3.1.8 \(\int \frac {\csc ^2(x)}{a-a \cos ^2(x)} \, dx\) [8]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {3254, 3852} \begin {gather*} -\frac {\cot ^3(x)}{3 a}-\frac {\cot (x)}{a} \end {gather*}

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 3254

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 3852

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 {\text {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 = 0.01, size = 21, normalized size = 1.11 \begin {gather*} \frac {-\frac {2 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x)}{a} \end {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 18, normalized size = 0.95

method	result	size
default	\(\frac {-\frac {1}{3 \tan \left (x \right )^{3}}-\frac {1}{\tan \left (x \right )}}{a}\)	\(18\)
risch	\(\frac {4 i \left (3 \,{\mathrm e}^{2 i x}-1\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3} a}\)	\(25\)
norman	\(\frac {-\frac {1}{24 a}-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8 a}+\frac {\tan ^{6}\left (\frac {x}{2}\right )}{24 a}}{\tan \left (\frac {x}{2}\right )^{3}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2/(a-a*cos(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 17, normalized size = 0.89 \begin {gather*} -\frac {3 \, \tan \left (x\right )^{2} + 1}{3 \, a \tan \left (x\right )^{3}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 29, normalized size = 1.53 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\csc ^{2}{\left (x \right )}}{\cos ^{2}{\left (x \right )} - 1}\, dx}{a} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.39, size = 17, normalized size = 0.89 \begin {gather*} -\frac {3 \, \tan \left (x\right )^{2} + 1}{3 \, a \tan \left (x\right )^{3}} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 2.03, size = 13, normalized size = 0.68 \begin {gather*} -\frac {\mathrm {cot}\left (x\right )\,\left ({\mathrm {cot}\left (x\right )}^2+3\right )}{3\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^2*(a - a*cos(x)^2)),x)

[Out]

-(cot(x)*(cot(x)^2 + 3))/(3*a)

________________________________________________________________________________________