Integrand size = 27, antiderivative size = 130 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a x}{2}+\frac {5 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {5 a \cos (c+d x)}{2 d}-\frac {5 a \cos ^3(c+d x)}{6 d}+\frac {5 a \cot (c+d x)}{2 d}-\frac {a \cos ^3(c+d x) \cot ^2(c+d x)}{2 d}-\frac {5 a \cot ^3(c+d x)}{6 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d} \]
5/2*a*x+5/2*a*arctanh(cos(d*x+c))/d-5/2*a*cos(d*x+c)/d-5/6*a*cos(d*x+c)^3/ d+5/2*a*cot(d*x+c)/d-1/2*a*cos(d*x+c)^3*cot(d*x+c)^2/d-5/6*a*cot(d*x+c)^3/ d+1/2*a*cos(d*x+c)^2*cot(d*x+c)^3/d
Time = 6.23 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.34 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a (c+d x)}{2 d}-\frac {9 a \cos (c+d x)}{4 d}-\frac {a \cos (3 (c+d x))}{12 d}+\frac {7 a \cot (c+d x)}{3 d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sin (2 (c+d x))}{4 d} \]
(5*a*(c + d*x))/(2*d) - (9*a*Cos[c + d*x])/(4*d) - (a*Cos[3*(c + d*x)])/(1 2*d) + (7*a*Cot[c + d*x])/(3*d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) + (5*a*Log[Cos[(c + d*x)/2]])/(2*d) - (5*a*Lo g[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + (a*Sin[2*(c + d*x)])/(4*d)
Time = 0.45 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3317, 3042, 25, 3071, 252, 254, 2009, 3072, 252, 254, 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.
\(\displaystyle \int \cos ^2(c+d x) \cot ^4(c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)}{\sin (c+d x)^4}dx\) |
\(\Big \downarrow \) 3317 |
\(\displaystyle a \int \cos ^3(c+d x) \cot ^3(c+d x)dx+a \int \cos ^2(c+d x) \cot ^4(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int -\sin \left (c+d x+\frac {\pi }{2}\right )^3 \tan \left (c+d x+\frac {\pi }{2}\right )^3dx+a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^4dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^4dx-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
\(\Big \downarrow \) 3071 |
\(\displaystyle -\frac {a \int \frac {\cot ^6(c+d x)}{\left (\cot ^2(c+d x)+1\right )^2}d\cot (c+d x)}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -\frac {a \left (\frac {5}{2} \int \frac {\cot ^4(c+d x)}{\cot ^2(c+d x)+1}d\cot (c+d x)-\frac {\cot ^5(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
\(\Big \downarrow \) 254 |
\(\displaystyle -\frac {a \left (\frac {5}{2} \int \left (\cot ^2(c+d x)+\frac {1}{\cot ^2(c+d x)+1}-1\right )d\cot (c+d x)-\frac {\cot ^5(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx-\frac {a \left (\frac {5}{2} \left (\arctan (\cot (c+d x))+\frac {1}{3} \cot ^3(c+d x)-\cot (c+d x)\right )-\frac {\cot ^5(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
\(\Big \downarrow \) 3072 |
\(\displaystyle -\frac {a \int \frac {\cos ^6(c+d x)}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)}{d}-\frac {a \left (\frac {5}{2} \left (\arctan (\cot (c+d x))+\frac {1}{3} \cot ^3(c+d x)-\cot (c+d x)\right )-\frac {\cot ^5(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle -\frac {a \left (\frac {\cos ^5(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {5}{2} \int \frac {\cos ^4(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)\right )}{d}-\frac {a \left (\frac {5}{2} \left (\arctan (\cot (c+d x))+\frac {1}{3} \cot ^3(c+d x)-\cot (c+d x)\right )-\frac {\cot ^5(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
\(\Big \downarrow \) 254 |
\(\displaystyle -\frac {a \left (\frac {\cos ^5(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {5}{2} \int \left (-\cos ^2(c+d x)+\frac {1}{1-\cos ^2(c+d x)}-1\right )d\cos (c+d x)\right )}{d}-\frac {a \left (\frac {5}{2} \left (\arctan (\cot (c+d x))+\frac {1}{3} \cot ^3(c+d x)-\cot (c+d x)\right )-\frac {\cot ^5(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \left (\frac {5}{2} \left (\arctan (\cot (c+d x))+\frac {1}{3} \cot ^3(c+d x)-\cot (c+d x)\right )-\frac {\cot ^5(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-\frac {a \left (\frac {\cos ^5(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {5}{2} \left (\text {arctanh}(\cos (c+d x))-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)\right )\right )}{d}\) |
-((a*(Cos[c + d*x]^5/(2*(1 - Cos[c + d*x]^2)) - (5*(ArcTanh[Cos[c + d*x]] - Cos[c + d*x] - Cos[c + d*x]^3/3))/2))/d) - (a*(-1/2*Cot[c + d*x]^5/(1 + Cot[c + d*x]^2) + (5*(ArcTan[Cot[c + d*x]] - Cot[c + d*x] + Cot[c + d*x]^3 /3))/2))/d
3.6.79.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 0.32 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(154\) |
default | \(\frac {a \left (-\frac {\cos ^{7}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{5}\left (d x +c \right )\right )}{2}-\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {5 \cos \left (d x +c \right )}{2}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {\cos ^{7}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )}+\frac {4 \left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}\) | \(154\) |
parallelrisch | \(\frac {a \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (60 \left (-3 \sin \left (d x +c \right )+\sin \left (3 d x +3 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 d x \sin \left (3 d x +3 c \right )+180 d x \sin \left (d x +c \right )+65 \sin \left (3 d x +3 c \right )+24 \sin \left (4 d x +4 c \right )+\sin \left (6 d x +6 c \right )+30 \cos \left (d x +c \right )-65 \cos \left (3 d x +3 c \right )+3 \cos \left (5 d x +5 c \right )-195 \sin \left (d x +c \right )-75 \sin \left (2 d x +2 c \right )\right )}{768 d}\) | \(163\) |
risch | \(\frac {5 a x}{2}-\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {9 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {9 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a \left (18 i {\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{5 i \left (d x +c \right )}-24 i {\mathrm e}^{2 i \left (d x +c \right )}+14 i-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}-\frac {a \cos \left (3 d x +3 c \right )}{12 d}\) | \(187\) |
norman | \(\frac {\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a}{24 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {25 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {25 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {5 a x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {15 a x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {15 a x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {5 a x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {55 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {65 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {75 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {5 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(279\) |
1/d*(a*(-1/2/sin(d*x+c)^2*cos(d*x+c)^7-1/2*cos(d*x+c)^5-5/6*cos(d*x+c)^3-5 /2*cos(d*x+c)-5/2*ln(csc(d*x+c)-cot(d*x+c)))+a*(-1/3/sin(d*x+c)^3*cos(d*x+ c)^7+4/3/sin(d*x+c)*cos(d*x+c)^7+4/3*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*c os(d*x+c))*sin(d*x+c)+5/2*d*x+5/2*c))
Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.40 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {6 \, a \cos \left (d x + c\right )^{5} - 40 \, a \cos \left (d x + c\right )^{3} - 15 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 15 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, a \cos \left (d x + c\right ) + 2 \, {\left (2 \, a \cos \left (d x + c\right )^{5} - 15 \, a d x \cos \left (d x + c\right )^{2} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a d x - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
-1/12*(6*a*cos(d*x + c)^5 - 40*a*cos(d*x + c)^3 - 15*(a*cos(d*x + c)^2 - a )*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(a*cos(d*x + c)^2 - a)*log (-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 30*a*cos(d*x + c) + 2*(2*a*cos(d* x + c)^5 - 15*a*d*x*cos(d*x + c)^2 + 10*a*cos(d*x + c)^3 + 15*a*d*x - 15*a *cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.94 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {{\left (4 \, \cos \left (d x + c\right )^{3} - \frac {6 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + 24 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a - 2 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a}{12 \, d} \]
-1/12*((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 - 2*(15*d* x + 15*c + (15*tan(d*x + c)^4 + 10*tan(d*x + c)^2 - 2)/(tan(d*x + c)^5 + t an(d*x + c)^3))*a)/d
Time = 0.34 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.69 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 180 \, {\left (d x + c\right )} a - 180 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 81 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {110 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 111 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 240 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 273 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 306 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 253 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3}}}{72 \, d} \]
1/72*(3*a*tan(1/2*d*x + 1/2*c)^3 + 9*a*tan(1/2*d*x + 1/2*c)^2 + 180*(d*x + c)*a - 180*a*log(abs(tan(1/2*d*x + 1/2*c))) - 81*a*tan(1/2*d*x + 1/2*c) + (110*a*tan(1/2*d*x + 1/2*c)^9 + 9*a*tan(1/2*d*x + 1/2*c)^8 - 111*a*tan(1/ 2*d*x + 1/2*c)^7 + 240*a*tan(1/2*d*x + 1/2*c)^6 - 273*a*tan(1/2*d*x + 1/2* c)^5 + 306*a*tan(1/2*d*x + 1/2*c)^4 - 253*a*tan(1/2*d*x + 1/2*c)^3 + 72*a* tan(1/2*d*x + 1/2*c)^2 - 9*a*tan(1/2*d*x + 1/2*c) - 3*a)/(tan(1/2*d*x + 1/ 2*c)^3 + tan(1/2*d*x + 1/2*c))^3)/d
Time = 9.92 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.38 \[ \int \cos ^2(c+d x) \cot ^4(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+49\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {80\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+67\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-34\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {121\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {9\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {5\,a\,\mathrm {atan}\left (\frac {25\,a^2}{25\,a^2+25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{25\,a^2+25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
(a*tan(c/2 + (d*x)/2)^2)/(8*d) - (a/3 + a*tan(c/2 + (d*x)/2) - 8*a*tan(c/2 + (d*x)/2)^2 + (121*a*tan(c/2 + (d*x)/2)^3)/3 - 34*a*tan(c/2 + (d*x)/2)^4 + 67*a*tan(c/2 + (d*x)/2)^5 - (80*a*tan(c/2 + (d*x)/2)^6)/3 + 49*a*tan(c/ 2 + (d*x)/2)^7 - a*tan(c/2 + (d*x)/2)^8)/(d*(8*tan(c/2 + (d*x)/2)^3 + 24*t an(c/2 + (d*x)/2)^5 + 24*tan(c/2 + (d*x)/2)^7 + 8*tan(c/2 + (d*x)/2)^9)) - (9*a*tan(c/2 + (d*x)/2))/(8*d) + (a*tan(c/2 + (d*x)/2)^3)/(24*d) - (5*a*l og(tan(c/2 + (d*x)/2)))/(2*d) - (5*a*atan((25*a^2)/(25*a^2 + 25*a^2*tan(c/ 2 + (d*x)/2)) - (25*a^2*tan(c/2 + (d*x)/2))/(25*a^2 + 25*a^2*tan(c/2 + (d* x)/2))))/d