### 3.659 $$\int \frac{1}{\coth (x)+\text{csch}(x)} \, dx$$

Optimal. Leaf size=5 $\log (\cosh (x)+1)$

[Out]

Log[1 + Cosh[x]]

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Rubi [A]  time = 0.0318504, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.429, Rules used = {3160, 2667, 31} $\log (\cosh (x)+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[(Coth[x] + Csch[x])^(-1),x]

[Out]

Log[1 + Cosh[x]]

Rule 3160

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(-1), x_Symbol] :> Int[Sin[d + e*x
]/(b + a*Sin[d + e*x] + c*Cos[d + e*x]), x] /; FreeQ[{a, b, c, d, e}, x]

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\coth (x)+\text{csch}(x)} \, dx &=i \int \frac{\sinh (x)}{i+i \cosh (x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{i+x} \, dx,x,i \cosh (x)\right )\\ &=\log (1+\cosh (x))\\ \end{align*}

Mathematica [A]  time = 0.0188026, size = 9, normalized size = 1.8 $2 \log \left (\cosh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(Coth[x] + Csch[x])^(-1),x]

[Out]

2*Log[Cosh[x/2]]

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Maple [B]  time = 0.027, size = 20, normalized size = 4. \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x)+csch(x)),x)

[Out]

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

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Maxima [B]  time = 1.05773, size = 15, normalized size = 3. \begin{align*} x + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x)),x, algorithm="maxima")

[Out]

x + 2*log(e^(-x) + 1)

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Fricas [B]  time = 2.03081, size = 47, normalized size = 9.4 \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(1/(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 \frac{1}{\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x)),x)

[Out]

Integral(1/(coth(x) + csch(x)), x)

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Giac [B]  time = 1.12061, size = 15, normalized size = 3. \begin{align*} -x + 2 \, \log \left (e^{x} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(coth(x)+csch(x)),x, algorithm="giac")

[Out]

-x + 2*log(e^x + 1)