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

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

[Out]

Log[Sin[x]]

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Rubi [A]  time = 0.0021964, 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 = 0.0019737, 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 = 0.002, 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(1/tan(x),x)

[Out]

ln(sin(x))

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Maxima [A]  time = 0.935286, 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(1/tan(x),x, algorithm="maxima")

[Out]

log(sin(x))

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Fricas [B]  time = 1.97721, size = 46, normalized size = 15.33 \begin{align*} \frac{1}{2} \, \log \left (\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} + 1}\right ) \end{align*}

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 = 0.066143, 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(1/tan(x),x)

[Out]

log(sin(x))

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Giac [B]  time = 1.08098, size = 23, normalized size = 7.67 \begin{align*} -\frac{1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \frac{1}{2} \, \log \left (\tan \left (x\right )^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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