3.473 \(\int \frac{1}{(\cos ^2(x)+\sin ^2(x))^2} \, dx\)

Optimal. Leaf size=1 \[ x \]

[Out]

x

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Rubi [A]  time = 0.0093517, antiderivative size = 1, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4380, 8} \[ x \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]^2 + Sin[x]^2)^(-2),x]

[Out]

x

Rule 4380

Int[(u_.)*((a_.) + cos[(d_.) + (e_.)*(x_)]^2*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]^2)^(p_.), x_Symbol] :> Dist
[(a + c)^p, Int[ActivateTrig[u], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b - c, 0]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0002989, size = 1, normalized size = 1. \[ x \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]^2 + Sin[x]^2)^(-2),x]

[Out]

x

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Maple [A]  time = 0.016, size = 2, normalized size = 2. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^2+sin(x)^2)^2,x)

[Out]

x

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Maxima [A]  time = 1.49555, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2+sin(x)^2)^2,x, algorithm="maxima")

[Out]

x

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Fricas [A]  time = 1.87529, size = 4, normalized size = 4. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2+sin(x)^2)^2,x, algorithm="fricas")

[Out]

x

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Sympy [B]  time = 1.08813, size = 22, normalized size = 22. \begin{align*} \frac{x}{\sin ^{4}{\left (x \right )} + 2 \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )} + \cos ^{4}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)**2+sin(x)**2)**2,x)

[Out]

x/(sin(x)**4 + 2*sin(x)**2*cos(x)**2 + cos(x)**4)

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Giac [A]  time = 1.1291, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(cos(x)^2+sin(x)^2)^2,x, algorithm="giac")

[Out]

x