\(\int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx\) [9]

Optimal result

Integrand size = 13, antiderivative size = 53 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {3 x}{2 a}-\frac {4 \cos (x)}{a}+\frac {4 \cos ^3(x)}{3 a}+\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^2(x)}{a+a \csc (x)} \]

[Out]

-3/2*x/a-4*cos(x)/a+4/3*cos(x)^3/a+3/2*cos(x)*sin(x)/a+cos(x)*sin(x)^2/(a+a*csc(x))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2713, 2715, 8} \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \cos ^3(x)}{3 a}-\frac {4 \cos (x)}{a}+\frac {3 \sin (x) \cos (x)}{2 a}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a} \]

[In]

Int[Sin[x]^3/(a + a*Csc[x]),x]

[Out]

(-3*x)/(2*a) - (4*Cos[x])/a + (4*Cos[x]^3)/(3*a) + (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x]^2)/(a + a*Csc[x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3904

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (x) \sin ^2(x)}{a+a \csc (x)}-\frac {\int (-4 a+3 a \csc (x)) \sin ^3(x) \, dx}{a^2} \\ & = \frac {\cos (x) \sin ^2(x)}{a+a \csc (x)}-\frac {3 \int \sin ^2(x) \, dx}{a}+\frac {4 \int \sin ^3(x) \, dx}{a} \\ & = \frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^2(x)}{a+a \csc (x)}-\frac {3 \int 1 \, dx}{2 a}-\frac {4 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{a} \\ & = -\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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=\frac {-21 \cos (x)+\cos (3 x)+3 \left (-6 x+\frac {8 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}+\sin (2 x)\right )}{12 a} \]

[In]

Integrate[Sin[x]^3/(a + a*Csc[x]),x]

[Out]

(-21*Cos[x] + Cos[3*x] + 3*(-6*x + (8*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + Sin[2*x]))/(12*a)

Maple [A] (verified)

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	\(\frac {-\frac {5 \tan \left (\frac {x}{2}\right )^{2}}{a}+\frac {5 \tan \left (\frac {x}{2}\right )^{5}}{a}-\frac {3 x}{2 a}-\frac {8}{3 a}-\frac {3 x \tan \left (\frac {x}{2}\right )}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{2}}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{3}}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{4}}{2 a}-\frac {9 x \tan \left (\frac {x}{2}\right )^{5}}{2 a}-\frac {3 x \tan \left (\frac {x}{2}\right )^{6}}{2 a}-\frac {3 x \tan \left (\frac {x}{2}\right )^{7}}{2 a}-\frac {\tan \left (\frac {x}{2}\right )^{6}}{3 a}+\frac {\tan \left (\frac {x}{2}\right )}{3 a}+\frac {8 \tan \left (\frac {x}{2}\right )^{7}}{3 a}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {x}{2}\right )+1\right )}\)	\(167\)

[In]

int(sin(x)^3/(a+a*csc(x)),x,method=_RETURNVERBOSE)

[Out]

1/12*(-18*x-32+cos(3*x)-21*cos(x)+3*sin(2*x)+12*tan(x)-12*sec(x))/a

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=\frac {2 \, \cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, {\left (3 \, x + 5\right )} \cos \left (x\right ) - 12 \, \cos \left (x\right )^{2} + {\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 9 \, x - 9 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) - 9 \, x - 6}{6 \, {\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \]

[In]

integrate(sin(x)^3/(a+a*csc(x)),x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=\frac {\int \frac {\sin ^{3}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(sin(x)**3/(a+a*csc(x)),x)

[Out]

Integral(sin(x)**3/(csc(x) + 1), x)/a

Maxima [B] (verification not implemented)

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 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {39 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {24 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {24 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {9 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {9 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 16}{3 \, {\left (a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {3 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \]

[In]

integrate(sin(x)^3/(a+a*csc(x)),x, algorithm="maxima")

[Out]

-1/3*(7*sin(x)/(cos(x) + 1) + 39*sin(x)^2/(cos(x) + 1)^2 + 24*sin(x)^3/(cos(x) + 1)^3 + 24*sin(x)^4/(cos(x) +
1)^4 + 9*sin(x)^5/(cos(x) + 1)^5 + 9*sin(x)^6/(cos(x) + 1)^6 + 16)/(a + a*sin(x)/(cos(x) + 1) + 3*a*sin(x)^2/(
cos(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*si
n(x)^6/(cos(x) + 1)^6 + a*sin(x)^7/(cos(x) + 1)^7) - 3*arctan(sin(x)/(cos(x) + 1))/a

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, x\right ) + 10}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a} \]

[In]

integrate(sin(x)^3/(a+a*csc(x)),x, algorithm="giac")

[Out]

-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)

Mupad [B] (verification not implemented)

Time = 19.65 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx=-\frac {3\,x}{2\,a}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {16}{3}}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]

[In]

int(sin(x)^3/(a + a/sin(x)),x)

[Out]

- (3*x)/(2*a) - ((7*tan(x/2))/3 + 13*tan(x/2)^2 + 8*tan(x/2)^3 + 8*tan(x/2)^4 + 3*tan(x/2)^5 + 3*tan(x/2)^6 +
16/3)/(a*(tan(x/2)^2 + 1)^3*(tan(x/2) + 1))