3.61 \(\int \cot (x) \, dx\)

Optimal. Leaf size=3 \[ \log (\sin (x)) \]

[Out]

Log[Sin[x]]

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Rubi [A]  time = 0.00456008, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x],x]

[Out]

Log[Sin[x]]

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Rubi in Sympy [A]  time = 0.030328, size = 3, normalized size = 1. \[ \log{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x),x)

[Out]

log(sin(x))

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Mathematica [A]  time = 0.00338638, size = 3, normalized size = 1. \[ \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x],x]

[Out]

Log[Sin[x]]

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Maple [A]  time = 0.001, size = 4, normalized size = 1.3 \[ \ln \left ( \sin \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x),x)

[Out]

ln(sin(x))

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Maxima [A]  time = 1.61971, size = 4, normalized size = 1.33 \[ \log \left (\sin \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sin(x))

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Fricas [A]  time = 0.219355, size = 15, normalized size = 5. \[ \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(-1/2*cos(2*x) + 1/2)

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Sympy [A]  time = 0.043976, size = 3, normalized size = 1. \[ \log{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x),x)

[Out]

log(sin(x))

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GIAC/XCAS [A]  time = 0.217179, size = 15, normalized size = 5. \[ \frac{1}{2} \,{\rm ln}\left (-\cos \left (x\right )^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(-cos(x)^2 + 1)