Integrand size = 19, antiderivative size = 140 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-a x-\frac {\cot ^9(c+d x) (a+a \sec (c+d x))}{9 d}+\frac {\cot ^7(c+d x) (9 a+8 a \sec (c+d x))}{63 d}-\frac {\cot ^5(c+d x) (21 a+16 a \sec (c+d x))}{105 d}+\frac {\cot ^3(c+d x) (105 a+64 a \sec (c+d x))}{315 d}-\frac {\cot (c+d x) (315 a+128 a \sec (c+d x))}{315 d} \]
-a*x-1/9*cot(d*x+c)^9*(a+a*sec(d*x+c))/d+1/63*cot(d*x+c)^7*(9*a+8*a*sec(d* x+c))/d-1/105*cot(d*x+c)^5*(21*a+16*a*sec(d*x+c))/d+1/315*cot(d*x+c)^3*(10 5*a+64*a*sec(d*x+c))/d-1/315*cot(d*x+c)*(315*a+128*a*sec(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.79 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a \csc (c+d x)}{d}+\frac {4 a \csc ^3(c+d x)}{3 d}-\frac {6 a \csc ^5(c+d x)}{5 d}+\frac {4 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d}-\frac {a \cot ^9(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},-\tan ^2(c+d x)\right )}{9 d} \]
-((a*Csc[c + d*x])/d) + (4*a*Csc[c + d*x]^3)/(3*d) - (6*a*Csc[c + d*x]^5)/ (5*d) + (4*a*Csc[c + d*x]^7)/(7*d) - (a*Csc[c + d*x]^9)/(9*d) - (a*Cot[c + d*x]^9*Hypergeometric2F1[-9/2, 1, -7/2, -Tan[c + d*x]^2])/(9*d)
Time = 0.74 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {3042, 4370, 25, 3042, 4370, 27, 3042, 4370, 25, 3042, 4370, 25, 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.
| \(\displaystyle \int \cot ^{10}(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}{\cot \left (c+d x+\frac {\pi }{2}\right )^{10}}dx\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {1}{9} \int -\cot ^8(c+d x) (8 \sec (c+d x) a+9 a)dx-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{9} \int \cot ^8(c+d x) (8 \sec (c+d x) a+9 a)dx-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{9} \int \frac {8 \csc \left (c+d x+\frac {\pi }{2}\right ) a+9 a}{\cot \left (c+d x+\frac {\pi }{2}\right )^8}dx-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {1}{9} \left (\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}-\frac {1}{7} \int -3 \cot ^6(c+d x) (16 \sec (c+d x) a+21 a)dx\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \int \cot ^6(c+d x) (16 \sec (c+d x) a+21 a)dx+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \int \frac {16 \csc \left (c+d x+\frac {\pi }{2}\right ) a+21 a}{\cot \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \int -\cot ^4(c+d x) (64 \sec (c+d x) a+105 a)dx-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \left (-\frac {1}{5} \int \cot ^4(c+d x) (64 \sec (c+d x) a+105 a)dx-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \left (-\frac {1}{5} \int \frac {64 \csc \left (c+d x+\frac {\pi }{2}\right ) a+105 a}{\cot \left (c+d x+\frac {\pi }{2}\right )^4}dx-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{3 d}-\frac {1}{3} \int -\cot ^2(c+d x) (128 \sec (c+d x) a+315 a)dx\right )-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \cot ^2(c+d x) (128 \sec (c+d x) a+315 a)dx+\frac {\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{3 d}\right )-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {128 \csc \left (c+d x+\frac {\pi }{2}\right ) a+315 a}{\cot \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{3 d}\right )-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle \frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int -315 adx-\frac {\cot (c+d x) (128 a \sec (c+d x)+315 a)}{d}\right )+\frac {\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{3 d}\right )-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )+\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{9} \left (\frac {\cot ^7(c+d x) (8 a \sec (c+d x)+9 a)}{7 d}+\frac {3}{7} \left (\frac {1}{5} \left (\frac {\cot ^3(c+d x) (64 a \sec (c+d x)+105 a)}{3 d}+\frac {1}{3} \left (-\frac {\cot (c+d x) (128 a \sec (c+d x)+315 a)}{d}-315 a x\right )\right )-\frac {\cot ^5(c+d x) (16 a \sec (c+d x)+21 a)}{5 d}\right )\right )-\frac {\cot ^9(c+d x) (a \sec (c+d x)+a)}{9 d}\) |
-1/9*(Cot[c + d*x]^9*(a + a*Sec[c + d*x]))/d + ((Cot[c + d*x]^7*(9*a + 8*a *Sec[c + d*x]))/(7*d) + (3*(-1/5*(Cot[c + d*x]^5*(21*a + 16*a*Sec[c + d*x] ))/d + ((Cot[c + d*x]^3*(105*a + 64*a*Sec[c + d*x]))/(3*d) + (-315*a*x - ( Cot[c + d*x]*(315*a + 128*a*Sec[c + d*x]))/d)/3)/5))/7)/9
3.1.18.3.1 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]
Time = 1.78 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.46
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{10}}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a \left (-\frac {\cot \left (d x +c \right )^{9}}{9}+\frac {\cot \left (d x +c \right )^{7}}{7}-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(205\) |
| default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{10}}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a \left (-\frac {\cot \left (d x +c \right )^{9}}{9}+\frac {\cot \left (d x +c \right )^{7}}{7}-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(205\) |
| risch | \(-a x -\frac {2 i a \left (315 \,{\mathrm e}^{15 i \left (d x +c \right )}+945 \,{\mathrm e}^{14 i \left (d x +c \right )}-3045 \,{\mathrm e}^{13 i \left (d x +c \right )}-1155 \,{\mathrm e}^{12 i \left (d x +c \right )}+10143 \,{\mathrm e}^{11 i \left (d x +c \right )}+1869 \,{\mathrm e}^{10 i \left (d x +c \right )}-18993 \,{\mathrm e}^{9 i \left (d x +c \right )}+4617 \,{\mathrm e}^{8 i \left (d x +c \right )}+20417 \,{\mathrm e}^{7 i \left (d x +c \right )}-6013 \,{\mathrm e}^{6 i \left (d x +c \right )}-13503 \,{\mathrm e}^{5 i \left (d x +c \right )}+6727 \,{\mathrm e}^{4 i \left (d x +c \right )}+4837 \,{\mathrm e}^{3 i \left (d x +c \right )}-2433 \,{\mathrm e}^{2 i \left (d x +c \right )}-811 \,{\mathrm e}^{i \left (d x +c \right )}+563\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(206\) |
1/d*(a*(-1/9/sin(d*x+c)^9*cos(d*x+c)^10+1/63/sin(d*x+c)^7*cos(d*x+c)^10-1/ 105/sin(d*x+c)^5*cos(d*x+c)^10+1/63/sin(d*x+c)^3*cos(d*x+c)^10-1/9/sin(d*x +c)*cos(d*x+c)^10-1/9*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+ c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c))+a*(-1/9*cot(d*x+c)^9+1/7*cot(d*x+c)^7 -1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c))
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (130) = 260\).
