\(\int \cos (\log (x)) \, dx\) [35]

Optimal result

Integrand size = 3, antiderivative size = 17 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4564} \[ \int \cos (\log (x)) \, dx=\frac {1}{2} x \sin (\log (x))+\frac {1}{2} x \cos (\log (x)) \]

[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*} \text {integral}& = \frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} x \cos (\log (x))+\frac {1}{2} x \sin (\log (x)) \]

[In]

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

[Out]

(x*Cos[Log[x]])/2 + (x*Sin[Log[x]])/2

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(\frac {x \left (\cos \left (\ln \left (x \right )\right )+\sin \left (\ln \left (x \right )\right )\right )}{2}\)	\(11\)
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	\(\left (\frac {1}{4}-\frac {i}{4}\right ) x \,x^{i}+\left (\frac {1}{4}+\frac {i}{4}\right ) x \,x^{-i}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \]

[In]

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

[Out]

1/2*x*cos(log(x)) + 1/2*x*sin(log(x))

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \cos (\log (x)) \, dx=\frac {x \sin {\left (\log {\left (x \right )} \right )}}{2} + \frac {x \cos {\left (\log {\left (x \right )} \right )}}{2} \]

[In]

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

[Out]

x*sin(log(x))/2 + x*cos(log(x))/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} \, x {\left (\cos \left (\log \left (x\right )\right ) + \sin \left (\log \left (x\right )\right )\right )} \]

[In]

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

[Out]

1/2*x*(cos(log(x)) + sin(log(x)))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (\log (x)) \, dx=\frac {1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac {1}{2} \, x \sin \left (\log \left (x\right )\right ) \]

[In]

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

[Out]

1/2*x*cos(log(x)) + 1/2*x*sin(log(x))

Mupad [B] (verification not implemented)

Time = 14.86 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (\log (x)) \, dx=\frac {\sqrt {2}\,x\,\sin \left (\frac {\pi }{4}+\ln \left (x\right )\right )}{2} \]

[In]

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

[Out]

(2^(1/2)*x*sin(pi/4 + log(x)))/2