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

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

[Out]

-ArcTanh[Cosh[x]] + Log[Sinh[x]]

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Rubi [A]  time = 0.0079258, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 5, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, 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*}

Mathematica [A]  time = 0.003432, 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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Maple [A]  time = 0.001, size = 10, normalized size = 1.1 \begin{align*} \ln \left ( \sinh \left ( x \right ) \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

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 = 1.04251, size = 12, normalized size = 1.33 \begin{align*} \log \left (\sinh \left (x\right )\right ) + \log \left (\tanh \left (\frac{1}{2} \, x\right )\right ) \end{align*}

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 = 2.05436, size = 47, normalized size = 5.22 \begin{align*} -x + 2 \, \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(coth(x)+csch(x),x, algorithm="fricas")

[Out]

-x + 2*log(cosh(x) + sinh(x) - 1)

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

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]  time = 1.13858, size = 34, normalized size = 3.78 \begin{align*} -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{align*}

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))