3.24 \(\int \sqrt{-\text{csch}^2(x)} \, dx\)

Optimal. Leaf size=3 \[ \sin ^{-1}(\coth (x)) \]

[Out]

ArcSin[Coth[x]]

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Rubi [A]  time = 0.0081699, antiderivative size = 3, 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, 216} \[ \sin ^{-1}(\coth (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[Coth[x]]

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 216

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

Rubi steps

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

Mathematica [B]  time = 0.0048837, size = 20, normalized size = 6.67 \[ \sinh (x) \sqrt{-\text{csch}^2(x)} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-Csch[x]^2]*Log[Tanh[x/2]]*Sinh[x]

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Maple [B]  time = 0.052, size = 67, normalized size = 22.3 \begin{align*} -{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}\ln \left ({{\rm e}^{x}}+1 \right ) +{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \sqrt{-{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}\ln \left ({{\rm e}^{x}}-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.74671, size = 26, normalized size = 8.67 \begin{align*} i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*log(e^(-x) + 1) - I*log(e^(-x) - 1)

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Fricas [C]  time = 1.50787, size = 46, normalized size = 15.33 \begin{align*} -i \, \log \left (e^{x} + 1\right ) + i \, \log \left (e^{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(e^x + 1) + I*log(e^x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-csch(x)**2), x)

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Giac [C]  time = 1.15374, size = 36, normalized size = 12. \begin{align*}{\left (i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \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((-csch(x)^2)^(1/2),x, algorithm="giac")

[Out]

(I*log(e^x + 1) - I*log(abs(e^x - 1)))*sgn(-e^(3*x) + e^x)