Integrand size = 29, antiderivative size = 159 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 a^2 x}{64}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d} \]
3/64*a^2*x-2/5*a^2*cos(d*x+c)^5/d+3/7*a^2*cos(d*x+c)^7/d-1/9*a^2*cos(d*x+c )^9/d+3/64*a^2*cos(d*x+c)*sin(d*x+c)/d+1/32*a^2*cos(d*x+c)^3*sin(d*x+c)/d- 1/8*a^2*cos(d*x+c)^5*sin(d*x+c)/d-1/4*a^2*cos(d*x+c)^5*sin(d*x+c)^3/d
Time = 0.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.54 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (7560 c+7560 d x-11340 \cos (c+d x)-3360 \cos (3 (c+d x))+1008 \cos (5 (c+d x))+450 \cos (7 (c+d x))-70 \cos (9 (c+d x))-2520 \sin (4 (c+d x))+315 \sin (8 (c+d x)))}{161280 d} \]
(a^2*(7560*c + 7560*d*x - 11340*Cos[c + d*x] - 3360*Cos[3*(c + d*x)] + 100 8*Cos[5*(c + d*x)] + 450*Cos[7*(c + d*x)] - 70*Cos[9*(c + d*x)] - 2520*Sin [4*(c + d*x)] + 315*Sin[8*(c + d*x)]))/(161280*d)
Time = 0.46 (sec) , antiderivative size = 159, 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.
| \(\displaystyle \int \sin ^3(c+d x) \cos ^4(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^3 \cos (c+d x)^4 (a \sin (c+d x)+a)^2dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \sin ^5(c+d x) \cos ^4(c+d x)+2 a^2 \sin ^4(c+d x) \cos ^4(c+d x)+a^2 \sin ^3(c+d x) \cos ^4(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{4 d}-\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {3 a^2 x}{64}\) |
(3*a^2*x)/64 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (3*a^2*Cos[c + d*x]^7)/(7*d) - (a^2*Cos[c + d*x]^9)/(9*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(32*d) - (a^2*Cos[c + d*x]^5*Sin[c + d* x])/(8*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(4*d)
3.4.79.3.1 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 = 0.59 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(-\frac {a^{2} \left (-7560 d x +11340 \cos \left (d x +c \right )+2520 \sin \left (4 d x +4 c \right )-315 \sin \left (8 d x +8 c \right )-1008 \cos \left (5 d x +5 c \right )-450 \cos \left (7 d x +7 c \right )+3360 \cos \left (3 d x +3 c \right )+70 \cos \left (9 d x +9 c \right )+13312\right )}{161280 d}\) | \(89\) |
| risch | \(\frac {3 a^{2} x}{64}-\frac {9 a^{2} \cos \left (d x +c \right )}{128 d}-\frac {a^{2} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a^{2} \sin \left (8 d x +8 c \right )}{512 d}+\frac {5 a^{2} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {a^{2} \cos \left (5 d x +5 c \right )}{160 d}-\frac {a^{2} \sin \left (4 d x +4 c \right )}{64 d}-\frac {a^{2} \cos \left (3 d x +3 c \right )}{48 d}\) | \(124\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+2 a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(162\) |
| default | \(\frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+2 a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(162\) |
| norman | \(\frac {-\frac {52 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {27 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {189 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {63 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {44 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {27 a^{2} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {27 a^{2} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {3 a^{2} x \left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {4 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a^{2} x}{64}-\frac {52 a^{2}}{315 d}-\frac {155 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {169 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {3 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d}+\frac {27 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {63 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {189 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {3 a^{2} \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {12 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {13 a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {169 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {68 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {4 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {13 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {155 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {164 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(468\) |
-1/161280*a^2*(-7560*d*x+11340*cos(d*x+c)+2520*sin(4*d*x+4*c)-315*sin(8*d* x+8*c)-1008*cos(5*d*x+5*c)-450*cos(7*d*x+7*c)+3360*cos(3*d*x+3*c)+70*cos(9 *d*x+9*c)+13312)/d
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.70 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2240 \, a^{2} \cos \left (d x + c\right )^{9} - 8640 \, a^{2} \cos \left (d x + c\right )^{7} + 8064 \, a^{2} \cos \left (d x + c\right )^{5} - 945 \, a^{2} d x - 315 \, {\left (16 \, a^{2} \cos \left (d x + c\right )^{7} - 24 \, a^{2} \cos \left (d x + c\right )^{5} + 2 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20160 \, d} \]
-1/20160*(2240*a^2*cos(d*x + c)^9 - 8640*a^2*cos(d*x + c)^7 + 8064*a^2*cos (d*x + c)^5 - 945*a^2*d*x - 315*(16*a^2*cos(d*x + c)^7 - 24*a^2*cos(d*x + c)^5 + 2*a^2*cos(d*x + c)^3 + 3*a^2*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (146) = 292\).
