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

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

[Out]

Log[Sin[x]]

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Rubi [A]  time = 0.00479751, 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 = 0.031322, 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(1/tan(x),x)

[Out]

log(sin(x))

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Mathematica [A]  time = 0.00716954, 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., 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(1/tan(x),x)

[Out]

ln(sin(x))

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Maxima [A]  time = 1.38813, 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(1/tan(x),x, algorithm="maxima")

[Out]

log(sin(x))

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Fricas [A]  time = 0.219127, size = 22, normalized size = 7.33 \[ \frac{1}{2} \, \log \left (\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/tan(x),x, algorithm="fricas")

[Out]

1/2*log(tan(x)^2/(tan(x)^2 + 1))

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Sympy [A]  time = 0.047514, size = 3, normalized size = 1. \[ \log{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/tan(x),x)

[Out]

log(sin(x))

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GIAC/XCAS [A]  time = 0.202874, size = 23, normalized size = 7.67 \[ -\frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )^{2} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/tan(x),x, algorithm="giac")

[Out]

-1/2*ln(tan(x)^2 + 1) + 1/2*ln(tan(x)^2)