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

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

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 8

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

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ \frac {x}{2}+\frac {1}{4} \sin (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^2,x]

[Out]

x/2 + Sin[2*x]/4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^2(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]^2,x]

[Out]

Could not integrate

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fricas [A]  time = 1.14, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cos(x)*sin(x) + 1/2*x

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giac [A]  time = 0.83, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, x + \frac {1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 1/4*sin(2*x)

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maple [A]  time = 0.03, size = 11, normalized size = 0.79

method	result	size
default	\(\frac {x}{2}+\frac {\cos \relax (x ) \sin \relax (x )}{2}\)	\(11\)
risch	\(\frac {x}{2}+\frac {\sin \left (2 x \right )}{4}\)	\(11\)
norman	\(\frac {x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {x}{2}-\left (\tan ^{3}\left (\frac {x}{2}\right )\right )+\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}+\tan \left (\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 10, normalized size = 0.71 \[ \frac {1}{2} \, x + \frac {1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 1/4*sin(2*x)

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mupad [B]  time = 0.03, size = 10, normalized size = 0.71 \[ \frac {x}{2}+\frac {\sin \left (2\,x\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2,x)

[Out]

x/2 + sin(2*x)/4

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sympy [A]  time = 0.06, size = 10, normalized size = 0.71 \[ \frac {x}{2} + \frac {\sin {\relax (x )} \cos {\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/2 + sin(x)*cos(x)/2

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