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

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

[Out]

Log[Sin[x]]

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Rubi [A]  time = 0.0024754, 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, Rules used = {3475} \[ \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x],x]

[Out]

Log[Sin[x]]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cot (x) \, dx &=\log (\sin (x))\\ \end{align*}

Mathematica [A]  time = 0.0019405, 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 \begin{align*} \ln \left ( \sin \left ( x \right ) \right ) \end{align*}

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 = 0.959389, size = 4, normalized size = 1.33 \begin{align*} \log \left (\sin \left (x\right )\right ) \end{align*}

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 [B]  time = 1.6188, size = 41, normalized size = 13.67 \begin{align*} \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \end{align*}

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 = 0.059446, size = 3, normalized size = 1. \begin{align*} \log{\left (\sin{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x),x)

[Out]

log(sin(x))

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Giac [B]  time = 1.11588, size = 15, normalized size = 5. \begin{align*} \frac{1}{2} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x),x, algorithm="giac")

[Out]

1/2*log(-cos(x)^2 + 1)