Integrand size = 29, antiderivative size = 178 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {135 a^4 x}{16}+\frac {6 a^4 \text {arctanh}(\cos (c+d x))}{d}-\frac {4 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^5(c+d x)}{5 d}-\frac {4 a^4 \cot (c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^4 \cot (c+d x) \csc (c+d x)}{d}-\frac {89 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^4 \cos (c+d x) \sin ^3(c+d x)}{24 d}+\frac {a^4 \cos (c+d x) \sin ^5(c+d x)}{6 d} \]
-135/16*a^4*x+6*a^4*arctanh(cos(d*x+c))/d-4*a^4*cos(d*x+c)/d+4/5*a^4*cos(d *x+c)^5/d-4*a^4*cot(d*x+c)/d-1/3*a^4*cot(d*x+c)^3/d-2*a^4*cot(d*x+c)*csc(d *x+c)/d-89/16*a^4*cos(d*x+c)*sin(d*x+c)/d+23/24*a^4*cos(d*x+c)*sin(d*x+c)^ 3/d+1/6*a^4*cos(d*x+c)*sin(d*x+c)^5/d
Time = 7.20 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.29 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 (1+\sin (c+d x))^4 \left (-8100 (c+d x)-3360 \cos (c+d x)+240 \cos (3 (c+d x))+48 \cos (5 (c+d x))-1760 \cot \left (\frac {1}{2} (c+d x)\right )-480 \csc ^2\left (\frac {1}{2} (c+d x)\right )+5760 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5760 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 \sec ^2\left (\frac {1}{2} (c+d x)\right )+320 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-20 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-2415 \sin (2 (c+d x))-135 \sin (4 (c+d x))+5 \sin (6 (c+d x))+1760 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{960 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^8} \]
(a^4*(1 + Sin[c + d*x])^4*(-8100*(c + d*x) - 3360*Cos[c + d*x] + 240*Cos[3 *(c + d*x)] + 48*Cos[5*(c + d*x)] - 1760*Cot[(c + d*x)/2] - 480*Csc[(c + d *x)/2]^2 + 5760*Log[Cos[(c + d*x)/2]] - 5760*Log[Sin[(c + d*x)/2]] + 480*S ec[(c + d*x)/2]^2 + 320*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 20*Csc[(c + d* x)/2]^4*Sin[c + d*x] - 2415*Sin[2*(c + d*x)] - 135*Sin[4*(c + d*x)] + 5*Si n[6*(c + d*x)] + 1760*Tan[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8)
Time = 0.50 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \cos ^2(c+d x) \cot ^4(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^4}{\sin (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (-\sin ^6(c+d x) a^{10}-4 \sin ^5(c+d x) a^{10}+\csc ^4(c+d x) a^{10}-3 \sin ^4(c+d x) a^{10}+4 \csc ^3(c+d x) a^{10}+8 \sin ^3(c+d x) a^{10}+3 \csc ^2(c+d x) a^{10}+14 \sin ^2(c+d x) a^{10}-8 \csc (c+d x) a^{10}-14 a^{10}\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {6 a^{10} \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a^{10} \cos ^5(c+d x)}{5 d}-\frac {4 a^{10} \cos (c+d x)}{d}-\frac {a^{10} \cot ^3(c+d x)}{3 d}-\frac {4 a^{10} \cot (c+d x)}{d}+\frac {a^{10} \sin ^5(c+d x) \cos (c+d x)}{6 d}+\frac {23 a^{10} \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {89 a^{10} \sin (c+d x) \cos (c+d x)}{16 d}-\frac {2 a^{10} \cot (c+d x) \csc (c+d x)}{d}-\frac {135 a^{10} x}{16}}{a^6}\) |
((-135*a^10*x)/16 + (6*a^10*ArcTanh[Cos[c + d*x]])/d - (4*a^10*Cos[c + d*x ])/d + (4*a^10*Cos[c + d*x]^5)/(5*d) - (4*a^10*Cot[c + d*x])/d - (a^10*Cot [c + d*x]^3)/(3*d) - (2*a^10*Cot[c + d*x]*Csc[c + d*x])/d - (89*a^10*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (23*a^10*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d ) + (a^10*Cos[c + d*x]*Sin[c + d*x]^5)/(6*d))/a^6
3.7.23.3.1 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 = 0.59 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.12
| method | result | size |
| parallelrisch | \(\frac {a^{4} \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-34560 \left (\sin \left (d x +c \right )-\frac {\sin \left (3 d x +3 c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48600 d x \sin \left (d x +c \right )+16200 d x \sin \left (3 d x +3 c \right )-35712 \sin \left (d x +c \right )-15072 \sin \left (2 d x +2 c \right )+11904 \sin \left (3 d x +3 c \right )+3936 \sin \left (4 d x +4 c \right )-96 \sin \left (6 d x +6 c \right )-48 \sin \left (8 d x +8 c \right )-14295 \cos \left (d x +c \right )+13875 \cos \left (3 d x +3 c \right )-1995 \cos \left (5 d x +5 c \right )-150 \cos \left (7 d x +7 c \right )+5 \cos \left (9 d x +9 c \right )\right )}{61440 d}\) | \(200\) |
