Integrand size = 29, antiderivative size = 152 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d} \]
5/64*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-1/9*a^2*cot(d*x+c)^9 /d+5/64*a^2*cot(d*x+c)*csc(d*x+c)/d-5/32*a^2*cot(d*x+c)*csc(d*x+c)^3/d+5/2 4*a^2*cot(d*x+c)^3*csc(d*x+c)^3/d-1/4*a^2*cot(d*x+c)^5*csc(d*x+c)^3/d
Leaf count is larger than twice the leaf count of optimal. \(313\) vs. \(2(152)=304\).
Time = 6.77 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.06 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^9(c+d x) \left (72576 \cos (c+d x)+37632 \cos (3 (c+d x))+6912 \cos (5 (c+d x))-1728 \cos (7 (c+d x))-704 \cos (9 (c+d x))-39690 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+39690 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+36540 \sin (2 (c+d x))+26460 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-26460 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+20916 \sin (4 (c+d x))-11340 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+11340 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+16044 \sin (6 (c+d x))+2835 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-2835 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))+630 \sin (8 (c+d x))-315 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))+315 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (9 (c+d x))\right )}{1032192 d} \]
-1/1032192*(a^2*Csc[c + d*x]^9*(72576*Cos[c + d*x] + 37632*Cos[3*(c + d*x) ] + 6912*Cos[5*(c + d*x)] - 1728*Cos[7*(c + d*x)] - 704*Cos[9*(c + d*x)] - 39690*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 39690*Log[Sin[(c + d*x)/2]]*Si n[c + d*x] + 36540*Sin[2*(c + d*x)] + 26460*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 26460*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 20916*Sin[4*(c + d*x)] - 11340*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 11340*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 16044*Sin[6*(c + d*x)] + 2835*Log[Cos[(c + d* x)/2]]*Sin[7*(c + d*x)] - 2835*Log[Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] + 63 0*Sin[8*(c + d*x)] - 315*Log[Cos[(c + d*x)/2]]*Sin[9*(c + d*x)] + 315*Log[ Sin[(c + d*x)/2]]*Sin[9*(c + d*x)]))/d
Time = 0.47 (sec) , antiderivative size = 152, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^2}{\sin (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \cot ^6(c+d x) \csc ^4(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^3(c+d x)+a^2 \cot ^6(c+d x) \csc ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 a^2 \text {arctanh}(\cos (c+d x))}{64 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{4 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{64 d}\) |
(5*a^2*ArcTanh[Cos[c + d*x]])/(64*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (a^2 *Cot[c + d*x]^9)/(9*d) + (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(64*d) - (5*a^2 *Cot[c + d*x]*Csc[c + d*x]^3)/(32*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^ 3)/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(4*d)
3.7.1.3.1 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 = 0.53 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03
| method | result | size |
| parallelrisch | \(-\frac {9 \left (\frac {5120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\left (\csc ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cos \left (d x +c \right )+\frac {14 \cos \left (3 d x +3 c \right )}{27}+\frac {2 \cos \left (5 d x +5 c \right )}{21}-\frac {\cos \left (7 d x +7 c \right )}{42}-\frac {11 \cos \left (9 d x +9 c \right )}{1134}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {1765 \cos \left (d x +c \right )}{432}+\frac {895 \cos \left (3 d x +3 c \right )}{432}+\frac {397 \cos \left (5 d x +5 c \right )}{432}+\frac {5 \cos \left (7 d x +7 c \right )}{144}\right ) \left (\sec ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a^{2}}{65536 d}\) | \(157\) |
| derivativedivides | \(\frac {-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(192\) |
| default | \(\frac {-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+2 a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos ^{7}\left (d x +c \right )}{128 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{128}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\) | \(192\) |
| risch | \(-\frac {a^{2} \left (16128 i {\mathrm e}^{14 i \left (d x +c \right )}+315 \,{\mathrm e}^{17 i \left (d x +c \right )}-4032 i {\mathrm e}^{16 i \left (d x +c \right )}+8022 \,{\mathrm e}^{15 i \left (d x +c \right )}-8064 i {\mathrm e}^{8 i \left (d x +c \right )}+10458 \,{\mathrm e}^{13 i \left (d x +c \right )}+48384 i {\mathrm e}^{6 i \left (d x +c \right )}+18270 \,{\mathrm e}^{11 i \left (d x +c \right )}-10752 i {\mathrm e}^{12 i \left (d x +c \right )}+80640 i {\mathrm e}^{10 i \left (d x +c \right )}-18270 \,{\mathrm e}^{7 i \left (d x +c \right )}-9216 i {\mathrm e}^{4 i \left (d x +c \right )}-10458 \,{\mathrm e}^{5 i \left (d x +c \right )}+2304 i {\mathrm e}^{2 i \left (d x +c \right )}-8022 \,{\mathrm e}^{3 i \left (d x +c \right )}-704 i-315 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2016 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}\) | \(250\) |
-9/65536*(5120/9*ln(tan(1/2*d*x+1/2*c))+csc(1/2*d*x+1/2*c)^8*(sec(1/2*d*x+ 1/2*c)*(cos(d*x+c)+14/27*cos(3*d*x+3*c)+2/21*cos(5*d*x+5*c)-1/42*cos(7*d*x +7*c)-11/1134*cos(9*d*x+9*c))*csc(1/2*d*x+1/2*c)+1765/432*cos(d*x+c)+895/4 32*cos(3*d*x+3*c)+397/432*cos(5*d*x+5*c)+5/144*cos(7*d*x+7*c))*sec(1/2*d*x +1/2*c)^8)*a^2/d
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (138) = 276\).
