3.6 \(\int \frac{\cot ^2(x)}{a+a \cos (x)} \, dx\)

Optimal. Leaf size=30 \[ -\frac{\cot ^3(x)}{3 a}+\frac{\csc ^3(x)}{3 a}-\frac{\csc (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0799864, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2607, 30, 2606} \[ -\frac{\cot ^3(x)}{3 a}+\frac{\csc ^3(x)}{3 a}-\frac{\csc (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2/(a + a*Cos[x]),x]

[Out]

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

Rule 2706

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

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

Mathematica [A]  time = 0.0523389, size = 25, normalized size = 0.83 \[ \frac{(-4 \cos (x)+\cos (2 x)-3) \csc (x)}{6 a (\cos (x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2/(a + a*Cos[x]),x]

[Out]

((-3 - 4*Cos[x] + Cos[2*x])*Csc[x])/(6*a*(1 + Cos[x]))

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Maple [A]  time = 0.04, size = 29, normalized size = 1. \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(cot(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 [A]  time = 1.00078, size = 57, normalized size = 1.9 \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(cot(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 = 1.38026, size = 74, normalized size = 2.47 \begin{align*} \frac{\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 2}{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(cot(x)^2/(a+a*cos(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cot ^{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(cot(x)**2/(a+a*cos(x)),x)

[Out]

Integral(cot(x)**2/(cos(x) + 1), x)/a

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Giac [A]  time = 1.2764, size = 50, normalized size = 1.67 \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(cot(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))