3.1.80 \(\int \cos (\log (x)) \, dx\) [80]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {4564} \begin {gather*} \frac {1}{2} x \sin (\log (x))+\frac {1}{2} x \cos (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Cos[Log[x]])/2 + (x*Sin[Log[x]])/2

Rule 4564

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*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \end {gather*}

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 = 0.02, size = 14, normalized size = 0.82

method	result	size
lookup	\(\frac {x \cos \left (\ln \left (x \right )\right )}{2}+\frac {x \sin \left (\ln \left (x \right )\right )}{2}\)	\(14\)
default	\(\frac {x \cos \left (\ln \left (x \right )\right )}{2}+\frac {x \sin \left (\ln \left (x \right )\right )}{2}\)	\(14\)
risch	\(\left (\frac {1}{4}-\frac {i}{4}\right ) x \,x^{i}+\left (\frac {1}{4}+\frac {i}{4}\right ) x \,x^{-i}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(ln(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.59, size = 10, normalized size = 0.59 \begin {gather*} \frac {1}{2} \, x {\left (\cos \left (\log \left (x\right )\right ) + \sin \left (\log \left (x\right )\right )\right )} \end {gather*}

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 = 0.64, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \end {gather*}

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 = 0.11, size = 15, normalized size = 0.88 \begin {gather*} \frac {x \sin {\left (\log {\left (x \right )} \right )}}{2} + \frac {x \cos {\left (\log {\left (x \right )} \right )}}{2} \end {gather*}

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 = 0.83, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \end {gather*}

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))

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Mupad [B]
time = 0.20, size = 13, normalized size = 0.76 \begin {gather*} \frac {\sqrt {2}\,x\,\sin \left (\frac {\pi }{4}+\ln \left (x\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(log(x)),x)

[Out]

(2^(1/2)*x*sin(pi/4 + log(x)))/2

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