3.281 \(\int \frac{\cos ^4(x)}{(a-a \sin ^2(x))^2} \, dx\)

Optimal. Leaf size=5 \[ \frac{x}{a^2} \]

[Out]

x/a^2

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Rubi [A]  time = 0.037163, 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} \[ \frac{x}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^4/(a - a*Sin[x]^2)^2,x]

[Out]

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 8

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

Rubi steps

\begin{align*} \int \frac{\cos ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int 1 \, dx}{a^2}\\ &=\frac{x}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0003858, size = 5, normalized size = 1. \[ \frac{x}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^4/(a - a*Sin[x]^2)^2,x]

[Out]

x/a^2

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Maple [C]  time = 0.035, size = 8, normalized size = 1.6 \begin{align*}{\frac{\arctan \left ( \tan \left ( x \right ) \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^4/(a-a*sin(x)^2)^2,x)

[Out]

1/a^2*arctan(tan(x))

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Maxima [A]  time = 1.44907, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

x/a^2

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Fricas [A]  time = 1.6706, size = 9, normalized size = 1.8 \begin{align*} \frac{x}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

x/a^2

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Sympy [A]  time = 19.1012, size = 3, normalized size = 0.6 \begin{align*} \frac{x}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**4/(a-a*sin(x)**2)**2,x)

[Out]

x/a**2

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Giac [A]  time = 1.09676, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

x/a^2