3.44 \(\int \cos (x) \log (\sin (x)) \, dx\)

Optimal. Leaf size=11 \[ \sin (x) \log (\sin (x))-\sin (x) \]

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

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Rubi [A]  time = 0.0160724, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \sin (x) \log (\sin (x))-\sin (x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Log[Sin[x]],x]

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

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Rubi in Sympy [A]  time = 1.06105, size = 10, normalized size = 0.91 \[ \log{\left (\sin{\left (x \right )} \right )} \sin{\left (x \right )} - \sin{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*ln(sin(x)),x)

[Out]

log(sin(x))*sin(x) - sin(x)

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Mathematica [A]  time = 0.00306192, size = 11, normalized size = 1. \[ \sin (x) \log (\sin (x))-\sin (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Log[Sin[x]],x]

[Out]

-Sin[x] + Log[Sin[x]]*Sin[x]

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Maple [A]  time = 0.006, size = 12, normalized size = 1.1 \[ -\sin \left ( x \right ) +\ln \left ( \sin \left ( x \right ) \right ) \sin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*ln(sin(x)),x)

[Out]

-sin(x)+ln(sin(x))*sin(x)

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Maxima [A]  time = 1.33769, size = 15, normalized size = 1.36 \[ \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*log(sin(x)),x, algorithm="maxima")

[Out]

log(sin(x))*sin(x) - sin(x)

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Fricas [A]  time = 0.240417, size = 15, normalized size = 1.36 \[ \log \left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*log(sin(x)),x, algorithm="fricas")

[Out]

log(sin(x))*sin(x) - sin(x)

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Sympy [A]  time = 1.37889, size = 10, normalized size = 0.91 \[ \log{\left (\sin{\left (x \right )} \right )} \sin{\left (x \right )} - \sin{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*ln(sin(x)),x)

[Out]

log(sin(x))*sin(x) - sin(x)

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GIAC/XCAS [A]  time = 0.198805, size = 15, normalized size = 1.36 \[{\rm ln}\left (\sin \left (x\right )\right ) \sin \left (x\right ) - \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(sin(x))*sin(x) - sin(x)