Time = 0.92 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.11 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {3 a^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {3 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {3 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {11 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} - \frac {a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {11 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} - \frac {4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {8 a^{2} \cos ^{9}{\left (c + d x \right )}}{315 d} - \frac {2 a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a**2*x*sin(c + d*x)**8/64 + 3*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 9*a**2*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 3*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 3*a**2*x*cos(c + d*x)**8/64 + 3*a**2*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 11*a**2*sin(c + d*x)**5*cos(c + d*x)**3/(6 4*d) - a**2*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 11*a**2*sin(c + d*x)** 3*cos(c + d*x)**5/(64*d) - 4*a**2*sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - a**2*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**7/(64*d) - 8*a**2*cos(c + d*x)**9/(315*d) - 2*a**2*cos(c + d*x)**7/( 35*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*sin(c)**3*cos(c)**4, True))
Time = 0.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {512 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{2} - 4608 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} - 315 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{161280 \, d} \]
-1/161280*(512*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5) *a^2 - 4608*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^2 - 315*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^2)/d
Time = 0.53 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.77 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3}{64} \, a^{2} x - \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {5 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a^{2} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {9 \, a^{2} \cos \left (d x + c\right )}{128 \, d} + \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \]
3/64*a^2*x - 1/2304*a^2*cos(9*d*x + 9*c)/d + 5/1792*a^2*cos(7*d*x + 7*c)/d + 1/160*a^2*cos(5*d*x + 5*c)/d - 1/48*a^2*cos(3*d*x + 3*c)/d - 9/128*a^2* cos(d*x + c)/d + 1/512*a^2*sin(8*d*x + 8*c)/d - 1/64*a^2*sin(4*d*x + 4*c)/ d
Time = 12.75 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.75 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3\,a^2\,x}{64}-\frac {\frac {3\,a^2\,\left (c+d\,x\right )}{64}+\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16}-\frac {155\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {169\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}-\frac {169\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{16}+\frac {155\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}-\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{16}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{32}-\frac {a^2\,\left (945\,c+945\,d\,x-3328\right )}{20160}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {27\,a^2\,\left (c+d\,x\right )}{64}-\frac {a^2\,\left (8505\,c+8505\,d\,x-29952\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {27\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (34020\,c+34020\,d\,x-39168\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {27\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (34020\,c+34020\,d\,x-80640\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {63\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (79380\,c+79380\,d\,x+16128\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {63\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (79380\,c+79380\,d\,x-295680\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {189\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (119070\,c+119070\,d\,x+241920\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {189\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (119070\,c+119070\,d\,x-661248\right )}{20160}\right )+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
(3*a^2*x)/64 - ((3*a^2*(c + d*x))/64 + (13*a^2*tan(c/2 + (d*x)/2)^3)/16 - (155*a^2*tan(c/2 + (d*x)/2)^5)/16 + (169*a^2*tan(c/2 + (d*x)/2)^7)/16 - (1 69*a^2*tan(c/2 + (d*x)/2)^11)/16 + (155*a^2*tan(c/2 + (d*x)/2)^13)/16 - (1 3*a^2*tan(c/2 + (d*x)/2)^15)/16 - (3*a^2*tan(c/2 + (d*x)/2)^17)/32 - (a^2* (945*c + 945*d*x - 3328))/20160 + tan(c/2 + (d*x)/2)^2*((27*a^2*(c + d*x)) /64 - (a^2*(8505*c + 8505*d*x - 29952))/20160) + tan(c/2 + (d*x)/2)^4*((27 *a^2*(c + d*x))/16 - (a^2*(34020*c + 34020*d*x - 39168))/20160) + tan(c/2 + (d*x)/2)^14*((27*a^2*(c + d*x))/16 - (a^2*(34020*c + 34020*d*x - 80640)) /20160) + tan(c/2 + (d*x)/2)^6*((63*a^2*(c + d*x))/16 - (a^2*(79380*c + 79 380*d*x + 16128))/20160) + tan(c/2 + (d*x)/2)^12*((63*a^2*(c + d*x))/16 - (a^2*(79380*c + 79380*d*x - 295680))/20160) + tan(c/2 + (d*x)/2)^10*((189* a^2*(c + d*x))/32 - (a^2*(119070*c + 119070*d*x + 241920))/20160) + tan(c/ 2 + (d*x)/2)^8*((189*a^2*(c + d*x))/32 - (a^2*(119070*c + 119070*d*x - 661 248))/20160) + (3*a^2*tan(c/2 + (d*x)/2))/32)/(d*(tan(c/2 + (d*x)/2)^2 + 1 )^9)