*f*tan(e/2 + f*x/2) - 35*a*c**4*f) - 350*tan(e/2 + f*x/2 )**5/(35*a*c**4*f*tan(e/2 + f*x/2)**8 - 210*a*c**4*f*tan(e/2 + f*x/2)**7 + 490*a*c**4*f*tan(e/2 + f*x/2)**6 - 490*a*c**4*f*tan(e/2 + f*x/2)**5 + 490 *a*c**4*f*tan(e/2 + f*x/2)**3 - 490*a*c**4*f*tan(e/2 + f*x/2)**2 + 210*a*c **4*f*tan(e/2 + f*x/2) - 35*a*c**4*f) + 210*tan(e/2 + f*x/2)**4/(35*a*c**4 *f*tan(e/2 + f*x/2)**8 - 210*a*c**4*f*tan(e/2 + f*x/2)**7 + 490*a*c**4*f*t an(e/2 + f*x/2)**6 - 490*a*c**4*f*tan(e/2 + f*x/2)**5 + 490*a*c**4*f*tan(e /2 + f*x/2)**3 - 490*a*c**4*f*tan(e/2 + f*x/2)**2 + 210*a*c**4*f*tan(e/2 + f*x/2) - 35*a*c**4*f) + 14*tan(e/2 + f*x/2)**3/(35*a*c**4*f*tan(e/2 + f*x /2)**8 - 210*a*c**4*f*tan(e/2 + f*x/2)**7 + 490*a*c**4*f*tan(e/2 + f*x/2)* *6 - 490*a*c**4*f*tan(e/2 + f*x/2)**5 + 490*a*c**4*f*tan(e/2 + f*x/2)**3 - 490*a*c**4*f*tan(e/2 + f*x/2)**2 + 210*a*c**4*f*tan(e/2 + f*x/2) - 35*a*c **4*f) - 154*tan(e/2 + f*x/2)**2/(35*a*c**4*f*tan(e/2 + f*x/2)**8 - 210...
Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (113) = 226\).
Time = 0.22 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (\frac {43 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {77 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {7 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {175 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {35 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - 13\right )}}{35 \, {\left (a c^{4} - \frac {6 \, a c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {6 \, a c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} f} \]
-2/35*(43*sin(f*x + e)/(cos(f*x + e) + 1) - 77*sin(f*x + e)^2/(cos(f*x + e ) + 1)^2 + 7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 105*sin(f*x + e)^4/(cos (f*x + e) + 1)^4 - 175*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 105*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 35*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 13)/ ((a*c^4 - 6*a*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 14*a*c^4*sin(f*x + e)^ 2/(cos(f*x + e) + 1)^2 - 14*a*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 14 *a*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 14*a*c^4*sin(f*x + e)^6/(cos( f*x + e) + 1)^6 + 6*a*c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - a*c^4*sin( f*x + e)^8/(cos(f*x + e) + 1)^8)*f)
Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx=-\frac {\frac {35}{a c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {525 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1960 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 4025 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4480 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3143 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1176 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 243}{a c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{280 \, f} \]
-1/280*(35/(a*c^4*(tan(1/2*f*x + 1/2*e) + 1)) + (525*tan(1/2*f*x + 1/2*e)^ 6 - 1960*tan(1/2*f*x + 1/2*e)^5 + 4025*tan(1/2*f*x + 1/2*e)^4 - 4480*tan(1 /2*f*x + 1/2*e)^3 + 3143*tan(1/2*f*x + 1/2*e)^2 - 1176*tan(1/2*f*x + 1/2*e ) + 243)/(a*c^4*(tan(1/2*f*x + 1/2*e) - 1)^7))/f
Time = 6.54 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx=\frac {\frac {2\,\sin \left (e+f\,x\right )}{5}+\frac {2\,\cos \left (2\,e+2\,f\,x\right )}{5}-\frac {\cos \left (4\,e+4\,f\,x\right )}{35}-\frac {6\,\sin \left (3\,e+3\,f\,x\right )}{35}}{a\,c^4\,f\,\left (\frac {7\,\cos \left (e+f\,x\right )}{4}-\frac {3\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {7\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {\sin \left (4\,e+4\,f\,x\right )}{8}\right )} \]