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 = 0.04, antiderivative size = 5, normalized size of antiderivative = 1.00, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 5, normalized size = 1.00 \[ \frac {x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a

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fricas [A]  time = 0.41, size = 5, normalized size = 1.00 \[ \frac {x}{a} \]

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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giac [B]  time = 0.14, size = 14, normalized size = 2.80 \[ \frac {\arctan \left (\frac {{\left | a \right |} \tan \relax (x)}{a}\right )}{{\left | a \right |}} \]

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)

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maple [C]  time = 0.19, size = 8, normalized size = 1.60 \[ \frac {\arctan \left (\tan \relax (x )\right )}{a} \]

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 = 0.48, size = 5, normalized size = 1.00 \[ \frac {x}{a} \]

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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mupad [B]  time = 13.81, size = 5, normalized size = 1.00 \[ \frac {x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2/(a - a*sin(x)^2),x)

[Out]

x/a

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sympy [A]  time = 1.60, size = 2, normalized size = 0.40 \[ \frac {x}{a} \]

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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