Integrand size = 27, antiderivative size = 149 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a x}{256}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d} \]
3/256*a*x-1/7*a*cos(d*x+c)^7/d+1/9*a*cos(d*x+c)^9/d+3/256*a*cos(d*x+c)*sin (d*x+c)/d+1/128*a*cos(d*x+c)^3*sin(d*x+c)/d+1/160*a*cos(d*x+c)^5*sin(d*x+c )/d-3/80*a*cos(d*x+c)^7*sin(d*x+c)/d-1/10*a*cos(d*x+c)^7*sin(d*x+c)^3/d
Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (7560 d x-15120 \cos (c+d x)-6720 \cos (3 (c+d x))+1080 \cos (7 (c+d x))+280 \cos (9 (c+d x))+1260 \sin (2 (c+d x))-2520 \sin (4 (c+d x))-630 \sin (6 (c+d x))+315 \sin (8 (c+d x))+126 \sin (10 (c+d x)))}{645120 d} \]
(a*(7560*d*x - 15120*Cos[c + d*x] - 6720*Cos[3*(c + d*x)] + 1080*Cos[7*(c + d*x)] + 280*Cos[9*(c + d*x)] + 1260*Sin[2*(c + d*x)] - 2520*Sin[4*(c + d *x)] - 630*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] + 126*Sin[10*(c + d*x)] ))/(645120*d)
Time = 0.77 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3317, 3042, 3045, 244, 2009, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 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.
\(\displaystyle \int \sin ^3(c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^6 (a \sin (c+d x)+a)dx\) |
\(\Big \downarrow \) 3317 |
\(\displaystyle a \int \cos ^6(c+d x) \sin ^4(c+d x)dx+a \int \cos ^6(c+d x) \sin ^3(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^3dx+a \int \cos (c+d x)^6 \sin (c+d x)^4dx\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^4dx-\frac {a \int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^4dx-\frac {a \int \left (\cos ^6(c+d x)-\cos ^8(c+d x)\right )d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^4dx-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle a \left (\frac {3}{10} \int \cos ^6(c+d x) \sin ^2(c+d x)dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {3}{10} \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle a \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {\sin ^3(c+d x) \cos ^7(c+d x)}{10 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
-((a*(Cos[c + d*x]^7/7 - Cos[c + d*x]^9/9))/d) + a*(-1/10*(Cos[c + d*x]^7* Sin[c + d*x]^3)/d + (3*(-1/8*(Cos[c + d*x]^7*Sin[c + d*x])/d + ((Cos[c + d *x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*( x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)/8))/10)
3.6.73.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 0.53 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(-\frac {\left (-3 d x +\sin \left (4 d x +4 c \right )+\frac {\sin \left (6 d x +6 c \right )}{4}-\frac {\sin \left (8 d x +8 c \right )}{8}-\frac {\sin \left (10 d x +10 c \right )}{20}+6 \cos \left (d x +c \right )+\frac {8 \cos \left (3 d x +3 c \right )}{3}-\frac {3 \cos \left (7 d x +7 c \right )}{7}-\frac {\cos \left (9 d x +9 c \right )}{9}-\frac {\sin \left (2 d x +2 c \right )}{2}+\frac {512}{63}\right ) a}{256 d}\) | \(107\) |
derivativedivides | \(\frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\) | \(116\) |
default | \(\frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\) | \(116\) |
risch | \(\frac {3 a x}{256}-\frac {3 a \cos \left (d x +c \right )}{128 d}+\frac {a \sin \left (10 d x +10 c \right )}{5120 d}+\frac {a \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a \sin \left (8 d x +8 c \right )}{2048 d}+\frac {3 a \cos \left (7 d x +7 c \right )}{1792 d}-\frac {a \sin \left (6 d x +6 c \right )}{1024 d}-\frac {a \sin \left (4 d x +4 c \right )}{256 d}-\frac {a \cos \left (3 d x +3 c \right )}{96 d}+\frac {a \sin \left (2 d x +2 c \right )}{512 d}\) | \(138\) |
-1/256*(-3*d*x+sin(4*d*x+4*c)+1/4*sin(6*d*x+6*c)-1/8*sin(8*d*x+8*c)-1/20*s in(10*d*x+10*c)+6*cos(d*x+c)+8/3*cos(3*d*x+3*c)-3/7*cos(7*d*x+7*c)-1/9*cos (9*d*x+9*c)-1/2*sin(2*d*x+2*c)+512/63)*a/d
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.64 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {8960 \, a \cos \left (d x + c\right )^{9} - 11520 \, a \cos \left (d x + c\right )^{7} + 945 \, a d x + 63 \, {\left (128 \, a \cos \left (d x + c\right )^{9} - 176 \, a \cos \left (d x + c\right )^{7} + 8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]
1/80640*(8960*a*cos(d*x + c)^9 - 11520*a*cos(d*x + c)^7 + 945*a*d*x + 63*( 128*a*cos(d*x + c)^9 - 176*a*cos(d*x + c)^7 + 8*a*cos(d*x + c)^5 + 10*a*co s(d*x + c)^3 + 15*a*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (138) = 276\).
