### 3.669 $$\int \frac{1}{-\coth (x)+\text{csch}(x)} \, dx$$

Optimal. Leaf size=9 $-\log (1-\cosh (x))$

[Out]

-Log[1 - Cosh[x]]

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Rubi [A]  time = 0.0342147, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {3160, 2667, 31} $-\log (1-\cosh (x))$

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 = 0.0212703, size = 9, normalized size = 1. $-2 \log \left (\sinh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-Coth[x] + Csch[x])^(-1),x]

[Out]

-2*Log[Sinh[x/2]]

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Maple [B]  time = 0.03, size = 23, normalized size = 2.6 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -2\,\ln \left ( \tanh \left ( x/2 \right ) \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)-2*ln(tanh(1/2*x))+ln(tanh(1/2*x)-1)

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Maxima [A]  time = 1.22438, size = 18, normalized size = 2. \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 [A]  time = 2.12802, size = 46, normalized size = 5.11 \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 [A]  time = 1.1381, size = 14, normalized size = 1.56 \begin{align*} x - 2 \, \log \left ({\left | e^{x} - 1 \right |}\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(abs(e^x - 1))