Integrand size = 27, antiderivative size = 82 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {5 \text {arctanh}(\cos (c+d x))}{8 a^2 d}+\frac {2 \cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d} \]
5/8*arctanh(cos(d*x+c))/a^2/d+2/3*cot(d*x+c)^3/a^2/d-3/8*cot(d*x+c)*csc(d* x+c)/a^2/d-1/4*cot(d*x+c)*csc(d*x+c)^3/a^2/d
Time = 1.61 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.41 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\left (\csc \left (\frac {1}{2} (c+d x)\right )+\sec \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (-33 \cos (c+d x)+60 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin ^4(c+d x)+\cos (3 (c+d x)) (9+16 \sin (c+d x))+24 \sin (2 (c+d x))\right )}{1536 a^2 d (1+\sin (c+d x))^2} \]
((Csc[(c + d*x)/2] + Sec[(c + d*x)/2])^4*(-33*Cos[c + d*x] + 60*(Log[Cos[( c + d*x)/2]] - Log[Sin[(c + d*x)/2]])*Sin[c + d*x]^4 + Cos[3*(c + d*x)]*(9 + 16*Sin[c + d*x]) + 24*Sin[2*(c + d*x)]))/(1536*a^2*d*(1 + Sin[c + d*x]) ^2)
Time = 0.50 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3354, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^5 (a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) \csc ^3(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^2 (a-a \sin (c+d x))^2}{\sin (c+d x)^5}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cot ^2(c+d x) \csc ^3(c+d x)-2 a^2 \cot ^2(c+d x) \csc ^2(c+d x)+a^2 \cot ^2(c+d x) \csc (c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{8 d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}}{a^4}\) |
((5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) + (2*a^2*Cot[c + d*x]^3)/(3*d) - (3*a ^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(4 *d))/a^4
3.7.41.3.1 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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 0.37 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.49
| method | result | size |
| parallelrisch | \(\frac {-3 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+48 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d \,a^{2}}\) | \(122\) |
| derivativedivides | \(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{16 d \,a^{2}}\) | \(124\) |
| default | \(\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-10 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{16 d \,a^{2}}\) | \(124\) |
| risch | \(\frac {-48 i {\mathrm e}^{6 i \left (d x +c \right )}+9 \,{\mathrm e}^{7 i \left (d x +c \right )}+48 i {\mathrm e}^{4 i \left (d x +c \right )}-33 \,{\mathrm e}^{5 i \left (d x +c \right )}-16 i {\mathrm e}^{2 i \left (d x +c \right )}-33 \,{\mathrm e}^{3 i \left (d x +c \right )}+16 i+9 \,{\mathrm e}^{i \left (d x +c \right )}}{12 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{2}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{2}}\) | \(146\) |
| norman | \(\frac {-\frac {1}{64 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{192 d a}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}-\frac {17 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}+\frac {17 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}-\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d a}-\frac {7 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {15 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}-\frac {55 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {185 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {125 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {205 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}\) | \(298\) |
1/192*(-3*cot(1/2*d*x+1/2*c)^4+3*tan(1/2*d*x+1/2*c)^4+16*cot(1/2*d*x+1/2*c )^3-16*tan(1/2*d*x+1/2*c)^3-24*cot(1/2*d*x+1/2*c)^2+24*tan(1/2*d*x+1/2*c)^ 2-48*cot(1/2*d*x+1/2*c)+48*tan(1/2*d*x+1/2*c)-120*ln(tan(1/2*d*x+1/2*c)))/ d/a^2
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.68 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {32 \, \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + 18 \, \cos \left (d x + c\right )^{3} + 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 30 \, \cos \left (d x + c\right )}{48 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}} \]
1/48*(32*cos(d*x + c)^3*sin(d*x + c) + 18*cos(d*x + c)^3 + 15*(cos(d*x + c )^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 15*(cos(d*x + c) ^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2) - 30*cos(d*x + c)) /(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)
Timed out. \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (74) = 148\).
Time = 0.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.37 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {\frac {48 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a^{2}} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {{\left (\frac {16 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {48 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a^{2} \sin \left (d x + c\right )^{4}}}{192 \, d} \]
1/192*((48*sin(d*x + c)/(cos(d*x + c) + 1) + 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^4/(cos (d*x + c) + 1)^4)/a^2 - 120*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 + (16 *sin(d*x + c)/(cos(d*x + c) + 1) - 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 48*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3)*(cos(d*x + c) + 1)^4/(a^2*si n(d*x + c)^4))/d
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (74) = 148\).
Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.93 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {250 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} - \frac {3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{8}}}{192 \, d} \]
-1/192*(120*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (250*tan(1/2*d*x + 1/2*c) ^4 - 48*tan(1/2*d*x + 1/2*c)^3 - 24*tan(1/2*d*x + 1/2*c)^2 + 16*tan(1/2*d* x + 1/2*c) - 3)/(a^2*tan(1/2*d*x + 1/2*c)^4) - (3*a^6*tan(1/2*d*x + 1/2*c) ^4 - 16*a^6*tan(1/2*d*x + 1/2*c)^3 + 24*a^6*tan(1/2*d*x + 1/2*c)^2 + 48*a^ 6*tan(1/2*d*x + 1/2*c))/a^8)/d
Time = 9.37 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.84 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12\,a^2\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^2\,d}-\frac {5\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a^2\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^2\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{4}\right )}{16\,a^2\,d} \]