Time = 0.28 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.91 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {1408 \, a^{2} \cos \left (d x + c\right )^{9} - 2304 \, a^{2} \cos \left (d x + c\right )^{7} + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 42 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{7} + 73 \, a^{2} \cos \left (d x + c\right )^{5} - 55 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
1/8064*(1408*a^2*cos(d*x + c)^9 - 2304*a^2*cos(d*x + c)^7 + 315*(a^2*cos(d *x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 315*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 42*(15*a^2*cos(d*x + c) ^7 + 73*a^2*cos(d*x + c)^5 - 55*a^2*cos(d*x + c)^3 + 15*a^2*cos(d*x + c))* sin(d*x + c))/((d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {21 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1152 \, a^{2}}{\tan \left (d x + c\right )^{7}} + \frac {128 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{8064 \, d} \]
-1/8064*(21*a^2*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c )^3 + 15*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c) ^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c ) - 1)) + 1152*a^2/tan(d*x + c)^7 + 128*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9)/d
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (138) = 276\).
Time = 0.42 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.13 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {14 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 3276 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {14258 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3276 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1848 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{64512 \, d} \]
1/64512*(14*a^2*tan(1/2*d*x + 1/2*c)^9 + 63*a^2*tan(1/2*d*x + 1/2*c)^8 + 1 8*a^2*tan(1/2*d*x + 1/2*c)^7 - 336*a^2*tan(1/2*d*x + 1/2*c)^6 - 504*a^2*ta n(1/2*d*x + 1/2*c)^5 + 504*a^2*tan(1/2*d*x + 1/2*c)^4 + 1848*a^2*tan(1/2*d *x + 1/2*c)^3 + 1008*a^2*tan(1/2*d*x + 1/2*c)^2 - 5040*a^2*log(abs(tan(1/2 *d*x + 1/2*c))) - 3276*a^2*tan(1/2*d*x + 1/2*c) + (14258*a^2*tan(1/2*d*x + 1/2*c)^9 + 3276*a^2*tan(1/2*d*x + 1/2*c)^8 - 1008*a^2*tan(1/2*d*x + 1/2*c )^7 - 1848*a^2*tan(1/2*d*x + 1/2*c)^6 - 504*a^2*tan(1/2*d*x + 1/2*c)^5 + 5 04*a^2*tan(1/2*d*x + 1/2*c)^4 + 336*a^2*tan(1/2*d*x + 1/2*c)^3 - 18*a^2*ta n(1/2*d*x + 1/2*c)^2 - 63*a^2*tan(1/2*d*x + 1/2*c) - 14*a^2)/tan(1/2*d*x + 1/2*c)^9)/d
Time = 10.78 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.35 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {11\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3584\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {5\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d}+\frac {13\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}-\frac {13\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]
(a^2*cot(c/2 + (d*x)/2)^5)/(128*d) - (11*a^2*cot(c/2 + (d*x)/2)^3)/(384*d) - (a^2*cot(c/2 + (d*x)/2)^4)/(128*d) - (a^2*cot(c/2 + (d*x)/2)^2)/(64*d) + (a^2*cot(c/2 + (d*x)/2)^6)/(192*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(3584*d) - (a^2*cot(c/2 + (d*x)/2)^8)/(1024*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(4608* d) + (a^2*tan(c/2 + (d*x)/2)^2)/(64*d) + (11*a^2*tan(c/2 + (d*x)/2)^3)/(38 4*d) + (a^2*tan(c/2 + (d*x)/2)^4)/(128*d) - (a^2*tan(c/2 + (d*x)/2)^5)/(12 8*d) - (a^2*tan(c/2 + (d*x)/2)^6)/(192*d) + (a^2*tan(c/2 + (d*x)/2)^7)/(35 84*d) + (a^2*tan(c/2 + (d*x)/2)^8)/(1024*d) + (a^2*tan(c/2 + (d*x)/2)^9)/( 4608*d) - (5*a^2*log(tan(c/2 + (d*x)/2)))/(64*d) + (13*a^2*cot(c/2 + (d*x) /2))/(256*d) - (13*a^2*tan(c/2 + (d*x)/2))/(256*d)