3.275 \(\int \frac{\cos ^5(x)}{(a-a \sin ^2(x))^2} \, dx\)

Optimal. Leaf size=6 \[ \frac{\sin (x)}{a^2} \]

[Out]

Sin[x]/a^2

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Rubi [A]  time = 0.0385285, antiderivative size = 6, 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, 2637} \[ \frac{\sin (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^5/(a - a*Sin[x]^2)^2,x]

[Out]

Sin[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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cos ^5(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \cos (x) \, dx}{a^2}\\ &=\frac{\sin (x)}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0017739, size = 6, normalized size = 1. \[ \frac{\sin (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^5/(a - a*Sin[x]^2)^2,x]

[Out]

Sin[x]/a^2

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Maple [A]  time = 0.03, size = 7, normalized size = 1.2 \begin{align*}{\frac{\sin \left ( x \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5/(a-a*sin(x)^2)^2,x)

[Out]

sin(x)/a^2

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Maxima [A]  time = 0.995567, size = 8, normalized size = 1.33 \begin{align*} \frac{\sin \left (x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

sin(x)/a^2

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Fricas [A]  time = 1.78371, size = 16, normalized size = 2.67 \begin{align*} \frac{\sin \left (x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

sin(x)/a^2

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Sympy [B]  time = 30.1794, size = 19, normalized size = 3.17 \begin{align*} \frac{2 \tan{\left (\frac{x}{2} \right )}}{a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**5/(a-a*sin(x)**2)**2,x)

[Out]

2*tan(x/2)/(a**2*tan(x/2)**2 + a**2)

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Giac [A]  time = 1.09659, size = 8, normalized size = 1.33 \begin{align*} \frac{\sin \left (x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

sin(x)/a^2