3.3.81 \(\int \frac {\cos ^4(x)}{(a-a \sin ^2(x))^2} \, dx\) [281]

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

[Out]

x/a^2

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Rubi [A]
time = 0.02, 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 = {3254, 8} \begin {gather*} \frac {x}{a^2} \end {gather*}

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 3254

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 \begin {gather*} \frac {x}{a^2} \end {gather*}

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] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.18, size = 8, normalized size = 1.60

method	result	size
risch	\(\frac {x}{a^{2}}\)	\(6\)
default	\(\frac {\arctan \left (\tan \left (x \right )\right )}{a^{2}}\)	\(8\)
norman	\(\frac {\frac {x \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{a}+\frac {x \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{a}-\frac {x}{a}-\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4} a \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a^{2}} \end {gather*}

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 = 0.37, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a^{2}} \end {gather*}

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 = 4.12, size = 3, normalized size = 0.60 \begin {gather*} \frac {x}{a^{2}} \end {gather*}

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 = 0.44, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a^{2}} \end {gather*}

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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Mupad [B]
time = 13.94, size = 5, normalized size = 1.00 \begin {gather*} \frac {x}{a^2} \end {gather*}

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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