Integrand size = 13, antiderivative size = 53 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {3 x}{2 a}-\frac {4 \cos (x)}{a}+\frac {4 \cos ^3(x)}{3 a}+\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^2(x)}{a+a \csc (x)} \]
Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=\frac {-21 \cos (x)+\cos (3 x)+3 \left (-6 x+\frac {8 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}+\sin (2 x)\right )}{12 a} \]
Time = 0.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 4306, 25, 3042, 4274, 3042, 3113, 2009, 3115, 24}
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.
\(\displaystyle \int \frac {\sin ^3(x)}{a \csc (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc (x)^3 (a \csc (x)+a)}dx\) |
\(\Big \downarrow \) 4306 |
\(\displaystyle \frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}-\frac {\int -\left ((4 a-3 a \csc (x)) \sin ^3(x)\right )dx}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int (4 a-3 a \csc (x)) \sin ^3(x)dx}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {4 a-3 a \csc (x)}{\csc (x)^3}dx}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {4 a \int \sin ^3(x)dx-3 a \int \sin ^2(x)dx}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \int \sin (x)^3dx-3 a \int \sin (x)^2dx}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle \frac {-3 a \int \sin (x)^2dx-4 a \int \left (1-\cos ^2(x)\right )d\cos (x)}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-3 a \int \sin (x)^2dx-4 a \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {-3 a \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-4 a \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {-4 a \left (\cos (x)-\frac {\cos ^3(x)}{3}\right )-3 a \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )}{a^2}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a}\) |
(Cos[x]*Sin[x]^2)/(a + a*Csc[x]) + (-4*a*(Cos[x] - Cos[x]^3/3) - 3*a*(x/2 - (Cos[x]*Sin[x])/2))/a^2
3.1.9.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(a + b*Csc[e + f*x]))), x] - Simp[1/a^2 Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0 ]
Time = 0.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.62
method | result | size |
parallelrisch | \(\frac {-18 x -32+\cos \left (3 x \right )-21 \cos \left (x \right )+3 \sin \left (2 x \right )+12 \tan \left (x \right )-12 \sec \left (x \right )}{12 a}\) | \(33\) |
risch | \(-\frac {3 x}{2 a}-\frac {7 \,{\mathrm e}^{i x}}{8 a}-\frac {7 \,{\mathrm e}^{-i x}}{8 a}-\frac {2}{\left (i+{\mathrm e}^{i x}\right ) a}+\frac {\cos \left (3 x \right )}{12 a}+\frac {\sin \left (2 x \right )}{4 a}\) | \(61\) |
default | \(\frac {-\frac {2}{\tan \left (\frac {x}{2}\right )+1}-\frac {2 \left (\frac {\tan \left (\frac {x}{2}\right )^{5}}{2}+\tan \left (\frac {x}{2}\right )^{4}+4 \tan \left (\frac {x}{2}\right )^{2}-\frac {\tan \left (\frac {x}{2}\right )}{2}+\frac {5}{3}\right )}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3}}-3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a}\) | \(66\) |
norman | \(\frac {-\frac {5 \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {5 \tan \left (\frac {x}{2}\right )^{5}}{a}-\frac {3 x}{2 a}-\frac {8}{3 a}-\frac {3 x \tan \left (\frac {x}{2}\right )}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{2}}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{3}}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{4}}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{5}}{2 a}-\frac {3 x \tan \left (\frac {x}{2}\right )^{6}}{2 a}-\frac {3 x \tan \left (\frac {x}{2}\right )^{7}}{2 a}-\frac {\tan \left (\frac {x}{2}\right )^{6}}{3 a}+\frac {\tan \left (\frac {x}{2}\right )}{3 a}+\frac {8 \tan \left (\frac {x}{2}\right )^{7}}{3 a}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(167\) |
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=\frac {2 \, \cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, {\left (3 \, x + 5\right )} \cos \left (x\right ) - 12 \, \cos \left (x\right )^{2} + {\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 9 \, x - 9 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) - 9 \, x - 6}{6 \, {\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \]
1/6*(2*cos(x)^4 - cos(x)^3 - 3*(3*x + 5)*cos(x) - 12*cos(x)^2 + (2*cos(x)^ 3 + 3*cos(x)^2 - 9*x - 9*cos(x) + 6)*sin(x) - 9*x - 6)/(a*cos(x) + a*sin(x ) + a)
\[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\sin ^{3}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (47) = 94\).
Time = 0.35 (sec) , antiderivative size = 180, normalized size of antiderivative = 3.40 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {39 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {24 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {24 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {9 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {9 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 16}{3 \, {\left (a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {3 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \]
-1/3*(7*sin(x)/(cos(x) + 1) + 39*sin(x)^2/(cos(x) + 1)^2 + 24*sin(x)^3/(co s(x) + 1)^3 + 24*sin(x)^4/(cos(x) + 1)^4 + 9*sin(x)^5/(cos(x) + 1)^5 + 9*s in(x)^6/(cos(x) + 1)^6 + 16)/(a + a*sin(x)/(cos(x) + 1) + 3*a*sin(x)^2/(co s(x) + 1)^2 + 3*a*sin(x)^3/(cos(x) + 1)^3 + 3*a*sin(x)^4/(cos(x) + 1)^4 + 3*a*sin(x)^5/(cos(x) + 1)^5 + a*sin(x)^6/(cos(x) + 1)^6 + a*sin(x)^7/(cos( x) + 1)^7) - 3*arctan(sin(x)/(cos(x) + 1))/a
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, x\right ) + 10}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a} \]
-3/2*x/a - 2/(a*(tan(1/2*x) + 1)) - 1/3*(3*tan(1/2*x)^5 + 6*tan(1/2*x)^4 + 24*tan(1/2*x)^2 - 3*tan(1/2*x) + 10)/((tan(1/2*x)^2 + 1)^3*a)
Time = 19.65 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {3\,x}{2\,a}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {16}{3}}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]