Integrand size = 29, antiderivative size = 60 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {x}{a^3}-\frac {7 \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d} \]
Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(60)=120\).
Time = 0.99 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.10 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (-8 (c+d x)+12 \cot \left (\frac {1}{2} (c+d x)\right )-\csc ^2\left (\frac {1}{2} (c+d x)\right )-28 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+28 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sec ^2\left (\frac {1}{2} (c+d x)\right )-12 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{8 d (a+a \sin (c+d x))^3} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-8*(c + d*x) + 12*Cot[(c + d*x)/ 2] - Csc[(c + d*x)/2]^2 - 28*Log[Cos[(c + d*x)/2]] + 28*Log[Sin[(c + d*x)/ 2]] + Sec[(c + d*x)/2]^2 - 12*Tan[(c + d*x)/2]))/(8*d*(a + a*Sin[c + d*x]) ^3)
Time = 0.40 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3348, 3042, 3236, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^3 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3348 |
| \(\displaystyle \frac {\int \csc ^3(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\sin (c+d x)^3}dx}{a^6}\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \frac {\int \left (\csc ^3(c+d x) a^3-3 \csc ^2(c+d x) a^3+3 \csc (c+d x) a^3-a^3\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {7 a^3 \text {arctanh}(\cos (c+d x))}{2 d}+\frac {3 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot (c+d x) \csc (c+d x)}{2 d}+a^3 (-x)}{a^6}\) |
(-(a^3*x) - (7*a^3*ArcTanh[Cos[c + d*x]])/(2*d) + (3*a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]*Csc[c + d*x])/(2*d))/a^6
3.7.49.3.1 Defintions of rubi rules used
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m) Int[(d* Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
Time = 0.43 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.20
| method | result | size |
| parallelrisch | \(\frac {-\left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-8 d x +28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{3} d}\) | \(72\) |
| derivativedivides | \(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+14 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(84\) |
| default | \(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+14 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(84\) |
| risch | \(-\frac {x}{a^{3}}+\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+6 i {\mathrm e}^{2 i \left (d x +c \right )}-6 i}{a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}\) | \(101\) |
| norman | \(\frac {-\frac {25 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {5 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {46 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {38 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {13 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {46 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {38 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {25 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {13 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1}{8 a d}-\frac {5 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {51 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {47 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {2245 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {1567 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {177 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {2079 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {717 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {2005 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {407 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}-\frac {7 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {229 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {29 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}}{\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 )^{2} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(540\) |
1/8*(-cot(1/2*d*x+1/2*c)^2+tan(1/2*d*x+1/2*c)^2-8*d*x+28*ln(tan(1/2*d*x+1/ 2*c))+12*cot(1/2*d*x+1/2*c)-12*tan(1/2*d*x+1/2*c))/a^3/d
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.82 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {4 \, d x \cos \left (d x + c\right )^{2} - 4 \, d x + 7 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 7 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 12 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \]
-1/4*(4*d*x*cos(d*x + c)^2 - 4*d*x + 7*(cos(d*x + c)^2 - 1)*log(1/2*cos(d* x + c) + 1/2) - 7*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) + 12*c os(d*x + c)*sin(d*x + c) - 2*cos(d*x + c))/(a^3*d*cos(d*x + c)^2 - a^3*d)
Timed out. \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (56) = 112\).
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.30 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{3}} + \frac {16 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {28 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {{\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{3} \sin \left (d x + c\right )^{2}}}{8 \, d} \]
-1/8*((12*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a^3 + 16*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3 - 28*log(sin(d *x + c)/(cos(d*x + c) + 1))/a^3 - (12*sin(d*x + c)/(cos(d*x + c) + 1) - 1) *(cos(d*x + c) + 1)^2/(a^3*sin(d*x + c)^2))/d
Time = 0.36 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.80 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {8 \, {\left (d x + c\right )}}{a^{3}} - \frac {28 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {42 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{8 \, d} \]
-1/8*(8*(d*x + c)/a^3 - 28*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 + (42*tan(1/ 2*d*x + 1/2*c)^2 - 12*tan(1/2*d*x + 1/2*c) + 1)/(a^3*tan(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 = 10.37 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.68 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {2\,\mathrm {atan}\left (\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{7\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a^3\,d}+\frac {7\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2\,a^3\,d}+\frac {3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \]
tan(c/2 + (d*x)/2)^2/(8*a^3*d) - cot(c/2 + (d*x)/2)^2/(8*a^3*d) + (2*atan( (2*cos(c/2 + (d*x)/2) - 7*sin(c/2 + (d*x)/2))/(7*cos(c/2 + (d*x)/2) + 2*si n(c/2 + (d*x)/2))))/(a^3*d) + (7*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2) ))/(2*a^3*d) + (3*cot(c/2 + (d*x)/2))/(2*a^3*d) - (3*tan(c/2 + (d*x)/2))/( 2*a^3*d)