Integrand size = 29, antiderivative size = 176 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {25 a^3 x}{8}+\frac {13 a^3 \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a^3 \cos (c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {3 a^3 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
-25/8*a^3*x+13/2*a^3*arctanh(cos(d*x+c))/d-5*a^3*cos(d*x+c)/d-2/3*a^3*cos( d*x+c)^3/d+1/5*a^3*cos(d*x+c)^5/d-a^3*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^3/d- 3/2*a^3*cot(d*x+c)*csc(d*x+c)/d-23/8*a^3*cos(d*x+c)*sin(d*x+c)/d+3/4*a^3*c os(d*x+c)*sin(d*x+c)^3/d
Time = 6.48 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.24 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (-1500 (c+d x)-2580 \cos (c+d x)-50 \cos (3 (c+d x))+6 \cos (5 (c+d x))-160 \cot \left (\frac {1}{2} (c+d x)\right )-180 \csc ^2\left (\frac {1}{2} (c+d x)\right )+3120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+180 \sec ^2\left (\frac {1}{2} (c+d x)\right )+160 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-10 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-600 \sin (2 (c+d x))-45 \sin (4 (c+d x))+160 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{480 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
(a^3*(1 + Sin[c + d*x])^3*(-1500*(c + d*x) - 2580*Cos[c + d*x] - 50*Cos[3* (c + d*x)] + 6*Cos[5*(c + d*x)] - 160*Cot[(c + d*x)/2] - 180*Csc[(c + d*x) /2]^2 + 3120*Log[Cos[(c + d*x)/2]] - 3120*Log[Sin[(c + d*x)/2]] + 180*Sec[ (c + d*x)/2]^2 + 160*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 10*Csc[(c + d*x)/ 2]^4*Sin[c + d*x] - 600*Sin[2*(c + d*x)] - 45*Sin[4*(c + d*x)] + 160*Tan[( c + d*x)/2]))/(480*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.44 (sec) , antiderivative size = 180, 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.
\(\displaystyle \int \cos ^2(c+d x) \cot ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^3}{\sin (c+d x)^4}dx\) |
\(\Big \downarrow \) 3351 |
\(\displaystyle \frac {\int \left (-\sin ^5(c+d x) a^9+\csc ^4(c+d x) a^9-3 \sin ^4(c+d x) a^9+3 \csc ^3(c+d x) a^9+8 \sin ^2(c+d x) a^9-8 \csc (c+d x) a^9+6 \sin (c+d x) a^9-6 a^9\right )dx}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {13 a^9 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a^9 \cos ^5(c+d x)}{5 d}-\frac {2 a^9 \cos ^3(c+d x)}{3 d}-\frac {5 a^9 \cos (c+d x)}{d}-\frac {a^9 \cot ^3(c+d x)}{3 d}-\frac {a^9 \cot (c+d x)}{d}+\frac {3 a^9 \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {23 a^9 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {3 a^9 \cot (c+d x) \csc (c+d x)}{2 d}-\frac {25 a^9 x}{8}}{a^6}\) |
((-25*a^9*x)/8 + (13*a^9*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a^9*Cos[c + d*x ])/d - (2*a^9*Cos[c + d*x]^3)/(3*d) + (a^9*Cos[c + d*x]^5)/(5*d) - (a^9*Co t[c + d*x])/d - (a^9*Cot[c + d*x]^3)/(3*d) - (3*a^9*Cot[c + d*x]*Csc[c + d *x])/(2*d) - (23*a^9*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^9*Cos[c + d*x ]*Sin[c + d*x]^3)/(4*d))/a^6
3.7.12.3.1 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 = 0.46 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {a^{3} \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (6240 \left (-3 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3000 d x \sin \left (3 d x +3 c \right )-9000 d x \sin \left (d x +c \right )-7884 \sin \left (2 d x +2 c \right )+7048 \sin \left (3 d x +3 c \right )+2412 \sin \left (4 d x +4 c \right )+68 \sin \left (6 d x +6 c \right )-6 \sin \left (8 d x +8 c \right )-3075 \cos \left (d x +c \right )+2305 \cos \left (3 d x +3 c \right )-465 \cos \left (5 d x +5 c \right )-45 \cos \left (7 d x +7 c \right )-21144 \sin \left (d x +c \right )\right )}{30720 d}\) | \(189\) |
