3.149 \(\int \cos (\log (x)) \, dx\)

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 = 0.00728153, 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 \[ \frac{1}{2} x \sin (\log (x))+\frac{1}{2} x \cos (\log (x)) \]

Antiderivative was successfully verified.

[In]  Int[Cos[Log[x]],x]

[Out]

(x*Cos[Log[x]])/2 + (x*Sin[Log[x]])/2

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Rubi in Sympy [A]  time = 0.473577, size = 15, normalized size = 0.88 \[ \frac{x \sin{\left (\log{\left (x \right )} \right )}}{2} + \frac{x \cos{\left (\log{\left (x \right )} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(ln(x)),x)

[Out]

x*sin(log(x))/2 + x*cos(log(x))/2

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Mathematica [A]  time = 0.00439241, 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 \[{\frac{x\cos \left ( \ln \left ( x \right ) \right ) }{2}}+{\frac{x\sin \left ( \ln \left ( x \right ) \right ) }{2}} \]

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 = 1.40124, size = 14, normalized size = 0.82 \[ \frac{1}{2} \, x{\left (\cos \left (\log \left (x\right )\right ) + \sin \left (\log \left (x\right )\right )\right )} \]

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 = 0.239583, size = 18, normalized size = 1.06 \[ \frac{1}{2} \, x \cos \left (\log \left (x\right )\right ) + \frac{1}{2} \, x \sin \left (\log \left (x\right )\right ) \]

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 = 0.521858, size = 15, normalized size = 0.88 \[ \frac{x \sin{\left (\log{\left (x \right )} \right )}}{2} + \frac{x \cos{\left (\log{\left (x \right )} \right )}}{2} \]

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/XCAS [A]  time = 0.215739, size = 18, normalized size = 1.06 \[ \frac{1}{2} \, x \cos \left ({\rm ln}\left (x\right )\right ) + \frac{1}{2} \, x \sin \left ({\rm ln}\left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(log(x)),x, algorithm="giac")

[Out]

1/2*x*cos(ln(x)) + 1/2*x*sin(ln(x))