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 = 0.04, 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} \[ \frac {x}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^4/(a - a*Sin[x]^2)^2,x]

[Out]

x/a^2

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

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Mathematica [A]  time = 0.00, size = 5, normalized size = 1.00 \[ \frac {x}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a^2

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fricas [A]  time = 0.39, size = 5, normalized size = 1.00 \[ \frac {x}{a^{2}} \]

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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giac [A]  time = 0.13, size = 5, normalized size = 1.00 \[ \frac {x}{a^{2}} \]

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

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maple [C]  time = 0.19, size = 8, normalized size = 1.60 \[ \frac {\arctan \left (\tan \relax (x )\right )}{a^{2}} \]

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 = 0.44, size = 5, normalized size = 1.00 \[ \frac {x}{a^{2}} \]

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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mupad [B]  time = 13.94, size = 5, normalized size = 1.00 \[ \frac {x}{a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^4/(a - a*sin(x)^2)^2,x)

[Out]

x/a^2

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sympy [A]  time = 13.00, size = 3, normalized size = 0.60 \[ \frac {x}{a^{2}} \]

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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