Time = 1.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.97 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {3 a x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {7 a \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a*x*sin(c + d*x)**10/256 + 15*a*x*sin(c + d*x)**8*cos(c + d*x )**2/256 + 15*a*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 15*a*x*sin(c + d*x )**4*cos(c + d*x)**6/128 + 15*a*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 3* a*x*cos(c + d*x)**10/256 + 3*a*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a* sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + a*sin(c + d*x)**5*cos(c + d*x)** 5/(10*d) - 7*a*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - a*sin(c + d*x)**2 *cos(c + d*x)**7/(7*d) - 3*a*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 2*a*co s(c + d*x)**9/(63*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**3*cos(c)**6, Tr ue))
Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.51 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {10240 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{645120 \, d} \]
1/645120*(10240*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a + 63*(32*sin(2*d*x + 2*c)^5 + 120*d*x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(4*d*x + 4*c))*a) /d
Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3}{256} \, a x + \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} + \frac {a \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
3/256*a*x + 1/2304*a*cos(9*d*x + 9*c)/d + 3/1792*a*cos(7*d*x + 7*c)/d - 1/ 96*a*cos(3*d*x + 3*c)/d - 3/128*a*cos(d*x + c)/d + 1/5120*a*sin(10*d*x + 1 0*c)/d + 1/2048*a*sin(8*d*x + 8*c)/d - 1/1024*a*sin(6*d*x + 6*c)/d - 1/256 *a*sin(4*d*x + 4*c)/d + 1/512*a*sin(2*d*x + 2*c)/d
Time = 14.31 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.78 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,x}{256}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\left (\frac {a\,\left (42525\,c+42525\,d\,x-322560\right )}{80640}-\frac {135\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {867\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}+\left (\frac {a\,\left (113400\,c+113400\,d\,x+215040\right )}{80640}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {519\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (198450\,c+198450\,d\,x-1075200\right )}{80640}-\frac {315\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {1879\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\left (\frac {a\,\left (238140\,c+238140\,d\,x-645120\right )}{80640}-\frac {189\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1879\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {519\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (113400\,c+113400\,d\,x-829440\right )}{80640}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {867\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\left (\frac {a\,\left (42525\,c+42525\,d\,x+92160\right )}{80640}-\frac {135\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\left (\frac {a\,\left (9450\,c+9450\,d\,x-51200\right )}{80640}-\frac {15\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {a\,\left (945\,c+945\,d\,x-5120\right )}{80640}-\frac {3\,a\,\left (c+d\,x\right )}{256}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
(3*a*x)/256 + ((a*(945*c + 945*d*x - 5120))/80640 - (3*a*tan(c/2 + (d*x)/2 ))/128 - (3*a*(c + d*x))/256 + tan(c/2 + (d*x)/2)^2*((a*(9450*c + 9450*d*x - 51200))/80640 - (15*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^4*((a*(42525 *c + 42525*d*x + 92160))/80640 - (135*a*(c + d*x))/256) + tan(c/2 + (d*x)/ 2)^16*((a*(42525*c + 42525*d*x - 322560))/80640 - (135*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^14*((a*(113400*c + 113400*d*x + 215040))/80640 - (45*a *(c + d*x))/32) + tan(c/2 + (d*x)/2)^6*((a*(113400*c + 113400*d*x - 829440 ))/80640 - (45*a*(c + d*x))/32) + tan(c/2 + (d*x)/2)^10*((a*(238140*c + 23 8140*d*x - 645120))/80640 - (189*a*(c + d*x))/64) + tan(c/2 + (d*x)/2)^12* ((a*(198450*c + 198450*d*x - 1075200))/80640 - (315*a*(c + d*x))/128) - (2 9*a*tan(c/2 + (d*x)/2)^3)/128 + (867*a*tan(c/2 + (d*x)/2)^5)/160 - (519*a* tan(c/2 + (d*x)/2)^7)/32 + (1879*a*tan(c/2 + (d*x)/2)^9)/64 - (1879*a*tan( c/2 + (d*x)/2)^11)/64 + (519*a*tan(c/2 + (d*x)/2)^13)/32 - (867*a*tan(c/2 + (d*x)/2)^15)/160 + (29*a*tan(c/2 + (d*x)/2)^17)/128 + (3*a*tan(c/2 + (d* x)/2)^19)/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^10)