Integrand size = 29, antiderivative size = 132 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=3 a^3 x+\frac {3 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {11 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d} \]
3*a^3*x+3/8*a^3*arctanh(cos(d*x+c))/d-a^3*cos(d*x+c)/d+3*a^3*cot(d*x+c)/d- a^3*cot(d*x+c)^3/d-1/5*a^3*cot(d*x+c)^5/d+11/8*a^3*cot(d*x+c)*csc(d*x+c)/d -3/4*a^3*cot(d*x+c)*csc(d*x+c)^3/d
Time = 1.01 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.64 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (960 c+960 d x-320 \cos (c+d x)+608 \cot \left (\frac {1}{2} (c+d x)\right )+110 \csc ^2\left (\frac {1}{2} (c+d x)\right )-15 \csc ^4\left (\frac {1}{2} (c+d x)\right )+120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-110 \sec ^2\left (\frac {1}{2} (c+d x)\right )+15 \sec ^4\left (\frac {1}{2} (c+d x)\right )+208 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+64 \csc ^5(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )-13 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-\csc ^6\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-608 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{320 d} \]
(a^3*(960*c + 960*d*x - 320*Cos[c + d*x] + 608*Cot[(c + d*x)/2] + 110*Csc[ (c + d*x)/2]^2 - 15*Csc[(c + d*x)/2]^4 + 120*Log[Cos[(c + d*x)/2]] - 120*L og[Sin[(c + d*x)/2]] - 110*Sec[(c + d*x)/2]^2 + 15*Sec[(c + d*x)/2]^4 + 20 8*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 64*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - 13*Csc[(c + d*x)/2]^4*Sin[c + d*x] - Csc[(c + d*x)/2]^6*Sin[c + d*x] - 608*Tan[(c + d*x)/2]))/(320*d)
Time = 0.43 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3351, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^6(c+d x) a^7+3 \csc ^5(c+d x) a^7+\csc ^4(c+d x) a^7-5 \csc ^3(c+d x) a^7-5 \csc ^2(c+d x) a^7+\csc (c+d x) a^7+\sin (c+d x) a^7+3 a^7\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^7 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^7 \cos (c+d x)}{d}-\frac {a^7 \cot ^5(c+d x)}{5 d}-\frac {a^7 \cot ^3(c+d x)}{d}+\frac {3 a^7 \cot (c+d x)}{d}-\frac {3 a^7 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {11 a^7 \cot (c+d x) \csc (c+d x)}{8 d}+3 a^7 x}{a^4}\) |
(3*a^7*x + (3*a^7*ArcTanh[Cos[c + d*x]])/(8*d) - (a^7*Cos[c + d*x])/d + (3 *a^7*Cot[c + d*x])/d - (a^7*Cot[c + d*x]^3)/d - (a^7*Cot[c + d*x]^5)/(5*d) + (11*a^7*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (3*a^7*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d))/a^4
3.5.1.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Time = 0.41 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.36
| method | result | size |
| parallelrisch | \(\frac {15 \left (\csc ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\left (\sin \left (3 d x +3 c \right )-\frac {\sin \left (5 d x +5 c \right )}{5}-2 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-8 d x -2\right ) \sin \left (3 d x +3 c \right )+\left (\frac {8 d x}{5}+\frac {2}{5}\right ) \sin \left (5 d x +5 c \right )-\frac {24 \cos \left (3 d x +3 c \right )}{5}+\frac {152 \cos \left (5 d x +5 c \right )}{75}-\frac {8 \sin \left (2 d x +2 c \right )}{5}-\frac {2 \sin \left (4 d x +4 c \right )}{5}-\frac {4 \sin \left (6 d x +6 c \right )}{15}+\left (16 d x +4\right ) \sin \left (d x +c \right )+\frac {16 \cos \left (d x +c \right )}{15}\right ) \left (\sec ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{4096 d}\) | \(180\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(190\) |
| default | \(\frac {a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{3} \left (-\frac {\cos ^{5}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{5}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 \sin \left (d x +c \right )^{5}}}{d}\) | \(190\) |
| risch | \(3 a^{3} x -\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {a^{3} \left (-200 i {\mathrm e}^{8 i \left (d x +c \right )}+55 \,{\mathrm e}^{9 i \left (d x +c \right )}+720 i {\mathrm e}^{6 i \left (d x +c \right )}+10 \,{\mathrm e}^{7 i \left (d x +c \right )}-800 i {\mathrm e}^{4 i \left (d x +c \right )}+560 i {\mathrm e}^{2 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}-152 i-55 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{20 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(198\) |
| norman | \(\frac {-\frac {a^{3}}{160 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {9 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {7 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {121 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {267 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {267 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {121 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}-\frac {7 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {9 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d}+\frac {3 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+3 a^{3} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9 a^{3} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+9 a^{3} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{3} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {25 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {55 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}\) | \(388\) |
15/4096*csc(1/2*d*x+1/2*c)^5*((sin(3*d*x+3*c)-1/5*sin(5*d*x+5*c)-2*sin(d*x +c))*ln(tan(1/2*d*x+1/2*c))+(-8*d*x-2)*sin(3*d*x+3*c)+(8/5*d*x+2/5)*sin(5* d*x+5*c)-24/5*cos(3*d*x+3*c)+152/75*cos(5*d*x+5*c)-8/5*sin(2*d*x+2*c)-2/5* sin(4*d*x+4*c)-4/15*sin(6*d*x+6*c)+(16*d*x+4)*sin(d*x+c)+16/15*cos(d*x+c)) *sec(1/2*d*x+1/2*c)^5*a^3/d
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (124) = 248\).
Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.91 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {304 \, a^{3} \cos \left (d x + c\right )^{5} - 560 \, a^{3} \cos \left (d x + c\right )^{3} + 240 \, a^{3} \cos \left (d x + c\right ) + 15 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 15 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 10 \, {\left (24 \, a^{3} d x \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{5} - 48 \, a^{3} d x \cos \left (d x + c\right )^{2} + 5 \, a^{3} \cos \left (d x + c\right )^{3} + 24 \, a^{3} d x - 3 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
1/80*(304*a^3*cos(d*x + c)^5 - 560*a^3*cos(d*x + c)^3 + 240*a^3*cos(d*x + c) + 15*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 15*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 10*(24*a^3*d*x*cos(d*x + c)^4 - 8*a^3*cos(d*x + c)^5 - 48*a^3*d*x*cos(d*x + c)^2 + 5*a^3*cos(d*x + c)^3 + 24*a^3*d*x - 3*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.36 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {80 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} - 15 \, a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 20 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {16 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{80 \, d} \]
1/80*(80*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^3 - 15*a^ 3*(2*(5*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^ 2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) + 20*a^3*(2*co s(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) - 16*a^3/tan(d*x + c)^5)/d
Time = 0.46 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.71 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 960 \, {\left (d x + c\right )} a^{3} - 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {640 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {274 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 580 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 80 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{320 \, d} \]
1/320*(2*a^3*tan(1/2*d*x + 1/2*c)^5 + 15*a^3*tan(1/2*d*x + 1/2*c)^4 + 30*a ^3*tan(1/2*d*x + 1/2*c)^3 - 80*a^3*tan(1/2*d*x + 1/2*c)^2 + 960*(d*x + c)* a^3 - 120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) - 580*a^3*tan(1/2*d*x + 1/2*c ) - 640*a^3/(tan(1/2*d*x + 1/2*c)^2 + 1) + (274*a^3*tan(1/2*d*x + 1/2*c)^5 + 580*a^3*tan(1/2*d*x + 1/2*c)^4 + 80*a^3*tan(1/2*d*x + 1/2*c)^3 - 30*a^3 *tan(1/2*d*x + 1/2*c)^2 - 15*a^3*tan(1/2*d*x + 1/2*c) - 2*a^3)/tan(1/2*d*x + 1/2*c)^5)/d
Time = 11.39 (sec) , antiderivative size = 554, normalized size of antiderivative = 4.20 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+65\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+550\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+80\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+560\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-550\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-65\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+1920\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1920\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}{320\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )} \]
-(a^3*(2*cos(c/2 + (d*x)/2)^12 - 2*sin(c/2 + (d*x)/2)^12 - 15*cos(c/2 + (d *x)/2)*sin(c/2 + (d*x)/2)^11 + 15*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 32*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 65*cos(c/2 + (d*x)/2)^3 *sin(c/2 + (d*x)/2)^9 + 550*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 80 *cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 560*cos(c/2 + (d*x)/2)^7*sin( c/2 + (d*x)/2)^5 - 550*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 65*cos( c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 32*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 120*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d* x)/2)^5*sin(c/2 + (d*x)/2)^7 + 120*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/ 2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 + 1920*atan((8*cos(c/2 + (d* x)/2) - sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2) + 8*sin(c/2 + (d*x)/2)))*c os(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 1920*atan((8*cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2) + 8*sin(c/2 + (d*x)/2)))*cos(c/ 2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5))/(320*d*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^5*(cos(c/2 + (d*x)/2)^2 + sin(c/2 + (d*x)/2)^2))