3.10 \(\int \frac{\cos (x) \sin (x)}{x} \, dx\)

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

[Out]

SinIntegral[2*x]/2

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Rubi [A]  time = 0.0285319, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4406, 12, 3299} \[ \frac{\text{Si}(2 x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]*Sin[x])/x,x]

[Out]

SinIntegral[2*x]/2

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{\cos (x) \sin (x)}{x} \, dx &=\int \frac{\sin (2 x)}{2 x} \, dx\\ &=\frac{1}{2} \int \frac{\sin (2 x)}{x} \, dx\\ &=\frac{\text{Si}(2 x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0058604, size = 8, normalized size = 1. \[ \frac{\text{Si}(2 x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[x]*Sin[x])/x,x]

[Out]

SinIntegral[2*x]/2

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Maple [A]  time = 0.026, size = 7, normalized size = 0.9 \begin{align*}{\frac{{\it Si} \left ( 2\,x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*Si(2*x)

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Maxima [C]  time = 1.21864, size = 18, normalized size = 2.25 \begin{align*} -\frac{1}{4} i \,{\rm Ei}\left (2 i \, x\right ) + \frac{1}{4} i \,{\rm Ei}\left (-2 i \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*I*Ei(2*I*x) + 1/4*I*Ei(-2*I*x)

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Fricas [A]  time = 0.451929, size = 31, normalized size = 3.88 \begin{align*} \frac{1}{2} \, \operatorname{Si}\left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sin_integral(2*x)

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Sympy [A]  time = 1.07542, size = 5, normalized size = 0.62 \begin{align*} \frac{\operatorname{Si}{\left (2 x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Si(2*x)/2

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Giac [A]  time = 1.11652, size = 8, normalized size = 1. \begin{align*} \frac{1}{2} \, \operatorname{Si}\left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sin_integral(2*x)