3.7 \(\int \frac{\csc ^2(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=24 \[ \frac{\csc (x)}{3 (a \cos (x)+a)}-\frac{2 \cot (x)}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.046257, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ \frac{\csc (x)}{3 (a \cos (x)+a)}-\frac{2 \cot (x)}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/(a + a*Cos[x]),x]

[Out]

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

Rule 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 3767

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

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0480526, size = 30, normalized size = 1.25 \[ -\frac{(2 \cos (x)+\cos (2 x)) \csc \left (\frac{x}{2}\right ) \sec ^3\left (\frac{x}{2}\right )}{12 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2/(a + a*Cos[x]),x]

[Out]

-((2*Cos[x] + Cos[2*x])*Csc[x/2]*Sec[x/2]^3)/(12*a)

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Maple [A]  time = 0.043, size = 29, normalized size = 1.2 \begin{align*}{\frac{1}{4\,a} \left ({\frac{1}{3} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{3}}+2\,\tan \left ( x/2 \right ) - \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a*(1/3*tan(1/2*x)^3+2*tan(1/2*x)-1/tan(1/2*x))

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Maxima [B]  time = 1.1243, size = 55, normalized size = 2.29 \begin{align*} \frac{\frac{6 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac{\sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{12 \, a} - \frac{\cos \left (x\right ) + 1}{4 \, a \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*sin(x)/(cos(x) + 1) + sin(x)^3/(cos(x) + 1)^3)/a - 1/4*(cos(x) + 1)/(a*sin(x))

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Fricas [A]  time = 1.57161, size = 78, normalized size = 3.25 \begin{align*} -\frac{2 \, \cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) - 1}{3 \,{\left (a \cos \left (x\right ) + a\right )} \sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**2/(cos(x) + 1), x)/a

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Giac [A]  time = 1.13177, size = 50, normalized size = 2.08 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, x\right )^{3} + 6 \, a^{2} \tan \left (\frac{1}{2} \, x\right )}{12 \, a^{3}} - \frac{1}{4 \, a \tan \left (\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(a^2*tan(1/2*x)^3 + 6*a^2*tan(1/2*x))/a^3 - 1/4/(a*tan(1/2*x))