3.8 \(\int \frac{\sin ^2(x)}{a+a \csc (x)} \, dx\)

Optimal. Leaf size=40 \[ \frac{3 x}{2 a}+\frac{2 \cos (x)}{a}-\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin (x) \cos (x)}{a \csc (x)+a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0612801, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2638} \[ \frac{3 x}{2 a}+\frac{2 \cos (x)}{a}-\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin (x) \cos (x)}{a \csc (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^2/(a + a*Csc[x]),x]

[Out]

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

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 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 8

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\sin ^2(x)}{a+a \csc (x)} \, dx &=\frac{\cos (x) \sin (x)}{a+a \csc (x)}-\frac{\int (-3 a+2 a \csc (x)) \sin ^2(x) \, dx}{a^2}\\ &=\frac{\cos (x) \sin (x)}{a+a \csc (x)}-\frac{2 \int \sin (x) \, dx}{a}+\frac{3 \int \sin ^2(x) \, dx}{a}\\ &=\frac{2 \cos (x)}{a}-\frac{3 \cos (x) \sin (x)}{2 a}+\frac{\cos (x) \sin (x)}{a+a \csc (x)}+\frac{3 \int 1 \, dx}{2 a}\\ &=\frac{3 x}{2 a}+\frac{2 \cos (x)}{a}-\frac{3 \cos (x) \sin (x)}{2 a}+\frac{\cos (x) \sin (x)}{a+a \csc (x)}\\ \end{align*}

Mathematica [A]  time = 0.129952, size = 42, normalized size = 1.05 \[ -\frac{-6 x+\sin (2 x)-4 \cos (x)+\frac{8 \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}}{4 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^2/(a + a*Csc[x]),x]

[Out]

-(-6*x - 4*Cos[x] + (8*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + Sin[2*x])/(4*a)

________________________________________________________________________________________

Maple [B]  time = 0.042, size = 100, normalized size = 2.5 \begin{align*}{\frac{1}{a} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+2\,{\frac{ \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{1}{a}\tan \left ({\frac{x}{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+2\,{\frac{1}{a \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}}+3\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}+2\,{\frac{1}{a \left ( \tan \left ( x/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^2/(a+a*csc(x)),x)

[Out]

1/a/(tan(1/2*x)^2+1)^2*tan(1/2*x)^3+2/a/(tan(1/2*x)^2+1)^2*tan(1/2*x)^2-1/a/(tan(1/2*x)^2+1)^2*tan(1/2*x)+2/a/
(tan(1/2*x)^2+1)^2+3/a*arctan(tan(1/2*x))+2/a/(tan(1/2*x)+1)

________________________________________________________________________________________

Maxima [B]  time = 1.45891, size = 173, normalized size = 4.32 \begin{align*} \frac{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{3 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 4}{a + \frac{a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{2 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{2 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac{a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}} + \frac{3 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(a+a*csc(x)),x, algorithm="maxima")

[Out]

(sin(x)/(cos(x) + 1) + 5*sin(x)^2/(cos(x) + 1)^2 + 3*sin(x)^3/(cos(x) + 1)^3 + 3*sin(x)^4/(cos(x) + 1)^4 + 4)/
(a + a*sin(x)/(cos(x) + 1) + 2*a*sin(x)^2/(cos(x) + 1)^2 + 2*a*sin(x)^3/(cos(x) + 1)^3 + a*sin(x)^4/(cos(x) +
1)^4 + a*sin(x)^5/(cos(x) + 1)^5) + 3*arctan(sin(x)/(cos(x) + 1))/a

________________________________________________________________________________________

Fricas [A]  time = 0.474, size = 166, normalized size = 4.15 \begin{align*} \frac{\cos \left (x\right )^{3} + 3 \,{\left (x + 1\right )} \cos \left (x\right ) + 2 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{2} - 3 \, x - \cos \left (x\right ) + 2\right )} \sin \left (x\right ) + 3 \, x + 2}{2 \,{\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(a+a*csc(x)),x, algorithm="fricas")

[Out]

1/2*(cos(x)^3 + 3*(x + 1)*cos(x) + 2*cos(x)^2 - (cos(x)^2 - 3*x - cos(x) + 2)*sin(x) + 3*x + 2)/(a*cos(x) + a*
sin(x) + a)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sin ^{2}{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**2/(a+a*csc(x)),x)

[Out]

Integral(sin(x)**2/(csc(x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.36766, size = 76, normalized size = 1.9 \begin{align*} \frac{3 \, x}{2 \, a} + \frac{\tan \left (\frac{1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac{1}{2} \, x\right )^{2} - \tan \left (\frac{1}{2} \, x\right ) + 2}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{2} a} + \frac{2}{a{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^2/(a+a*csc(x)),x, algorithm="giac")

[Out]

3/2*x/a + (tan(1/2*x)^3 + 2*tan(1/2*x)^2 - tan(1/2*x) + 2)/((tan(1/2*x)^2 + 1)^2*a) + 2/(a*(tan(1/2*x) + 1))