3.16 \(\int \cos (x) \sin (x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\sin ^2(x)}{2} \]

[Out]

Sin[x]^2/2

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Rubi [A]  time = 0.0066878, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2564, 30} \[ \frac{\sin ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sin[x],x]

[Out]

Sin[x]^2/2

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \cos (x) \sin (x) \, dx &=\operatorname{Subst}(\int x \, dx,x,\sin (x))\\ &=\frac{\sin ^2(x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0009839, size = 8, normalized size = 1. \[ -\frac{1}{2} \cos ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sin[x],x]

[Out]

-Cos[x]^2/2

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Maple [A]  time = 0.001, size = 7, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(x),x)

[Out]

1/2*sin(x)^2

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Maxima [A]  time = 0.912815, size = 8, normalized size = 1. \begin{align*} -\frac{1}{2} \, \cos \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x),x, algorithm="maxima")

[Out]

-1/2*cos(x)^2

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Fricas [A]  time = 1.9149, size = 20, normalized size = 2.5 \begin{align*} -\frac{1}{2} \, \cos \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x),x, algorithm="fricas")

[Out]

-1/2*cos(x)^2

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Sympy [A]  time = 0.053909, size = 5, normalized size = 0.62 \begin{align*} \frac{\sin ^{2}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)**2/2

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Giac [A]  time = 1.08135, size = 8, normalized size = 1. \begin{align*} -\frac{1}{2} \, \cos \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x),x, algorithm="giac")

[Out]

-1/2*cos(x)^2