3.575 \(\int \text{csch}(x) \, dx\)

Optimal. Leaf size=5 \[ -\tanh ^{-1}(\cosh (x)) \]

[Out]

-ArcTanh[Cosh[x]]

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Rubi [A]  time = 0.0036311, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3770} \[ -\tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csch[x],x]

[Out]

-ArcTanh[Cosh[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 \text{csch}(x) \, dx &=-\tanh ^{-1}(\cosh (x))\\ \end{align*}

Mathematica [A]  time = 0.0041317, size = 7, normalized size = 1.4 \[ \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x],x]

[Out]

Log[Tanh[x/2]]

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Maple [A]  time = 0.001, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x),x)

[Out]

ln(tanh(1/2*x))

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Maxima [A]  time = 0.936344, size = 7, normalized size = 1.4 \begin{align*} \log \left (\tanh \left (\frac{1}{2} \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tanh(1/2*x))

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Fricas [B]  time = 2.12289, size = 78, normalized size = 15.6 \begin{align*} -\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(cosh(x) + sinh(x) + 1) + log(cosh(x) + sinh(x) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x),x)

[Out]

Integral(csch(x), x)

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Giac [B]  time = 1.13312, size = 19, normalized size = 3.8 \begin{align*} -\log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(e^x + 1) + log(abs(e^x - 1))