risch | \(-\frac {25 a^{3} x}{8}+\frac {a^{3} {\mathrm e}^{5 i \left (d x +c \right )}}{160 d}+\frac {5 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {43 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {43 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}-\frac {5 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{3} {\mathrm e}^{-5 i \left (d x +c \right )}}{160 d}+\frac {a^{3} \left (9 \,{\mathrm e}^{5 i \left (d x +c \right )}+12 i {\mathrm e}^{2 i \left (d x +c \right )}-4 i-9 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{32 d}-\frac {5 a^{3} \cos \left (3 d x +3 c \right )}{48 d}\) | \(244\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(272\) |
default | \(\frac {a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a^{3} \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(272\) |
norman | \(\frac {-\frac {a^{3}}{24 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {7 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {23 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {31 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {31 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {23 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {7 a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {3 a^{3} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{3} \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {25 a^{3} x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {125 a^{3} x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {125 a^{3} x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {125 a^{3} x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {125 a^{3} x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {25 a^{3} x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {18 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {665 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {519 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {881 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}-\frac {1967 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {13 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(422\) |
1/30720*a^3*csc(1/2*d*x+1/2*c)^3*sec(1/2*d*x+1/2*c)^3*(6240*(-3*sin(d*x+c) +sin(3*d*x+3*c))*ln(tan(1/2*d*x+1/2*c))+3000*d*x*sin(3*d*x+3*c)-9000*d*x*s in(d*x+c)-7884*sin(2*d*x+2*c)+7048*sin(3*d*x+3*c)+2412*sin(4*d*x+4*c)+68*s in(6*d*x+6*c)-6*sin(8*d*x+8*c)-3075*cos(d*x+c)+2305*cos(3*d*x+3*c)-465*cos (5*d*x+5*c)-45*cos(7*d*x+7*c)-21144*sin(d*x+c))/d
Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.31 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {90 \, a^{3} \cos \left (d x + c\right )^{7} + 75 \, a^{3} \cos \left (d x + c\right )^{5} - 500 \, a^{3} \cos \left (d x + c\right )^{3} + 375 \, a^{3} \cos \left (d x + c\right ) + 390 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 390 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + {\left (24 \, a^{3} \cos \left (d x + c\right )^{7} - 104 \, a^{3} \cos \left (d x + c\right )^{5} - 375 \, a^{3} d x \cos \left (d x + c\right )^{2} - 520 \, a^{3} \cos \left (d x + c\right )^{3} + 375 \, a^{3} d x + 780 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
1/120*(90*a^3*cos(d*x + c)^7 + 75*a^3*cos(d*x + c)^5 - 500*a^3*cos(d*x + c )^3 + 375*a^3*cos(d*x + c) + 390*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d* x + c) + 1/2)*sin(d*x + c) - 390*(a^3*cos(d*x + c)^2 - a^3)*log(-1/2*cos(d *x + c) + 1/2)*sin(d*x + c) + (24*a^3*cos(d*x + c)^7 - 104*a^3*cos(d*x + c )^5 - 375*a^3*d*x*cos(d*x + c)^2 - 520*a^3*cos(d*x + c)^3 + 375*a^3*d*x + 780*a^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.43 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.40 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {4 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 30 \, {\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a^{3} - 45 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} + 20 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3}}{120 \, d} \]
