Integrand size = 26, antiderivative size = 129 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {x}{a^3 c^4}-\frac {\cot ^7(e+f x) (1+\sec (e+f x))}{7 a^3 c^4 f}+\frac {\cot ^5(e+f x) (7+6 \sec (e+f x))}{35 a^3 c^4 f}+\frac {\cot (e+f x) (35+16 \sec (e+f x))}{35 a^3 c^4 f}-\frac {\cot ^3(e+f x) (35+24 \sec (e+f x))}{105 a^3 c^4 f} \]
x/a^3/c^4-1/7*cot(f*x+e)^7*(1+sec(f*x+e))/a^3/c^4/f+1/35*cot(f*x+e)^5*(7+6 *sec(f*x+e))/a^3/c^4/f+1/35*cot(f*x+e)*(35+16*sec(f*x+e))/a^3/c^4/f-1/105* cot(f*x+e)^3*(35+24*sec(f*x+e))/a^3/c^4/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 5.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=-\frac {\csc ^7(e+f x) \left (-106+301 \cos (2 (e+f x))-70 \cos (4 (e+f x))+35 \cos (6 (e+f x))+160 \cos ^7(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(e+f x)\right )\right )}{1120 a^3 c^4 f} \]
-1/1120*(Csc[e + f*x]^7*(-106 + 301*Cos[2*(e + f*x)] - 70*Cos[4*(e + f*x)] + 35*Cos[6*(e + f*x)] + 160*Cos[e + f*x]^7*Hypergeometric2F1[-7/2, 1, -5/ 2, -Tan[e + f*x]^2]))/(a^3*c^4*f)
Time = 0.69 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3042, 4392, 3042, 4370, 25, 3042, 4370, 25, 3042, 4370, 27, 3042, 4370, 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.
| \(\displaystyle \int \frac {1}{(a \sec (e+f x)+a)^3 (c-c \sec (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4392 |
| \(\displaystyle \frac {\int \cot ^8(e+f x) (\sec (e+f x) a+a)dx}{a^4 c^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}{\cot \left (e+f x+\frac {\pi }{2}\right )^8}dx}{a^4 c^4}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {\frac {1}{7} \int -\cot ^6(e+f x) (6 \sec (e+f x) a+7 a)dx-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {1}{7} \int \cot ^6(e+f x) (6 \sec (e+f x) a+7 a)dx-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{7} \int \frac {6 \csc \left (e+f x+\frac {\pi }{2}\right ) a+7 a}{\cot \left (e+f x+\frac {\pi }{2}\right )^6}dx-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}-\frac {1}{5} \int -\cot ^4(e+f x) (24 \sec (e+f x) a+35 a)dx\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \cot ^4(e+f x) (24 \sec (e+f x) a+35 a)dx+\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \frac {24 \csc \left (e+f x+\frac {\pi }{2}\right ) a+35 a}{\cot \left (e+f x+\frac {\pi }{2}\right )^4}dx+\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int -3 \cot ^2(e+f x) (16 \sec (e+f x) a+35 a)dx-\frac {\cot ^3(e+f x) (24 a \sec (e+f x)+35 a)}{3 f}\right )+\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (-\int \cot ^2(e+f x) (16 \sec (e+f x) a+35 a)dx-\frac {\cot ^3(e+f x) (24 a \sec (e+f x)+35 a)}{3 f}\right )+\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (-\int \frac {16 \csc \left (e+f x+\frac {\pi }{2}\right ) a+35 a}{\cot \left (e+f x+\frac {\pi }{2}\right )^2}dx-\frac {\cot ^3(e+f x) (24 a \sec (e+f x)+35 a)}{3 f}\right )+\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (-\int -35 adx-\frac {\cot ^3(e+f x) (24 a \sec (e+f x)+35 a)}{3 f}+\frac {\cot (e+f x) (16 a \sec (e+f x)+35 a)}{f}\right )+\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {\cot ^5(e+f x) (6 a \sec (e+f x)+7 a)}{5 f}+\frac {1}{5} \left (-\frac {\cot ^3(e+f x) (24 a \sec (e+f x)+35 a)}{3 f}+\frac {\cot (e+f x) (16 a \sec (e+f x)+35 a)}{f}+35 a x\right )\right )-\frac {\cot ^7(e+f x) (a \sec (e+f x)+a)}{7 f}}{a^4 c^4}\) |
(-1/7*(Cot[e + f*x]^7*(a + a*Sec[e + f*x]))/f + ((Cot[e + f*x]^5*(7*a + 6* a*Sec[e + f*x]))/(5*f) + (35*a*x + (Cot[e + f*x]*(35*a + 16*a*Sec[e + f*x] ))/f - (Cot[e + f*x]^3*(35*a + 24*a*Sec[e + f*x]))/(3*f))/5)/7)/(a^4*c^4)
3.1.39.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-(e*Cot[c + d*x])^(m + 1))*((a + b*Csc[c + d*x])/( d*e*(m + 1))), x] - Simp[1/(e^2*(m + 1)) Int[(e*Cot[c + d*x])^(m + 2)*(a* (m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && L tQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[Cot[e + f*x]^(2*m)*( c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !( IntegerQ[n] && GtQ[m - n, 0])
