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

Optimal. Leaf size=1 \[ x \]

[Out]

x

________________________________________________________________________________________

Rubi [A]  time = 0.0092337, 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)^(-1),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}{\cos ^2(x)+\sin ^2(x)} \, dx &=\int 1 \, dx\\ &=x\\ \end{align*}

Mathematica [A]  time = 0.0004585, size = 1, normalized size = 1. \[ x \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]^2 + Sin[x]^2)^(-1),x]

[Out]

x

________________________________________________________________________________________

Maple [A]  time = 0.014, 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),x)

[Out]

x

________________________________________________________________________________________

Maxima [A]  time = 1.47255, 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),x, algorithm="maxima")

[Out]

x

________________________________________________________________________________________

Fricas [A]  time = 1.5574, 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),x, algorithm="fricas")

[Out]

x

________________________________________________________________________________________

Sympy [B]  time = 0.423272, size = 10, normalized size = 10. \begin{align*} \frac{x}{\sin ^{2}{\left (x \right )} + \cos ^{2}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sin(x)**2 + cos(x)**2)

________________________________________________________________________________________

Giac [A]  time = 1.14716, 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),x, algorithm="giac")

[Out]

x