Integrand size = 25, antiderivative size = 68 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac {5 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))} \]
-arctanh(cos(d*x+c))/a^3/d+2/3*cos(d*x+c)/a^3/d/(1+sin(d*x+c))^2+5/3*cos(d *x+c)/a^3/d/(1+sin(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(185\) vs. \(2(68)=136\).
Time = 0.68 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.72 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-4 \sin \left (\frac {1}{2} (c+d x)\right )+2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-10 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2-3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3+3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3\right )}{3 d (a+a \sin (c+d x))^3} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-4*Sin[(c + d*x)/2] + 2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 10*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Si n[(c + d*x)/2])^2 - 3*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d *x)/2])^3 + 3*Log[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^ 3))/(3*d*(a + a*Sin[c + d*x])^3)
Time = 0.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3353, 3042, 3445, 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.
\(\displaystyle \int \frac {\cos (c+d x) \cot (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x) (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3353 |
\(\displaystyle \frac {\int \frac {\csc (c+d x) (a-a \sin (c+d x))}{(\sin (c+d x) a+a)^2}dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a-a \sin (c+d x)}{\sin (c+d x) (\sin (c+d x) a+a)^2}dx}{a^2}\) |
\(\Big \downarrow \) 3445 |
\(\displaystyle \frac {\int \left (\frac {\csc (c+d x)}{a}-\frac {1}{a (\sin (c+d x)+1)}-\frac {2}{a (\sin (c+d x)+1)^2}\right )dx}{a^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {5 \cos (c+d x)}{3 a d (\sin (c+d x)+1)}+\frac {2 \cos (c+d x)}{3 a d (\sin (c+d x)+1)^2}}{a^2}\) |
(-(ArcTanh[Cos[c + d*x]]/(a*d)) + (2*Cos[c + d*x])/(3*a*d*(1 + Sin[c + d*x ])^2) + (5*Cos[c + d*x])/(3*a*d*(1 + Sin[c + d*x])))/a^2
3.4.18.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/b^2 Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] || !IGtQ[ n, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[si n[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; FreeQ[{ a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ [m] && IntegerQ[n]
Time = 0.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(64\) |
default | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{d \,a^{3}}\) | \(64\) |
parallelrisch | \(\frac {3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+14}{3 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(73\) |
risch | \(\frac {8 i {\mathrm e}^{i \left (d x +c \right )}+2 \,{\mathrm e}^{2 i \left (d x +c \right )}-\frac {10}{3}}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(88\) |
norman | \(\frac {\frac {6 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {20 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {14}{3 a d}+\frac {52 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}+\frac {94 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {98 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {112 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(172\) |
1/d/a^3*(ln(tan(1/2*d*x+1/2*c))+8/3/(tan(1/2*d*x+1/2*c)+1)^3-4/(tan(1/2*d* x+1/2*c)+1)^2+6/(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (64) = 128\).
Time = 0.26 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.85 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {10 \, \cos \left (d x + c\right )^{2} + 3 \, {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (5 \, \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 14 \, \cos \left (d x + c\right ) + 4}{6 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
-1/6*(10*cos(d*x + c)^2 + 3*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*log(1/2*cos(d*x + c) + 1/2) - 3*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*log(-1/2*cos(d*x + c) + 1/2) + 2*(5*cos(d*x + c) - 2)*sin(d*x + c) + 14*cos(d*x + c) + 4)/(a^3*d *cos(d*x + c)^2 - a^3*d*cos(d*x + c) - 2*a^3*d - (a^3*d*cos(d*x + c) + 2*a ^3*d)*sin(d*x + c))
\[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Integral(cos(c + d*x)**2*csc(c + d*x)/(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. 143 vs. \(2 (64) = 128\).
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.10 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 7\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}} + \frac {3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]
1/3*(2*(12*sin(d*x + c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^2/(cos(d*x + c ) + 1)^2 + 7)/(a^3 + 3*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 3*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) + 3*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.62 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.97 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {2 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{3 \, d} \]
1/3*(3*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + 2*(9*tan(1/2*d*x + 1/2*c)^2 + 12*tan(1/2*d*x + 1/2*c) + 7)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^3))/d
Time = 9.85 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int \frac {\cos (c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}+\frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {14}{3}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^3} \]