3.838 \(\int (\cos ^2(x)+\sin ^2(x)) \, dx\)

Optimal. Leaf size=1 \[ x \]

[Out]

x

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Rubi [A]  time = 0.0103859, antiderivative size = 1, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2635, 8} \[ x \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^2 + Sin[x]^2,x]

[Out]

x

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.000303, size = 1, normalized size = 1. \[ x \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^2 + Sin[x]^2,x]

[Out]

x

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Maple [A]  time = 0.003, size = 2, normalized size = 2. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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Maxima [A]  time = 0.963386, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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Fricas [A]  time = 1.82605, size = 4, normalized size = 4. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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Sympy [A]  time = 0.059741, size = 0, normalized size = 0. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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Giac [A]  time = 1.068, size = 1, normalized size = 1. \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x