Integrand size = 29, antiderivative size = 182 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {21 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \]
21/256*a^3*x-4/5*a^3*cos(d*x+c)^5/d+a^3*cos(d*x+c)^7/d-1/3*a^3*cos(d*x+c)^ 9/d+21/256*a^3*cos(d*x+c)*sin(d*x+c)/d+7/128*a^3*cos(d*x+c)^3*sin(d*x+c)/d -7/32*a^3*cos(d*x+c)^5*sin(d*x+c)/d-7/16*a^3*cos(d*x+c)^5*sin(d*x+c)^3/d-1 /10*a^3*cos(d*x+c)^5*sin(d*x+c)^5/d
Time = 0.58 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (2700 c+2520 d x-3600 \cos (c+d x)-960 \cos (3 (c+d x))+384 \cos (5 (c+d x))+120 \cos (7 (c+d x))-40 \cos (9 (c+d x))-60 \sin (2 (c+d x))-840 \sin (4 (c+d x))+30 \sin (6 (c+d x))+105 \sin (8 (c+d x))-6 \sin (10 (c+d x)))}{30720 d} \]
(a^3*(2700*c + 2520*d*x - 3600*Cos[c + d*x] - 960*Cos[3*(c + d*x)] + 384*C os[5*(c + d*x)] + 120*Cos[7*(c + d*x)] - 40*Cos[9*(c + d*x)] - 60*Sin[2*(c + d*x)] - 840*Sin[4*(c + d*x)] + 30*Sin[6*(c + d*x)] + 105*Sin[8*(c + d*x )] - 6*Sin[10*(c + d*x)]))/(30720*d)
Time = 0.56 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 2009}
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.
| \(\displaystyle \int \sin ^3(c+d x) \cos ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^4 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \sin ^6(c+d x) \cos ^4(c+d x)+3 a^3 \sin ^5(c+d x) \cos ^4(c+d x)+3 a^3 \sin ^4(c+d x) \cos ^4(c+d x)+a^3 \sin ^3(c+d x) \cos ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {21 a^3 x}{256}\) |
(21*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (a^3*Cos[c + d*x]^7)/d - ( a^3*Cos[c + d*x]^9)/(3*d) + (21*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + ( 7*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (7*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (7*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)
3.4.93.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Time = 0.67 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.66
| method | result | size |
| parallelrisch | \(-\frac {\left (-42 d x +\sin \left (2 d x +2 c \right )+14 \sin \left (4 d x +4 c \right )-\frac {\sin \left (6 d x +6 c \right )}{2}-\frac {7 \sin \left (8 d x +8 c \right )}{4}+\frac {\sin \left (10 d x +10 c \right )}{10}+60 \cos \left (d x +c \right )+16 \cos \left (3 d x +3 c \right )-\frac {32 \cos \left (5 d x +5 c \right )}{5}-2 \cos \left (7 d x +7 c \right )+\frac {2 \cos \left (9 d x +9 c \right )}{3}+\frac {1024}{15}\right ) a^{3}}{512 d}\) | \(120\) |
| risch | \(\frac {21 a^{3} x}{256}-\frac {15 a^{3} \cos \left (d x +c \right )}{128 d}-\frac {a^{3} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {a^{3} \cos \left (9 d x +9 c \right )}{768 d}+\frac {7 a^{3} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {a^{3} \cos \left (7 d x +7 c \right )}{256 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {a^{3} \cos \left (5 d x +5 c \right )}{80 d}-\frac {7 a^{3} \sin \left (4 d x +4 c \right )}{256 d}-\frac {a^{3} \cos \left (3 d x +3 c \right )}{32 d}-\frac {a^{3} \sin \left (2 d x +2 c \right )}{512 d}\) | \(175\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(252\) |
| default | \(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(252\) |
-1/512*(-42*d*x+sin(2*d*x+2*c)+14*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)-7/4*si n(8*d*x+8*c)+1/10*sin(10*d*x+10*c)+60*cos(d*x+c)+16*cos(3*d*x+3*c)-32/5*co s(5*d*x+5*c)-2*cos(7*d*x+7*c)+2/3*cos(9*d*x+9*c)+1024/15)*a^3/d
Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.68 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1280 \, a^{3} \cos \left (d x + c\right )^{9} - 3840 \, a^{3} \cos \left (d x + c\right )^{7} + 3072 \, a^{3} \cos \left (d x + c\right )^{5} - 315 \, a^{3} d x + 3 \, {\left (128 \, a^{3} \cos \left (d x + c\right )^{9} - 816 \, a^{3} \cos \left (d x + c\right )^{7} + 968 \, a^{3} \cos \left (d x + c\right )^{5} - 70 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3840 \, d} \]
-1/3840*(1280*a^3*cos(d*x + c)^9 - 3840*a^3*cos(d*x + c)^7 + 3072*a^3*cos( d*x + c)^5 - 315*a^3*d*x + 3*(128*a^3*cos(d*x + c)^9 - 816*a^3*cos(d*x + c )^7 + 968*a^3*cos(d*x + c)^5 - 70*a^3*cos(d*x + c)^3 - 105*a^3*cos(d*x + c ))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (172) = 344\).
Time = 1.37 (sec) , antiderivative size = 595, normalized size of antiderivative = 3.27 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {9 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {27 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {9 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {33 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {33 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {12 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \cos ^{9}{\left (c + d x \right )}}{105 d} - \frac {2 a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{3}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a**3*x*sin(c + d*x)**10/256 + 15*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 9*a**3*x*sin(c + d*x)**8/128 + 15*a**3*x*sin(c + d*x)**6* cos(c + d*x)**4/128 + 9*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*a** 3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 27*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 9*a**3*x*si n(c + d*x)**2*cos(c + d*x)**6/32 + 3*a**3*x*cos(c + d*x)**10/256 + 9*a**3* x*cos(c + d*x)**8/128 + 3*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a* *3*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + 9*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) - a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 33*a**3*sin (c + d*x)**5*cos(c + d*x)**3/(128*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x) **5/(5*d) - 7*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 33*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 12*a**3*sin(c + d*x)**2*cos(c + d*x)* *7/(35*d) - a**3*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*a**3*sin(c + d* x)*cos(c + d*x)**9/(256*d) - 9*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 8*a**3*cos(c + d*x)**9/(105*d) - 2*a**3*cos(c + d*x)**7/(35*d), Ne(d, 0)) , (x*(a*sin(c) + a)**3*sin(c)**3*cos(c)**4, True))
Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2048 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 6144 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} + 21 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 630 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{215040 \, d} \]
-1/215040*(2048*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5 )*a^3 - 6144*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^3 + 21*(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^3 - 630*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^3)/d
Time = 0.62 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21}{256} \, a^{3} x - \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{768 \, d} + \frac {a^{3} \cos \left (7 \, d x + 7 \, c\right )}{256 \, d} + \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a^{3} \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {15 \, a^{3} \cos \left (d x + c\right )}{128 \, d} - \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {7 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
