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

Optimal. Leaf size=13 \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]

[Out]

Coth[x]/Sqrt[-Csch[x]^2]

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Rubi [A]  time = 0.0217987, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 191} \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - Coth[x]^2],x]

[Out]

Coth[x]/Sqrt[-Csch[x]^2]

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-\coth ^2(x)}} \, dx &=\int \frac{1}{\sqrt{-\text{csch}^2(x)}} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.006995, size = 13, normalized size = 1. \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - Coth[x]^2],x]

[Out]

Coth[x]/Sqrt[-Csch[x]^2]

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Maple [A]  time = 0.009, size = 14, normalized size = 1.1 \begin{align*}{{\rm coth} \left (x\right ){\frac{1}{\sqrt{1- \left ({\rm coth} \left (x\right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.88455, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{2} i \, e^{\left (-x\right )} + \frac{1}{2} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*e^(-x) + 1/2*I*e^x

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Fricas [A]  time = 2.59502, size = 4, normalized size = 0.31 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{1 - \coth ^{2}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(1 - coth(x)**2), x)

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Giac [C]  time = 1.13184, size = 32, normalized size = 2.46 \begin{align*} -\frac{-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm{sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-I*e^(-x) - I*e^x)/sgn(-e^(2*x) + 1)