Time = 0.73 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.81
| method | result | size |
| parallelrisch | \(\frac {-15 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-21 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+168 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+280 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-1015 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+6720 f x -3045 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+6720 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )}{6720 f \,a^{3} c^{4}}\) | \(104\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-29 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {8}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {29}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {64}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+128 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f \,c^{4} a^{3}}\) | \(114\) |
| default | \(\frac {-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{5}+\frac {8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3}-29 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {8}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {29}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {64}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+128 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f \,c^{4} a^{3}}\) | \(114\) |
| risch | \(\frac {x}{a^{3} c^{4}}+\frac {2 i \left (105 \,{\mathrm e}^{11 i \left (f x +e \right )}+210 \,{\mathrm e}^{10 i \left (f x +e \right )}-735 \,{\mathrm e}^{9 i \left (f x +e \right )}+1638 \,{\mathrm e}^{7 i \left (f x +e \right )}-196 \,{\mathrm e}^{6 i \left (f x +e \right )}-1882 \,{\mathrm e}^{5 i \left (f x +e \right )}+880 \,{\mathrm e}^{4 i \left (f x +e \right )}+1025 \,{\mathrm e}^{3 i \left (f x +e \right )}-494 \,{\mathrm e}^{2 i \left (f x +e \right )}-247 \,{\mathrm e}^{i \left (f x +e \right )}+176\right )}{105 f \,c^{4} a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5}}\) | \(160\) |
| norman | \(\frac {\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a c f}+\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{c a}-\frac {1}{448 a c f}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{40 a c f}-\frac {29 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{192 a c f}-\frac {29 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{64 a c f}+\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{24 a c f}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{320 a c f}}{a^{2} c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\) | \(181\) |
1/6720*(-15*cot(1/2*f*x+1/2*e)^7-21*tan(1/2*f*x+1/2*e)^5+168*cot(1/2*f*x+1 /2*e)^5+280*tan(1/2*f*x+1/2*e)^3-1015*cot(1/2*f*x+1/2*e)^3+6720*f*x-3045*t an(1/2*f*x+1/2*e)+6720*cot(1/2*f*x+1/2*e))/f/a^3/c^4
Time = 0.25 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.80 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {176 \, \cos \left (f x + e\right )^{6} - 71 \, \cos \left (f x + e\right )^{5} - 335 \, \cos \left (f x + e\right )^{4} + 125 \, \cos \left (f x + e\right )^{3} + 225 \, \cos \left (f x + e\right )^{2} + 105 \, {\left (f x \cos \left (f x + e\right )^{5} - f x \cos \left (f x + e\right )^{4} - 2 \, f x \cos \left (f x + e\right )^{3} + 2 \, f x \cos \left (f x + e\right )^{2} + f x \cos \left (f x + e\right ) - f x\right )} \sin \left (f x + e\right ) - 57 \, \cos \left (f x + e\right ) - 48}{105 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{5} - a^{3} c^{4} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} c^{4} f \cos \left (f x + e\right )^{3} + 2 \, a^{3} c^{4} f \cos \left (f x + e\right )^{2} + a^{3} c^{4} f \cos \left (f x + e\right ) - a^{3} c^{4} f\right )} \sin \left (f x + e\right )} \]
