Integrand size = 27, antiderivative size = 78 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {9 \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))} \]
-9/2*arctanh(cos(d*x+c))/a^3/d+3*cot(d*x+c)/a^3/d-1/2*cot(d*x+c)*csc(d*x+c )/a^3/d+4*cos(d*x+c)/a^3/d/(1+sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(213\) vs. \(2(78)=156\).
Time = 4.42 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.73 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\left (\csc ^2\left (\frac {1}{2} (c+d x)\right )+2 \csc (c+d x)\right )^5 \left (\csc ^6\left (\frac {1}{2} (c+d x)\right ) (-6+\csc (c+d x))-8 (-6+\csc (c+d x)) \csc ^3(c+d x)+2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \csc (c+d x) \left (-6+\csc (c+d x)+18 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-18 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-4 \csc ^2\left (\frac {1}{2} (c+d x)\right ) \csc ^2(c+d x) \left (-38+\csc (c+d x)-18 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+18 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \sin ^8\left (\frac {1}{2} (c+d x)\right ) \sin ^7(c+d x)}{512 a^3 d (1+\sin (c+d x))^3} \]
-1/512*((Csc[(c + d*x)/2]^2 + 2*Csc[c + d*x])^5*(Csc[(c + d*x)/2]^6*(-6 + Csc[c + d*x]) - 8*(-6 + Csc[c + d*x])*Csc[c + d*x]^3 + 2*Csc[(c + d*x)/2]^ 4*Csc[c + d*x]*(-6 + Csc[c + d*x] + 18*Log[Cos[(c + d*x)/2]] - 18*Log[Sin[ (c + d*x)/2]]) - 4*Csc[(c + d*x)/2]^2*Csc[c + d*x]^2*(-38 + Csc[c + d*x] - 18*Log[Cos[(c + d*x)/2]] + 18*Log[Sin[(c + d*x)/2]]))*Sin[(c + d*x)/2]^8* Sin[c + d*x]^7)/(a^3*d*(1 + Sin[c + d*x])^3)
Time = 0.47 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95, 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, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^3 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \csc ^3(c+d x) \sec ^2(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\cos (c+d x)^2 \sin (c+d x)^3}dx}{a^6}\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (a \csc ^3(c+d x)-3 a \csc ^2(c+d x)+4 a \csc (c+d x)-\frac {4 a}{\sin (c+d x)+1}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {9 a \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a \cot (c+d x)}{d}+\frac {4 a \cos (c+d x)}{d (\sin (c+d x)+1)}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}}{a^4}\) |
((-9*a*ArcTanh[Cos[c + d*x]])/(2*d) + (3*a*Cot[c + d*x])/d - (a*Cot[c + d* x]*Csc[c + d*x])/(2*d) + (4*a*Cos[c + d*x])/(d*(1 + Sin[c + d*x])))/a^4
3.5.37.3.1 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]))
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 = 0.44 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+18 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {32}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{4 d \,a^{3}}\) | \(87\) |
| default | \(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+18 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {32}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{4 d \,a^{3}}\) | \(87\) |
| parallelrisch | \(\frac {36 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )-\left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-11 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-88 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(105\) |
| risch | \(\frac {9 \,{\mathrm e}^{4 i \left (d x +c \right )}-21 \,{\mathrm e}^{2 i \left (d x +c \right )}+7 i {\mathrm e}^{3 i \left (d x +c \right )}+14-5 i {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) d \,a^{3}}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}\) | \(124\) |
| norman | \(\frac {-\frac {1}{8 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {7 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {81 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {51 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {303 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {689 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {1093 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {1141 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {1289 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {9 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(260\) |
1/4/d/a^3*(1/2*tan(1/2*d*x+1/2*c)^2-6*tan(1/2*d*x+1/2*c)-1/2/tan(1/2*d*x+1 /2*c)^2+6/tan(1/2*d*x+1/2*c)+18*ln(tan(1/2*d*x+1/2*c))+32/(tan(1/2*d*x+1/2 *c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (74) = 148\).
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.15 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {28 \, \cos \left (d x + c\right )^{3} + 18 \, \cos \left (d x + c\right )^{2} - 9 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (14 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) - 8\right )} \sin \left (d x + c\right ) - 26 \, \cos \left (d x + c\right ) - 16}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
1/4*(28*cos(d*x + c)^3 + 18*cos(d*x + c)^2 - 9*(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)*log(1/2*cos( d*x + c) + 1/2) + 9*(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1 )*sin(d*x + c) - cos(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) - 2*(14*co s(d*x + c)^2 + 5*cos(d*x + c) - 8)*sin(d*x + c) - 26*cos(d*x + c) - 16)/(a ^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 - a^3*d*cos(d*x + c) - a^3*d + (a^3*d*cos(d*x + c)^2 - a^3*d)*sin(d*x + c))
\[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{4}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Integral(cos(c + d*x)**4*csc(c + d*x)**3/(sin(c + d*x)**3 + 3*sin(c + d*x) **2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (74) = 148\).
Time = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.06 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {11 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1}{\frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} - \frac {\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{3}} + \frac {36 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{8 \, d} \]
1/8*((11*sin(d*x + c)/(cos(d*x + c) + 1) + 76*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)/(a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^3 /(cos(d*x + c) + 1)^3) - (12*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c )^2/(cos(d*x + c) + 1)^2)/a^3 + 36*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^ 3)/d
Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.49 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {36 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {64}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} - \frac {54 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{8 \, d} \]
1/8*(36*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + 64/(a^3*(tan(1/2*d*x + 1/2*c) + 1)) - (54*tan(1/2*d*x + 1/2*c)^2 - 12*tan(1/2*d*x + 1/2*c) + 1)/(a^3*ta n(1/2*d*x + 1/2*c)^2) + (a^3*tan(1/2*d*x + 1/2*c)^2 - 12*a^3*tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 9.56 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.54 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {9\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^3\,d}+\frac {38\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {1}{2}}{d\,\left (4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \]