3.8 \(\int \frac{\csc ^2(x)}{a-a \cos ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\cot ^3(x)}{3 a}-\frac{\cot (x)}{a} \]

[Out]

-(Cot[x]/a) - Cot[x]^3/(3*a)

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Rubi [A]  time = 0.0477914, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 3767} \[ -\frac{\cot ^3(x)}{3 a}-\frac{\cot (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/(a - a*Cos[x]^2),x]

[Out]

-(Cot[x]/a) - Cot[x]^3/(3*a)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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]

Rubi steps

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

Mathematica [A]  time = 0.0032647, size = 21, normalized size = 1.11 \[ \frac{-\frac{2 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2/(a - a*Cos[x]^2),x]

[Out]

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

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Maple [A]  time = 0.03, size = 18, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{3\, \left ( \tan \left ( x \right ) \right ) ^{3}}}- \left ( \tan \left ( x \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)^2),x)

[Out]

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

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Maxima [A]  time = 0.941513, size = 23, normalized size = 1.21 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, a \tan \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*tan(x)^2 + 1)/(a*tan(x)^3)

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Fricas [A]  time = 1.82509, size = 76, normalized size = 4. \begin{align*} -\frac{2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \,{\left (a \cos \left (x\right )^{2} - 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)^2),x, algorithm="fricas")

[Out]

-1/3*(2*cos(x)^3 - 3*cos(x))/((a*cos(x)^2 - 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 ^{2}{\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)**2),x)

[Out]

-Integral(csc(x)**2/(cos(x)**2 - 1), x)/a

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Giac [A]  time = 1.17476, size = 23, normalized size = 1.21 \begin{align*} -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, a \tan \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*tan(x)^2 + 1)/(a*tan(x)^3)