3.573 \(\int \coth (x) \, dx\)

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

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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 = {3475} \[ \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x],x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[Coth[x],x]

[Out]

Log[Sinh[x]]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth (x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Coth[x],x]

[Out]

Could not integrate

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fricas [B]  time = 1.12, size = 18, normalized size = 6.00 \[ -x + \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]

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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giac [B]  time = 0.62, size = 12, normalized size = 4.00 \[ -x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]

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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maple [A]  time = 0.02, size = 4, normalized size = 1.33

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

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

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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mupad [B]  time = 0.03, size = 3, normalized size = 1.00 \[ \ln \left (\mathrm {sinh}\relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x),x)

[Out]

log(sinh(x))

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sympy [B]  time = 0.31, size = 12, normalized size = 4.00 \[ x - \log {\left (\tanh {\relax (x )} + 1 \right )} + \log {\left (\tanh {\relax (x )} \right )} \]

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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