| risch | \(-\frac {135 a^{4} x}{16}+\frac {a^{4} {\mathrm e}^{5 i \left (d x +c \right )}}{40 d}-\frac {161 i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {7 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{4 d}-\frac {7 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{4 d}-\frac {9 i a^{4} {\mathrm e}^{-4 i \left (d x +c \right )}}{128 d}+\frac {9 i a^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{128 d}+\frac {a^{4} {\mathrm e}^{-5 i \left (d x +c \right )}}{40 d}+\frac {161 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {2 a^{4} \left (-9 i {\mathrm e}^{4 i \left (d x +c \right )}+6 \,{\mathrm e}^{5 i \left (d x +c \right )}+24 i {\mathrm e}^{2 i \left (d x +c \right )}-11 i-6 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {a^{4} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{4 d}\) | \(292\) |
| derivativedivides | \(\frac {a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+4 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(320\) |
| default | \(\frac {a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+6 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+4 a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{4} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(320\) |
1/61440*a^4*csc(1/2*d*x+1/2*c)^3*sec(1/2*d*x+1/2*c)^3*(-34560*(sin(d*x+c)- 1/3*sin(3*d*x+3*c))*ln(tan(1/2*d*x+1/2*c))-48600*d*x*sin(d*x+c)+16200*d*x* sin(3*d*x+3*c)-35712*sin(d*x+c)-15072*sin(2*d*x+2*c)+11904*sin(3*d*x+3*c)+ 3936*sin(4*d*x+4*c)-96*sin(6*d*x+6*c)-48*sin(8*d*x+8*c)-14295*cos(d*x+c)+1 3875*cos(3*d*x+3*c)-1995*cos(5*d*x+5*c)-150*cos(7*d*x+7*c)+5*cos(9*d*x+9*c ))/d
Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.38 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {40 \, a^{4} \cos \left (d x + c\right )^{9} - 390 \, a^{4} \cos \left (d x + c\right )^{7} - 405 \, a^{4} \cos \left (d x + c\right )^{5} + 2700 \, a^{4} \cos \left (d x + c\right )^{3} - 2025 \, a^{4} \cos \left (d x + c\right ) - 720 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 720 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (64 \, a^{4} \cos \left (d x + c\right )^{7} - 64 \, a^{4} \cos \left (d x + c\right )^{5} - 675 \, a^{4} d x \cos \left (d x + c\right )^{2} - 320 \, a^{4} \cos \left (d x + c\right )^{3} + 675 \, a^{4} d x + 480 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
-1/240*(40*a^4*cos(d*x + c)^9 - 390*a^4*cos(d*x + c)^7 - 405*a^4*cos(d*x + c)^5 + 2700*a^4*cos(d*x + c)^3 - 2025*a^4*cos(d*x + c) - 720*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 720*(a^4*cos(d*x + c)^2 - a^4)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*(64*a^4*cos(d *x + c)^7 - 64*a^4*cos(d*x + c)^5 - 675*a^4*d*x*cos(d*x + c)^2 - 320*a^4*c os(d*x + c)^3 + 675*a^4*d*x + 480*a^4*cos(d*x + c))*sin(d*x + c))/((d*cos( d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.65 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {128 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} - 320 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 720 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{4} + 160 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{4}}{960 \, d} \]
1/960*(128*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*lo g(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^4 - 320*(4*cos(d*x + c)^ 3 - 6*cos(d*x + c)/(cos(d*x + c)^2 - 1) + 24*cos(d*x + c) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^4 - 5*(4*sin(2*d*x + 2*c)^3 - 60* d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^4 - 720*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)^5 + 2*ta n(d*x + c)^3 + tan(d*x + c)))*a^4 + 160*(15*d*x + 15*c + (15*tan(d*x + c)^ 4 + 10*tan(d*x + c)^2 - 2)/(tan(d*x + c)^5 + tan(d*x + c)^3))*a^4)/d
