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

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

[Out]

ln(sin(x))

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Rubi [A]  time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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*}

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Mathematica [A]  time = 0.00, size = 3, normalized size = 1.00 \[ \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x],x]

[Out]

Log[Sin[x]]

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fricas [B]  time = 0.43, size = 16, normalized size = 5.33 \[ \frac {1}{2} \, \log \left (\frac {\tan \relax (x)^{2}}{\tan \relax (x)^{2} + 1}\right ) \]

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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giac [B]  time = 1.13, size = 17, normalized size = 5.67 \[ -\frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) + \frac {1}{2} \, \log \left (\tan \relax (x)^{2}\right ) \]

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)

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maple [A]  time = 0.00, size = 4, normalized size = 1.33 \[ \ln \left (\sin \relax (x )\right ) \]

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 = 0.42, size = 3, normalized size = 1.00 \[ \log \left (\sin \relax (x)\right ) \]

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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mupad [B]  time = 0.20, size = 13, normalized size = 4.33 \[ \ln \left (\mathrm {tan}\relax (x)\right )-\frac {\ln \left ({\mathrm {tan}\relax (x)}^2+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/tan(x),x)

[Out]

log(tan(x)) - log(tan(x)^2 + 1)/2

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sympy [A]  time = 0.07, size = 3, normalized size = 1.00 \[ \log {\left (\sin {\relax (x )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/tan(x),x)

[Out]

log(sin(x))

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