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

Optimal. Leaf size=5 \[ \frac{x}{a} \]

[Out]

x/a

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Rubi [A]  time = 0.040586, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 8} \[ \frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^2/(a - a*Sin[x]^2),x]

[Out]

x/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 8

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

Rubi steps

\begin{align*} \int \frac{\cos ^2(x)}{a-a \sin ^2(x)} \, dx &=\frac{\int 1 \, dx}{a}\\ &=\frac{x}{a}\\ \end{align*}

Mathematica [A]  time = 0.0007805, size = 5, normalized size = 1. \[ \frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a

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Maple [C]  time = 0.033, size = 8, normalized size = 1.6 \begin{align*}{\frac{\arctan \left ( \tan \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*arctan(tan(x))

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Maxima [A]  time = 1.47286, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Fricas [A]  time = 1.58364, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Sympy [A]  time = 2.11331, size = 2, normalized size = 0.4 \begin{align*} \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Giac [B]  time = 1.10916, size = 19, normalized size = 3.8 \begin{align*} \frac{\arctan \left (\frac{{\left | a \right |} \tan \left (x\right )}{a}\right )}{{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(abs(a)*tan(x)/a)/abs(a)