21/256*a^3*x - 1/768*a^3*cos(9*d*x + 9*c)/d + 1/256*a^3*cos(7*d*x + 7*c)/d + 1/80*a^3*cos(5*d*x + 5*c)/d - 1/32*a^3*cos(3*d*x + 3*c)/d - 15/128*a^3* cos(d*x + c)/d - 1/5120*a^3*sin(10*d*x + 10*c)/d + 7/2048*a^3*sin(8*d*x + 8*c)/d + 1/1024*a^3*sin(6*d*x + 6*c)/d - 7/256*a^3*sin(4*d*x + 4*c)/d - 1/ 512*a^3*sin(2*d*x + 2*c)/d
Time = 11.89 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.14 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21\,a^3\,x}{256}-\frac {\frac {203\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}-\frac {1973\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {463\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {3231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {3231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {463\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {1973\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {203\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {a^3\,\left (315\,c+315\,d\,x\right )}{3840}-\frac {a^3\,\left (315\,c+315\,d\,x-1024\right )}{3840}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{384}-\frac {a^3\,\left (3150\,c+3150\,d\,x\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{384}-\frac {a^3\,\left (3150\,c+3150\,d\,x-10240\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{256}-\frac {a^3\,\left (14175\,c+14175\,d\,x-15360\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{256}-\frac {a^3\,\left (14175\,c+14175\,d\,x-30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^3\,\left (37800\,c+37800\,d\,x+30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {7\,a^3\,\left (315\,c+315\,d\,x\right )}{128}-\frac {a^3\,\left (66150\,c+66150\,d\,x+30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^3\,\left (37800\,c+37800\,d\,x-153600\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {21\,a^3\,\left (315\,c+315\,d\,x\right )}{320}-\frac {a^3\,\left (79380\,c+79380\,d\,x-129024\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {7\,a^3\,\left (315\,c+315\,d\,x\right )}{128}-\frac {a^3\,\left (66150\,c+66150\,d\,x-245760\right )}{3840}\right )+\frac {21\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]
(21*a^3*x)/256 - ((203*a^3*tan(c/2 + (d*x)/2)^3)/128 - (1973*a^3*tan(c/2 + (d*x)/2)^5)/160 - (463*a^3*tan(c/2 + (d*x)/2)^7)/32 + (3231*a^3*tan(c/2 + (d*x)/2)^9)/64 - (3231*a^3*tan(c/2 + (d*x)/2)^11)/64 + (463*a^3*tan(c/2 + (d*x)/2)^13)/32 + (1973*a^3*tan(c/2 + (d*x)/2)^15)/160 - (203*a^3*tan(c/2 + (d*x)/2)^17)/128 - (21*a^3*tan(c/2 + (d*x)/2)^19)/128 + (a^3*(315*c + 3 15*d*x))/3840 - (a^3*(315*c + 315*d*x - 1024))/3840 + tan(c/2 + (d*x)/2)^1 8*((a^3*(315*c + 315*d*x))/384 - (a^3*(3150*c + 3150*d*x))/3840) + tan(c/2 + (d*x)/2)^2*((a^3*(315*c + 315*d*x))/384 - (a^3*(3150*c + 3150*d*x - 102 40))/3840) + tan(c/2 + (d*x)/2)^16*((3*a^3*(315*c + 315*d*x))/256 - (a^3*( 14175*c + 14175*d*x - 15360))/3840) + tan(c/2 + (d*x)/2)^4*((3*a^3*(315*c + 315*d*x))/256 - (a^3*(14175*c + 14175*d*x - 30720))/3840) + tan(c/2 + (d *x)/2)^6*((a^3*(315*c + 315*d*x))/32 - (a^3*(37800*c + 37800*d*x + 30720)) /3840) + tan(c/2 + (d*x)/2)^12*((7*a^3*(315*c + 315*d*x))/128 - (a^3*(6615 0*c + 66150*d*x + 30720))/3840) + tan(c/2 + (d*x)/2)^14*((a^3*(315*c + 315 *d*x))/32 - (a^3*(37800*c + 37800*d*x - 153600))/3840) + tan(c/2 + (d*x)/2 )^10*((21*a^3*(315*c + 315*d*x))/320 - (a^3*(79380*c + 79380*d*x - 129024) )/3840) + tan(c/2 + (d*x)/2)^8*((7*a^3*(315*c + 315*d*x))/128 - (a^3*(6615 0*c + 66150*d*x - 245760))/3840) + (21*a^3*tan(c/2 + (d*x)/2))/128)/(d*(ta n(c/2 + (d*x)/2)^2 + 1)^10)