3.282 \(\int \frac{\cos ^2(x)}{(a-a \sin ^2(x))^2} \, dx\)

Optimal. Leaf size=6 \[ \frac{\tan (x)}{a^2} \]

[Out]

Tan[x]/a^2

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Rubi [A]  time = 0.0430509, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3175, 3767, 8} \[ \frac{\tan (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^2/(a - a*Sin[x]^2)^2,x]

[Out]

Tan[x]/a^2

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]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{\cos ^2(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \sec ^2(x) \, dx}{a^2}\\ &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,-\tan (x))}{a^2}\\ &=\frac{\tan (x)}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0022568, size = 6, normalized size = 1. \[ \frac{\tan (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[x]/a^2

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Maple [A]  time = 0.033, size = 7, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( x \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2/(a-a*sin(x)^2)^2,x)

[Out]

tan(x)/a^2

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Maxima [A]  time = 0.989701, size = 8, normalized size = 1.33 \begin{align*} \frac{\tan \left (x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)/a^2

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Fricas [A]  time = 1.85202, size = 28, normalized size = 4.67 \begin{align*} \frac{\sin \left (x\right )}{a^{2} \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

sin(x)/(a^2*cos(x))

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Sympy [B]  time = 7.59203, size = 20, normalized size = 3.33 \begin{align*} - \frac{2 \tan{\left (\frac{x}{2} \right )}}{a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} - a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**2/(a-a*sin(x)**2)**2,x)

[Out]

-2*tan(x/2)/(a**2*tan(x/2)**2 - a**2)

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Giac [A]  time = 1.0964, size = 8, normalized size = 1.33 \begin{align*} \frac{\tan \left (x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(x)/a^2