Time = 0.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.99 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {563 \, a \cos \left (d x + c\right )^{8} - 248 \, a \cos \left (d x + c\right )^{7} - 1498 \, a \cos \left (d x + c\right )^{6} + 658 \, a \cos \left (d x + c\right )^{5} + 1610 \, a \cos \left (d x + c\right )^{4} - 602 \, a \cos \left (d x + c\right )^{3} - 763 \, a \cos \left (d x + c\right )^{2} + 187 \, a \cos \left (d x + c\right ) + 315 \, {\left (a d x \cos \left (d x + c\right )^{7} - a d x \cos \left (d x + c\right )^{6} - 3 \, a d x \cos \left (d x + c\right )^{5} + 3 \, a d x \cos \left (d x + c\right )^{4} + 3 \, a d x \cos \left (d x + c\right )^{3} - 3 \, a d x \cos \left (d x + c\right )^{2} - a d x \cos \left (d x + c\right ) + a d x\right )} \sin \left (d x + c\right ) + 128 \, a}{315 \, {\left (d \cos \left (d x + c\right )^{7} - d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} + 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \]
-1/315*(563*a*cos(d*x + c)^8 - 248*a*cos(d*x + c)^7 - 1498*a*cos(d*x + c)^ 6 + 658*a*cos(d*x + c)^5 + 1610*a*cos(d*x + c)^4 - 602*a*cos(d*x + c)^3 - 763*a*cos(d*x + c)^2 + 187*a*cos(d*x + c) + 315*(a*d*x*cos(d*x + c)^7 - a* d*x*cos(d*x + c)^6 - 3*a*d*x*cos(d*x + c)^5 + 3*a*d*x*cos(d*x + c)^4 + 3*a *d*x*cos(d*x + c)^3 - 3*a*d*x*cos(d*x + c)^2 - a*d*x*cos(d*x + c) + a*d*x) *sin(d*x + c) + 128*a)/((d*cos(d*x + c)^7 - d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^5 + 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 - d *cos(d*x + c) + d)*sin(d*x + c))
Timed out. \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.85 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {{\left (315 \, d x + 315 \, c + \frac {315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a + \frac {{\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a}{\sin \left (d x + c\right )^{9}}}{315 \, d} \]
-1/315*((315*d*x + 315*c + (315*tan(d*x + c)^8 - 105*tan(d*x + c)^6 + 63*t an(d*x + c)^4 - 45*tan(d*x + c)^2 + 35)/tan(d*x + c)^9)*a + (315*sin(d*x + c)^8 - 420*sin(d*x + c)^6 + 378*sin(d*x + c)^4 - 180*sin(d*x + c)^2 + 35) *a/sin(d*x + c)^9)/d
Time = 0.43 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4830 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80640 \, {\left (d x + c\right )} a - 40950 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {80640 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 13650 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2898 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 450 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{80640 \, d} \]
-1/80640*(45*a*tan(1/2*d*x + 1/2*c)^7 - 630*a*tan(1/2*d*x + 1/2*c)^5 + 483 0*a*tan(1/2*d*x + 1/2*c)^3 + 80640*(d*x + c)*a - 40950*a*tan(1/2*d*x + 1/2 *c) + (80640*a*tan(1/2*d*x + 1/2*c)^8 - 13650*a*tan(1/2*d*x + 1/2*c)^6 + 2 898*a*tan(1/2*d*x + 1/2*c)^4 - 450*a*tan(1/2*d*x + 1/2*c)^2 + 35*a)/tan(1/ 2*d*x + 1/2*c)^9)/d
Time = 15.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.80 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a\,\left (35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+45\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4830\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-40950\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+80640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-13650\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2898\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-450\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+80640\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )\right )}{80640\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
-(a*(35*cos(c/2 + (d*x)/2)^16 + 45*sin(c/2 + (d*x)/2)^16 - 630*cos(c/2 + ( d*x)/2)^2*sin(c/2 + (d*x)/2)^14 + 4830*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x )/2)^12 - 40950*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 80640*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8 - 13650*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 + 2898*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 - 450*cos(c/ 2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 + 80640*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^9*(c + d*x)))/(80640*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^ 9)