3.6.73 \(\int \coth (x) \, dx\) [573]

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

[Out]

ln(sinh(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 = {3556} \begin {gather*} \log (\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x],x]

[Out]

Log[Sinh[x]]

Rule 3556

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 \coth (x) \, dx &=\log (\sinh (x))\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 3, normalized size = 1.00 \begin {gather*} \log (\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x],x]

[Out]

Log[Sinh[x]]

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Maple [A]
time = 0.01, size = 4, normalized size = 1.33

method	result	size
lookup	\(\ln \left (\sinh \left (x \right )\right )\)	\(4\)
default	\(\ln \left (\sinh \left (x \right )\right )\)	\(4\)
risch	\(-x +\ln \left ({\mathrm e}^{2 x}-1\right )\)	\(12\)
derivativedivides	\(-\frac {\ln \left (\coth \left (x \right )-1\right )}{2}-\frac {\ln \left (\coth \left (x \right )+1\right )}{2}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x),x,method=_RETURNVERBOSE)

[Out]

ln(sinh(x))

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Maxima [A]
time = 3.06, size = 3, normalized size = 1.00 \begin {gather*} \log \left (\sinh \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x),x, algorithm="maxima")

[Out]

log(sinh(x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (3) = 6\).
time = 0.48, size = 18, normalized size = 6.00 \begin {gather*} -x + \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x),x, algorithm="fricas")

[Out]

-x + log(2*sinh(x)/(cosh(x) - sinh(x)))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (3) = 6\).
time = 0.13, size = 12, normalized size = 4.00 \begin {gather*} x - \log {\left (\tanh {\left (x \right )} + 1 \right )} + \log {\left (\tanh {\left (x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x),x)

[Out]

x - log(tanh(x) + 1) + log(tanh(x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 12 vs. \(2 (3) = 6\).
time = 1.27, size = 12, normalized size = 4.00 \begin {gather*} -x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x),x, algorithm="giac")

[Out]

-x + log(abs(e^(2*x) - 1))

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Mupad [B]
time = 0.03, size = 3, normalized size = 1.00 \begin {gather*} \ln \left (\mathrm {sinh}\left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x),x)

[Out]

log(sinh(x))

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