3.31 \(\int \sqrt{a \text{csch}^2(x)} \, dx\)

Optimal. Leaf size=26 \[ -\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (x)}{\sqrt{a \text{csch}^2(x)}}\right ) \]

[Out]

-(Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*Csch[x]^2]])

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Rubi [A]  time = 0.018236, antiderivative size = 26, 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 = {4122, 217, 206} \[ -\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (x)}{\sqrt{a \text{csch}^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Csch[x]^2],x]

[Out]

-(Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*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 217

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

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

Mathematica [A]  time = 0.0051363, size = 20, normalized size = 0.77 \[ \sinh (x) \sqrt{a \text{csch}^2(x)} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Csch[x]^2],x]

[Out]

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

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Maple [B]  time = 0.072, size = 67, normalized size = 2.6 \begin{align*} \sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{x}}-1 \right ) -\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{x}}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csch(x)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.58171, size = 32, normalized size = 1.23 \begin{align*} \sqrt{a} \log \left (e^{\left (-x\right )} + 1\right ) - \sqrt{a} \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a)*log(e^(-x) + 1) - sqrt(a)*log(e^(-x) - 1)

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Fricas [B]  time = 1.90139, size = 296, normalized size = 11.38 \begin{align*} \left [\sqrt{\frac{a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}}{\left (e^{\left (2 \, x\right )} - 1\right )} \log \left (\frac{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}\right ), 2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}}{\left (e^{\left (2 \, x\right )} - 1\right )} e^{x}}{a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*(e^(2*x) - 1)*log((cosh(x) + sinh(x) - 1)/(cosh(x) + sinh(x) + 1)), 2*sqrt(
-a)*arctan(sqrt(-a)*sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*(e^(2*x) - 1)*e^x/(a*cosh(x)*e^x + a*e^x*sinh(x)))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csch(x)**2), x)

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Giac [A]  time = 1.17051, size = 39, normalized size = 1.5 \begin{align*} -\sqrt{a}{\left (\log \left (e^{x} + 1\right ) - \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((a*csch(x)^2)^(1/2),x, algorithm="giac")

[Out]

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