Time = 0.53 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.82 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {10 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2025 \, {\left (d x + c\right )} a^{4} - 1440 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 450 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {10 \, {\left (264 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{4}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (1335 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3085 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3840 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1110 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7680 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1110 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 7680 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3085 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4608 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1335 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 768 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
1/240*(10*a^4*tan(1/2*d*x + 1/2*c)^3 + 120*a^4*tan(1/2*d*x + 1/2*c)^2 - 20 25*(d*x + c)*a^4 - 1440*a^4*log(abs(tan(1/2*d*x + 1/2*c))) + 450*a^4*tan(1 /2*d*x + 1/2*c) + 10*(264*a^4*tan(1/2*d*x + 1/2*c)^3 - 45*a^4*tan(1/2*d*x + 1/2*c)^2 - 12*a^4*tan(1/2*d*x + 1/2*c) - a^4)/tan(1/2*d*x + 1/2*c)^3 + 2 *(1335*a^4*tan(1/2*d*x + 1/2*c)^11 + 3085*a^4*tan(1/2*d*x + 1/2*c)^9 - 384 0*a^4*tan(1/2*d*x + 1/2*c)^8 + 1110*a^4*tan(1/2*d*x + 1/2*c)^7 - 7680*a^4* tan(1/2*d*x + 1/2*c)^6 - 1110*a^4*tan(1/2*d*x + 1/2*c)^5 - 7680*a^4*tan(1/ 2*d*x + 1/2*c)^4 - 3085*a^4*tan(1/2*d*x + 1/2*c)^3 - 4608*a^4*tan(1/2*d*x + 1/2*c)^2 - 1335*a^4*tan(1/2*d*x + 1/2*c) - 768*a^4)/(tan(1/2*d*x + 1/2*c )^2 + 1)^6)/d
Time = 10.67 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.66 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d}+\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {6\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {135\,a^4\,\mathrm {atan}\left (\frac {18225\,a^8}{64\,\left (\frac {405\,a^8}{2}-\frac {18225\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}+\frac {405\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {405\,a^8}{2}-\frac {18225\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}\right )}\right )}{8\,d}-\frac {-74\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-\frac {346\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+280\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+153\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+572\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+379\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+592\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {1312\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {1836\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+184\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {376\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+17\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+160\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {15\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]
(a^4*tan(c/2 + (d*x)/2)^2)/(2*d) + (a^4*tan(c/2 + (d*x)/2)^3)/(24*d) - (6* a^4*log(tan(c/2 + (d*x)/2)))/d - (135*a^4*atan((18225*a^8)/(64*((405*a^8)/ 2 - (18225*a^8*tan(c/2 + (d*x)/2))/64)) + (405*a^8*tan(c/2 + (d*x)/2))/(2* ((405*a^8)/2 - (18225*a^8*tan(c/2 + (d*x)/2))/64))))/(8*d) - (17*a^4*tan(c /2 + (d*x)/2)^2 + (376*a^4*tan(c/2 + (d*x)/2)^3)/5 + 184*a^4*tan(c/2 + (d* x)/2)^4 + (1836*a^4*tan(c/2 + (d*x)/2)^5)/5 + (1312*a^4*tan(c/2 + (d*x)/2) ^6)/3 + 592*a^4*tan(c/2 + (d*x)/2)^7 + 379*a^4*tan(c/2 + (d*x)/2)^8 + 572* a^4*tan(c/2 + (d*x)/2)^9 + 153*a^4*tan(c/2 + (d*x)/2)^10 + 280*a^4*tan(c/2 + (d*x)/2)^11 - (346*a^4*tan(c/2 + (d*x)/2)^12)/3 + 4*a^4*tan(c/2 + (d*x) /2)^13 - 74*a^4*tan(c/2 + (d*x)/2)^14 + a^4/3 + 4*a^4*tan(c/2 + (d*x)/2))/ (d*(8*tan(c/2 + (d*x)/2)^3 + 48*tan(c/2 + (d*x)/2)^5 + 120*tan(c/2 + (d*x) /2)^7 + 160*tan(c/2 + (d*x)/2)^9 + 120*tan(c/2 + (d*x)/2)^11 + 48*tan(c/2 + (d*x)/2)^13 + 8*tan(c/2 + (d*x)/2)^15)) + (15*a^4*tan(c/2 + (d*x)/2))/(8 *d)