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

Optimal. Leaf size=53 \[ -\frac {3 x}{2 a}+\frac {4 \cos ^3(x)}{3 a}-\frac {4 \cos (x)}{a}+\frac {3 \sin (x) \cos (x)}{2 a}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a} \]

[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))

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Rubi [A]  time = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3819, 3787, 2633, 2635, 8} \[ -\frac {3 x}{2 a}+\frac {4 \cos ^3(x)}{3 a}-\frac {4 \cos (x)}{a}+\frac {3 \sin (x) \cos (x)}{2 a}+\frac {\sin ^2(x) \cos (x)}{a \csc (x)+a} \]

Antiderivative was successfully verified.

[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 2633

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 2635

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] && Integer
Q[2*n]

Rule 3787

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 3819

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*} \int \frac {\sin ^3(x)}{a+a \csc (x)} \, dx &=\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 \operatorname {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*}

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Mathematica [A]  time = 0.17, size = 49, normalized size = 0.92 \[ \frac {-21 \cos (x)+\cos (3 x)+3 \left (-6 x+\sin (2 x)+\frac {8 \sin \left (\frac {x}{2}\right )}{\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}{12 a} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.45, size = 70, normalized size = 1.32 \[ \frac {2 \, \cos \relax (x)^{4} - \cos \relax (x)^{3} - 3 \, {\left (3 \, x + 5\right )} \cos \relax (x) - 12 \, \cos \relax (x)^{2} + {\left (2 \, \cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} - 9 \, x - 9 \, \cos \relax (x) + 6\right )} \sin \relax (x) - 9 \, x - 6}{6 \, {\left (a \cos \relax (x) + a \sin \relax (x) + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.29, size = 67, normalized size = 1.26 \[ -\frac {3 \, x}{2 \, a} - \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, x\right ) + 10}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.51, size = 121, normalized size = 2.28 \[ -\frac {2}{a \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {\tan ^{5}\left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {8 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\tan \left (\frac {x}{2}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {10}{3 a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a/(tan(1/2*x)+1)-1/a/(tan(1/2*x)^2+1)^3*tan(1/2*x)^5-2/a/(tan(1/2*x)^2+1)^3*tan(1/2*x)^4-8/a/(tan(1/2*x)^2+
1)^3*tan(1/2*x)^2+1/a/(tan(1/2*x)^2+1)^3*tan(1/2*x)-10/3/a/(tan(1/2*x)^2+1)^3-3/a*arctan(tan(1/2*x))

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maxima [B]  time = 0.43, size = 180, normalized size = 3.40 \[ -\frac {\frac {7 \, \sin \relax (x)}{\cos \relax (x) + 1} + \frac {39 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {24 \, \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {24 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {9 \, \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {9 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + 16}{3 \, {\left (a + \frac {a \sin \relax (x)}{\cos \relax (x) + 1} + \frac {3 \, a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {3 \, a \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} + \frac {3 \, a \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {3 \, a \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {a \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {a \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}}\right )}} - \frac {3 \, \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.35, size = 78, normalized size = 1.47 \[ -\frac {3\,x}{2\,a}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {16}{3}}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sin ^{3}{\relax (x )}}{\csc {\relax (x )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x)**3/(csc(x) + 1), x)/a

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