3.7.58 \(\int (\coth (x)+\text {csch}(x)) \, dx\) [658]

Optimal. Leaf size=9 \[ -\tanh ^{-1}(\cosh (x))+\log (\sinh (x)) \]

[Out]

-arctanh(cosh(x))+ln(sinh(x))

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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3556, 3855} \begin {gather*} \log (\sinh (x))-\tanh ^{-1}(\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x] + Csch[x],x]

[Out]

-ArcTanh[Cosh[x]] + 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]

Rule 3855

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

Rubi steps

\begin {align*} \int (\coth (x)+\text {csch}(x)) \, dx &=\int \coth (x) \, dx+\int \text {csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))+\log (\sinh (x))\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.22 \begin {gather*} \log (\sinh (x))+\log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x] + Csch[x],x]

[Out]

Log[Sinh[x]] + Log[Tanh[x/2]]

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Maple [A]
time = 0.45, size = 10, normalized size = 1.11

method	result	size
default	\(\ln \left (\sinh \left (x \right )\right )+\ln \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(10\)
risch	\(-x +\ln \left ({\mathrm e}^{2 x}-1\right )-\ln \left ({\mathrm e}^{x}+1\right )+\ln \left ({\mathrm e}^{x}-1\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(sinh(x))+ln(tanh(1/2*x))

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Maxima [A]
time = 0.26, size = 9, normalized size = 1.00 \begin {gather*} \log \left (\sinh \left (x\right )\right ) + \log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sinh(x)) + log(tanh(1/2*x))

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Fricas [A]
time = 0.41, size = 13, normalized size = 1.44 \begin {gather*} -x + 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(cosh(x) + sinh(x) - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)+csch(x),x)

[Out]

Integral(coth(x) + csch(x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (9) = 18\).
time = 0.40, size = 25, normalized size = 2.78 \begin {gather*} -x - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x - log(e^x + 1) + log(abs(e^(2*x) - 1)) + log(abs(e^x - 1))

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Mupad [B]
time = 0.04, size = 11, normalized size = 1.22 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^x-1\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x) + 1/sinh(x),x)

[Out]

2*log(exp(x) - 1) - x

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