1/120*(4*(6*cos(d*x + c)^5 + 10*cos(d*x + c)^3 + 30*cos(d*x + c) - 15*log( cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1))*a^3 - 30*(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^3 - 45*(15*d*x + 15*c + (15*tan(d*x + c)^4 + 25*tan(d*x + c)^2 + 8)/(tan(d*x + c)^5 + 2*tan(d*x + c)^3 + tan(d *x + c)))*a^3 + 20*(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^3)/d
Time = 0.43 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.66 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 375 \, {\left (d x + c\right )} a^{3} - 780 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {5 \, {\left (286 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {2 \, {\left (345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 330 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 2880 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 330 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 656 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
1/120*(5*a^3*tan(1/2*d*x + 1/2*c)^3 + 45*a^3*tan(1/2*d*x + 1/2*c)^2 - 375* (d*x + c)*a^3 - 780*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 45*a^3*tan(1/2*d* x + 1/2*c) + 5*(286*a^3*tan(1/2*d*x + 1/2*c)^3 - 9*a^3*tan(1/2*d*x + 1/2*c )^2 - 9*a^3*tan(1/2*d*x + 1/2*c) - a^3)/tan(1/2*d*x + 1/2*c)^3 + 2*(345*a^ 3*tan(1/2*d*x + 1/2*c)^9 - 720*a^3*tan(1/2*d*x + 1/2*c)^8 + 330*a^3*tan(1/ 2*d*x + 1/2*c)^7 - 2880*a^3*tan(1/2*d*x + 1/2*c)^6 - 3680*a^3*tan(1/2*d*x + 1/2*c)^4 - 330*a^3*tan(1/2*d*x + 1/2*c)^3 - 2560*a^3*tan(1/2*d*x + 1/2*c )^2 - 345*a^3*tan(1/2*d*x + 1/2*c) - 656*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1) ^5)/d
Time = 11.16 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.44 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {13\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-43\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+99\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-\frac {86\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+399\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {95\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {1562\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {232\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {1114\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {193\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {1537\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}+\frac {14\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+40\,{\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 {25\,a^3\,\mathrm {atan}\left (\frac {625\,a^6}{16\,\left (\frac {325\,a^6}{4}-\frac {625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {325\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {325\,a^6}{4}-\frac {625\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]
(3*a^3*tan(c/2 + (d*x)/2)^2)/(8*d) + (a^3*tan(c/2 + (d*x)/2)^3)/(24*d) - ( 13*a^3*log(tan(c/2 + (d*x)/2)))/(2*d) - ((14*a^3*tan(c/2 + (d*x)/2)^2)/3 + (1537*a^3*tan(c/2 + (d*x)/2)^3)/15 + (193*a^3*tan(c/2 + (d*x)/2)^4)/3 + ( 1114*a^3*tan(c/2 + (d*x)/2)^5)/3 + (232*a^3*tan(c/2 + (d*x)/2)^6)/3 + (156 2*a^3*tan(c/2 + (d*x)/2)^7)/3 + (95*a^3*tan(c/2 + (d*x)/2)^8)/3 + 399*a^3* tan(c/2 + (d*x)/2)^9 - (86*a^3*tan(c/2 + (d*x)/2)^10)/3 + 99*a^3*tan(c/2 + (d*x)/2)^11 - 43*a^3*tan(c/2 + (d*x)/2)^12 + a^3/3 + 3*a^3*tan(c/2 + (d*x )/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 40*tan(c/2 + (d*x)/2)^5 + 80*tan(c/2 + (d*x)/2)^7 + 80*tan(c/2 + (d*x)/2)^9 + 40*tan(c/2 + (d*x)/2)^11 + 8*tan(c/ 2 + (d*x)/2)^13)) - (25*a^3*atan((625*a^6)/(16*((325*a^6)/4 - (625*a^6*tan (c/2 + (d*x)/2))/16)) + (325*a^6*tan(c/2 + (d*x)/2))/(4*((325*a^6)/4 - (62 5*a^6*tan(c/2 + (d*x)/2))/16))))/(4*d) + (3*a^3*tan(c/2 + (d*x)/2))/(8*d)