Integrand size = 27, antiderivative size = 45 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {x}{a^3}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {4 \cos (c+d x)}{a^3 d (1+\sin (c+d x))} \]
Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(45)=90\).
Time = 0.76 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.71 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (c+d x)\right ) \left (c+d x-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\left (-8+c+d x-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d (1+\sin (c+d x))^3} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(Cos[(c + d*x)/2]*(c + d*x - Log[ Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]) + (-8 + c + d*x - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])*Sin[(c + d*x)/2]))/(a^3*d*(1 + Sin[c + d *x])^3)
Time = 0.39 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96, 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.
\(\displaystyle \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x) (a \sin (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3354 |
\(\displaystyle \frac {\int \csc (c+d x) \sec ^2(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(a-a \sin (c+d x))^3}{\cos (c+d x)^2 \sin (c+d x)}dx}{a^6}\) |
\(\Big \downarrow \) 3351 |
\(\displaystyle \frac {\int \left (\csc (c+d x) a-\frac {4 a}{\sin (c+d x)+1}+a\right )dx}{a^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {a \text {arctanh}(\cos (c+d x))}{d}+\frac {4 a \cos (c+d x)}{d (\sin (c+d x)+1)}+a x}{a^4}\) |
3.5.35.3.1 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 = 0.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(46\) |
default | \(\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(46\) |
parallelrisch | \(\frac {\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d x +d x +8}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(60\) |
risch | \(\frac {x}{a^{3}}+\frac {8}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\) | \(68\) |
norman | \(\frac {\frac {x}{a}+\frac {x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {8}{a d}+\frac {8 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {32 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {72 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {72 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {128 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {13 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {25 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {38 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {46 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {46 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {38 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {25 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {13 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {128 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {176 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {176 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {192 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\) | \(437\) |
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (45) = 90\).
Time = 0.27 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.60 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, d x + 2 \, {\left (d x + 4\right )} \cos \left (d x + c\right ) - {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (d x - 4\right )} \sin \left (d x + c\right ) + 8}{2 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
1/2*(2*d*x + 2*(d*x + 4)*cos(d*x + c) - (cos(d*x + c) + sin(d*x + c) + 1)* log(1/2*cos(d*x + c) + 1/2) + (cos(d*x + c) + sin(d*x + c) + 1)*log(-1/2*c os(d*x + c) + 1/2) + 2*(d*x - 4)*sin(d*x + c) + 8)/(a^3*d*cos(d*x + c) + a ^3*d*sin(d*x + c) + a^3*d)
\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Integral(cos(c + d*x)**4*csc(c + d*x)/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {8}{a^{3} + \frac {a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}} + \frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{d} \]
(8/(a^3 + a^3*sin(d*x + c)/(cos(d*x + c) + 1)) + 2*arctan(sin(d*x + c)/(co s(d*x + c) + 1))/a^3 + log(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {d x + c}{a^{3}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {8}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{d} \]
Time = 9.63 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.56 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {8}{d\,\left (a^3+a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}+\frac {2\,\mathrm {atan}\left (\frac {4\,a^3}{4\,a^3-4\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {4\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^3-4\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a^3\,d} \]