Integrand size = 29, antiderivative size = 100 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {7 \text {arctanh}(\cos (c+d x))}{8 a^3 d}-\frac {4 \cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d} \]
-7/8*arctanh(cos(d*x+c))/a^3/d-4/3*cot(d*x+c)^3/a^3/d-1/5*cot(d*x+c)^5/a^3 /d+1/8*cot(d*x+c)*csc(d*x+c)/a^3/d+3/4*cot(d*x+c)*csc(d*x+c)^3/a^3/d
Time = 2.18 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.89 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\csc ^5(c+d x) \left (560 \cos (c+d x)-40 \cos (3 (c+d x))-136 \cos (5 (c+d x))+1050 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-1050 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-780 \sin (2 (c+d x))-525 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+525 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+30 \sin (4 (c+d x))+105 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-105 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{1920 a^3 d} \]
-1/1920*(Csc[c + d*x]^5*(560*Cos[c + d*x] - 40*Cos[3*(c + d*x)] - 136*Cos[ 5*(c + d*x)] + 1050*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] - 1050*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] - 780*Sin[2*(c + d*x)] - 525*Log[Cos[(c + d*x)/2]]* Sin[3*(c + d*x)] + 525*Log[Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 30*Sin[4*( c + d*x)] + 105*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] - 105*Log[Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(a^3*d)
Time = 0.56 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, 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.
\(\displaystyle \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^6 (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3354 |
\(\displaystyle \frac {\int \cot ^2(c+d x) \csc ^4(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\cos (c+d x)^2 (a-a \sin (c+d x))^3}{\sin (c+d x)^6}dx}{a^6}\) |
\(\Big \downarrow \) 3352 |
\(\displaystyle \frac {\int \left (a^3 \cot ^2(c+d x) \csc ^4(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^2(c+d x)-a^3 \cot ^2(c+d x) \csc (c+d x)\right )dx}{a^6}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {7 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {4 a^3 \cot ^3(c+d x)}{3 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc (c+d x)}{8 d}}{a^6}\) |
((-7*a^3*ArcTanh[Cos[c + d*x]])/(8*d) - (4*a^3*Cot[c + d*x]^3)/(3*d) - (a^ 3*Cot[c + d*x]^5)/(5*d) + (a^3*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (3*a^3*C ot[c + d*x]*Csc[c + d*x]^3)/(4*d))/a^6
3.8.47.3.1 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 = 0.39 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.48
method | result | size |
parallelrisch | \(\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+130 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-130 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-420 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+420 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{960 d \,a^{3}}\) | \(148\) |
derivativedivides | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{32 d \,a^{3}}\) | \(150\) |
default | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {13}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{32 d \,a^{3}}\) | \(150\) |
risch | \(-\frac {-360 i {\mathrm e}^{8 i \left (d x +c \right )}+15 \,{\mathrm e}^{9 i \left (d x +c \right )}+960 i {\mathrm e}^{6 i \left (d x +c \right )}-390 \,{\mathrm e}^{7 i \left (d x +c \right )}-400 i {\mathrm e}^{4 i \left (d x +c \right )}+320 i {\mathrm e}^{2 i \left (d x +c \right )}+390 \,{\mathrm e}^{3 i \left (d x +c \right )}-136 i-15 \,{\mathrm e}^{i \left (d x +c \right )}}{60 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d \,a^{3}}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d \,a^{3}}\) | \(158\) |
1/960*(6*tan(1/2*d*x+1/2*c)^5-6*cot(1/2*d*x+1/2*c)^5-45*tan(1/2*d*x+1/2*c) ^4+45*cot(1/2*d*x+1/2*c)^4+130*tan(1/2*d*x+1/2*c)^3-130*cot(1/2*d*x+1/2*c) ^3-120*tan(1/2*d*x+1/2*c)^2+120*cot(1/2*d*x+1/2*c)^2-420*tan(1/2*d*x+1/2*c )+840*ln(tan(1/2*d*x+1/2*c))+420*cot(1/2*d*x+1/2*c))/d/a^3
Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {272 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 105 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 30 \, {\left (\cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )} \]
1/240*(272*cos(d*x + c)^5 - 320*cos(d*x + c)^3 - 105*(cos(d*x + c)^4 - 2*c os(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 105*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 30*(cos(d*x + c)^3 - 7*cos(d*x + c))*sin(d*x + c))/((a^3*d*cos(d*x + c)^ 4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c))
Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (90) = 180\).
Time = 0.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.35 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {420 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {120 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {130 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {130 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {420 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 6\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{960 \, d} \]
-1/960*((420*sin(d*x + c)/(cos(d*x + c) + 1) + 120*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 130*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 45*sin(d*x + c)^4 /(cos(d*x + c) + 1)^4 - 6*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 840*l og(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - (45*sin(d*x + c)/(cos(d*x + c) + 1) - 130*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 120*sin(d*x + c)^3/(cos(d* x + c) + 1)^3 + 420*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 6)*(cos(d*x + c) + 1)^5/(a^3*sin(d*x + c)^5))/d
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (90) = 180\).
Time = 0.40 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.86 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {1918 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 130 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {6 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 130 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 420 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{960 \, d} \]
1/960*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (1918*tan(1/2*d*x + 1/2*c) ^5 - 420*tan(1/2*d*x + 1/2*c)^4 - 120*tan(1/2*d*x + 1/2*c)^3 + 130*tan(1/2 *d*x + 1/2*c)^2 - 45*tan(1/2*d*x + 1/2*c) + 6)/(a^3*tan(1/2*d*x + 1/2*c)^5 ) + (6*a^12*tan(1/2*d*x + 1/2*c)^5 - 45*a^12*tan(1/2*d*x + 1/2*c)^4 + 130* a^12*tan(1/2*d*x + 1/2*c)^3 - 120*a^12*tan(1/2*d*x + 1/2*c)^2 - 420*a^12*t an(1/2*d*x + 1/2*c))/a^15)/d
Time = 10.46 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.91 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-45\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+130\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-130\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{960\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
(6*sin(c/2 + (d*x)/2)^10 - 6*cos(c/2 + (d*x)/2)^10 - 45*cos(c/2 + (d*x)/2) *sin(c/2 + (d*x)/2)^9 + 45*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2) + 130*c os(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^8 - 120*cos(c/2 + (d*x)/2)^3*sin(c/ 2 + (d*x)/2)^7 - 420*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^6 + 420*cos(c /2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^4 + 120*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^3 - 130*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^2 + 840*log(sin(c /2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^ 5)/(960*a^3*d*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^5)