3.633 \(\int \frac{\cot (x)}{\log (\sin (x))} \, dx\)

Optimal. Leaf size=4 \[ \log (\log (\sin (x))) \]

[Out]

Log[Log[Sin[x]]]

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Rubi [A]  time = 0.0304205, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \log (\log (\sin (x))) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]/Log[Sin[x]],x]

[Out]

Log[Log[Sin[x]]]

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Rubi in Sympy [A]  time = 2.18413, size = 5, normalized size = 1.25 \[ \log{\left (\log{\left (\sin{\left (x \right )} \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)/ln(sin(x)),x)

[Out]

log(log(sin(x)))

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Mathematica [A]  time = 0.0042337, size = 4, normalized size = 1. \[ \log (\log (\sin (x))) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]/Log[Sin[x]],x]

[Out]

Log[Log[Sin[x]]]

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Maple [A]  time = 0.012, size = 5, normalized size = 1.3 \[ \ln \left ( \ln \left ( \sin \left ( x \right ) \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)/ln(sin(x)),x)

[Out]

ln(ln(sin(x)))

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Maxima [A]  time = 1.44547, size = 5, normalized size = 1.25 \[ \log \left (\log \left (\sin \left (x\right )\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)/log(sin(x)),x, algorithm="maxima")

[Out]

log(log(sin(x)))

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Fricas [A]  time = 0.215226, size = 5, normalized size = 1.25 \[ \log \left (\log \left (\sin \left (x\right )\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)/log(sin(x)),x, algorithm="fricas")

[Out]

log(log(sin(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\cot{\left (x \right )}}{\log{\left (\sin{\left (x \right )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)/ln(sin(x)),x)

[Out]

Integral(cot(x)/log(sin(x)), x)

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GIAC/XCAS [A]  time = 0.221748, size = 7, normalized size = 1.75 \[{\rm ln}\left ({\left |{\rm ln}\left (\sin \left (x\right )\right ) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)/log(sin(x)),x, algorithm="giac")

[Out]

ln(abs(ln(sin(x))))