### 3.48 $$\int \cos (\log (x)) \, dx$$

Optimal. Leaf size=17 $\frac{1}{2} x \sin (\log (x))+\frac{1}{2} x \cos (\log (x))$

[Out]

(x*Cos[Log[x]])/2 + (x*Sin[Log[x]])/2

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Rubi [A]  time = 0.0029784, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 3, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {4476} $\frac{1}{2} x \sin (\log (x))+\frac{1}{2} x \cos (\log (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Cos[Log[x]],x]

[Out]

(x*Cos[Log[x]])/2 + (x*Sin[Log[x]])/2

Rule 4476

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(x*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2
*n^2 + 1), x] + Simp[(b*d*n*x*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] /; FreeQ[{a, b, c, d, n}, x] &&
NeQ[b^2*d^2*n^2 + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0026487, size = 17, normalized size = 1. $\frac{1}{2} x \sin (\log (x))+\frac{1}{2} x \cos (\log (x))$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cos[Log[x]],x]

[Out]

(x*Cos[Log[x]])/2 + (x*Sin[Log[x]])/2

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Maple [A]  time = 0., size = 14, normalized size = 0.8 \begin{align*}{\frac{x\cos \left ( \ln \left ( x \right ) \right ) }{2}}+{\frac{x\sin \left ( \ln \left ( x \right ) \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cos(ln(x)),x)

[Out]

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

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Maxima [A]  time = 0.936863, size = 14, normalized size = 0.82 \begin{align*} \frac{1}{2} \, x{\left (\cos \left (\log \left (x\right )\right ) + \sin \left (\log \left (x\right )\right )\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(log(x)),x, algorithm="maxima")

[Out]

1/2*x*(cos(log(x)) + sin(log(x)))

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Fricas [A]  time = 2.37387, size = 53, normalized size = 3.12 \begin{align*} \frac{1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac{1}{2} \, x \sin \left (\log \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(log(x)),x, algorithm="fricas")

[Out]

1/2*x*cos(log(x)) + 1/2*x*sin(log(x))

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Sympy [A]  time = 0.375787, size = 15, normalized size = 0.88 \begin{align*} \frac{x \sin{\left (\log{\left (x \right )} \right )}}{2} + \frac{x \cos{\left (\log{\left (x \right )} \right )}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(ln(x)),x)

[Out]

x*sin(log(x))/2 + x*cos(log(x))/2

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Giac [A]  time = 1.05204, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac{1}{2} \, x \sin \left (\log \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(log(x)),x, algorithm="giac")

[Out]

1/2*x*cos(log(x)) + 1/2*x*sin(log(x))