3.9 \(\int \frac{1}{1-2 \coth ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )-x \]

[Out]

-x + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

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Rubi [A]  time = 0.0514689, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3660, 3675, 206} \[ \sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )-x \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*Coth[x]^2)^(-1),x]

[Out]

-x + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

Rule 3660

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> Simp[x/(a - b), x] - Dist[b/(a - b), Int[Sec[e
+ f*x]^2/(a + b*Tan[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a, b]

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{1-2 \coth ^2(x)} \, dx &=-x-2 \int \frac{\text{csch}^2(x)}{1-2 \coth ^2(x)} \, dx\\ &=-x+2 \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\coth (x)\right )\\ &=-x+\sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0915005, size = 19, normalized size = 1. \[ \sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )-x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*Coth[x]^2)^(-1),x]

[Out]

-x + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]

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Maple [A]  time = 0.019, size = 27, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{2}}+\sqrt{2}{\it Artanh} \left ( \sqrt{2}{\rm coth} \left (x\right ) \right ) +{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*coth(x)^2),x)

[Out]

-1/2*ln(1+coth(x))+2^(1/2)*arctanh(2^(1/2)*coth(x))+1/2*ln(coth(x)-1)

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Maxima [B]  time = 1.9028, size = 51, normalized size = 2.68 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (-2 \, x\right )} + 3}\right ) - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*coth(x)^2),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*log(-(2*sqrt(2) - e^(-2*x) - 3)/(2*sqrt(2) + e^(-2*x) + 3)) - x

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Fricas [B]  time = 2.56611, size = 220, normalized size = 11.58 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{3 \,{\left (2 \, \sqrt{2} - 3\right )} \cosh \left (x\right )^{2} - 4 \,{\left (3 \, \sqrt{2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (2 \, \sqrt{2} - 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*coth(x)^2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*log(-(3*(2*sqrt(2) - 3)*cosh(x)^2 - 4*(3*sqrt(2) - 4)*cosh(x)*sinh(x) + 3*(2*sqrt(2) - 3)*sinh(x)^
2 + 2*sqrt(2) - 3)/(cosh(x)^2 + sinh(x)^2 + 3)) - x

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Sympy [A]  time = 1.0863, size = 34, normalized size = 1.79 \begin{align*} - x - \frac{\sqrt{2} \log{\left (\tanh{\left (x \right )} - \sqrt{2} \right )}}{2} + \frac{\sqrt{2} \log{\left (\tanh{\left (x \right )} + \sqrt{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*coth(x)**2),x)

[Out]

-x - sqrt(2)*log(tanh(x) - sqrt(2))/2 + sqrt(2)*log(tanh(x) + sqrt(2))/2

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Giac [B]  time = 1.15317, size = 51, normalized size = 2.68 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}\right ) - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*coth(x)^2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*log(-(2*sqrt(2) - e^(2*x) - 3)/(2*sqrt(2) + e^(2*x) + 3)) - x