3.25 \(\int \frac{1}{\sqrt{-\text{csch}^2(x)}} \, dx\)

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 = 0.0097834, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 191} \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-Csch[x]^2],x]

[Out]

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

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{-\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 = 0.0046467, size = 13, normalized size = 1. \[ \frac{\coth (x)}{\sqrt{-\text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-Csch[x]^2],x]

[Out]

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

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Maple [B]  time = 0.049, size = 58, normalized size = 4.5 \begin{align*}{\frac{{{\rm e}^{2\,x}}}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{1}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(exp(2*x)-1)*exp(2*x)+1/2/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(exp(2*x)-1)

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Maxima [C]  time = 1.6128, 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/(-csch(x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*I*e^(-x) + 1/2*I*e^x

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Fricas [C]  time = 1.50158, size = 39, normalized size = 3. \begin{align*} \frac{1}{2} \,{\left (-i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-I*e^(2*x) - I)*e^(-x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-csch(x)**2), x)

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Giac [C]  time = 1.15287, size = 34, normalized size = 2.62 \begin{align*} -\frac{-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-I*e^(-x) - I*e^x)/sgn(-e^(3*x) + e^x)