3.88 \(\int \cos ^2(x) \, dx\)

Optimal. Leaf size=14 \[ \frac{x}{2}+\frac{1}{2} \sin (x) \cos (x) \]

[Out]

x/2 + (Cos[x]*Sin[x])/2

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Rubi [A]  time = 0.0064947, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{x}{2}+\frac{1}{2} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^2,x]

[Out]

x/2 + (Cos[x]*Sin[x])/2

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

Mathematica [A]  time = 0.0017493, size = 14, normalized size = 1. \[ \frac{x}{2}+\frac{1}{4} \sin (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^2,x]

[Out]

x/2 + Sin[2*x]/4

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{x}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sec(x)^2,x)

[Out]

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

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Maxima [A]  time = 0.928115, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, x + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(x)^2,x, algorithm="maxima")

[Out]

1/2*x + 1/4*sin(2*x)

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Fricas [A]  time = 2.2063, size = 36, normalized size = 2.57 \begin{align*} \frac{1}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{2} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(x)^2,x, algorithm="fricas")

[Out]

1/2*cos(x)*sin(x) + 1/2*x

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Sympy [A]  time = 0.061051, size = 10, normalized size = 0.71 \begin{align*} \frac{x}{2} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(x)**2,x)

[Out]

x/2 + sin(x)*cos(x)/2

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Giac [A]  time = 1.06457, size = 22, normalized size = 1.57 \begin{align*} \frac{1}{2} \, x + \frac{\tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(x)^2,x, algorithm="giac")

[Out]

1/2*x + 1/2*tan(x)/(tan(x)^2 + 1)