1/105*(176*cos(f*x + e)^6 - 71*cos(f*x + e)^5 - 335*cos(f*x + e)^4 + 125*c os(f*x + e)^3 + 225*cos(f*x + e)^2 + 105*(f*x*cos(f*x + e)^5 - f*x*cos(f*x + e)^4 - 2*f*x*cos(f*x + e)^3 + 2*f*x*cos(f*x + e)^2 + f*x*cos(f*x + e) - f*x)*sin(f*x + e) - 57*cos(f*x + e) - 48)/((a^3*c^4*f*cos(f*x + e)^5 - a^ 3*c^4*f*cos(f*x + e)^4 - 2*a^3*c^4*f*cos(f*x + e)^3 + 2*a^3*c^4*f*cos(f*x + e)^2 + a^3*c^4*f*cos(f*x + e) - a^3*c^4*f)*sin(f*x + e))
\[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {\int \frac {1}{\sec ^{7}{\left (e + f x \right )} - \sec ^{6}{\left (e + f x \right )} - 3 \sec ^{5}{\left (e + f x \right )} + 3 \sec ^{4}{\left (e + f x \right )} + 3 \sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} - \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{4}} \]
Integral(1/(sec(e + f*x)**7 - sec(e + f*x)**6 - 3*sec(e + f*x)**5 + 3*sec( e + f*x)**4 + 3*sec(e + f*x)**3 - 3*sec(e + f*x)**2 - sec(e + f*x) + 1), x )/(a**3*c**4)
Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=-\frac {\frac {7 \, {\left (\frac {435 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {40 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{4}} - \frac {13440 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{4}} - \frac {{\left (\frac {168 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {1015 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {6720 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{a^{3} c^{4} \sin \left (f x + e\right )^{7}}}{6720 \, f} \]
-1/6720*(7*(435*sin(f*x + e)/(cos(f*x + e) + 1) - 40*sin(f*x + e)^3/(cos(f *x + e) + 1)^3 + 3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)/(a^3*c^4) - 13440* arctan(sin(f*x + e)/(cos(f*x + e) + 1))/(a^3*c^4) - (168*sin(f*x + e)^2/(c os(f*x + e) + 1)^2 - 1015*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 6720*sin(f *x + e)^6/(cos(f*x + e) + 1)^6 - 15)*(cos(f*x + e) + 1)^7/(a^3*c^4*sin(f*x + e)^7))/f
Time = 0.40 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=\frac {\frac {6720 \, {\left (f x + e\right )}}{a^{3} c^{4}} + \frac {6720 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1015 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 168 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15}{a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}} - \frac {7 \, {\left (3 \, a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 40 \, a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 435 \, a^{12} c^{16} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{20}}}{6720 \, f} \]
1/6720*(6720*(f*x + e)/(a^3*c^4) + (6720*tan(1/2*f*x + 1/2*e)^6 - 1015*tan (1/2*f*x + 1/2*e)^4 + 168*tan(1/2*f*x + 1/2*e)^2 - 15)/(a^3*c^4*tan(1/2*f* x + 1/2*e)^7) - 7*(3*a^12*c^16*tan(1/2*f*x + 1/2*e)^5 - 40*a^12*c^16*tan(1 /2*f*x + 1/2*e)^3 + 435*a^12*c^16*tan(1/2*f*x + 1/2*e))/(a^15*c^20))/f
Time = 15.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4} \, dx=-\frac {15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+21\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-280\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+3045\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-6720\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+1015\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-168\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-6720\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (e+f\,x\right )}{6720\,a^3\,c^4\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7} \]
-(15*cos(e/2 + (f*x)/2)^12 + 21*sin(e/2 + (f*x)/2)^12 - 280*cos(e/2 + (f*x )/2)^2*sin(e/2 + (f*x)/2)^10 + 3045*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2 )^8 - 6720*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f*x)/2)^6 + 1015*cos(e/2 + (f*x )/2)^8*sin(e/2 + (f*x)/2)^4 - 168*cos(e/2 + (f*x)/2)^10*sin(e/2 + (f*x)/2) ^2 - 6720*cos(e/2 + (f*x)/2)^5*sin(e/2 + (f*x)/2)^7*(e + f*x))/(6720*a^3*c ^4*f*cos(e/2 + (f*x)/2)^5*sin(e/2 + (f*x)/2)^7)