3.658 \(\int (\coth (x)+\text {csch}(x)) \, dx\)

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

[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 = {3475, 3770} \[ \log (\sinh (x))-\tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x] + Csch[x],x]

[Out]

-ArcTanh[Cosh[x]] + 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]

Rule 3770

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 \[ \log (\sinh (x))+\log \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x] + Csch[x],x]

[Out]

Log[Sinh[x]] + Log[Tanh[x/2]]

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fricas [A]  time = 0.40, size = 13, normalized size = 1.44 \[ -x + 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]

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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giac [B]  time = 0.11, size = 25, normalized size = 2.78 \[ -x - \log \left (e^{x} + 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]

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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maple [A]  time = 0.03, size = 10, normalized size = 1.11 \[ \ln \left (\sinh \relax (x )\right )+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)+csch(x),x)

[Out]

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

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maxima [A]  time = 0.55, size = 9, normalized size = 1.00 \[ \log \left (\sinh \relax (x)\right ) + \log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \]

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )\, dx \]

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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