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

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

[Out]

Coth[x]/Sqrt[a*Csch[x]^2]

________________________________________________________________________________________

Rubi [A]  time = 0.0135736, 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} \[ \frac{\coth (x)}{\sqrt{a \text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Csch[x]^2],x]

[Out]

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

Mathematica [A]  time = 0.0057898, size = 13, normalized size = 1. \[ \frac{\coth (x)}{\sqrt{a \text{csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Csch[x]^2],x]

[Out]

Coth[x]/Sqrt[a*Csch[x]^2]

________________________________________________________________________________________

Maple [B]  time = 0.065, size = 58, normalized size = 4.5 \begin{align*}{\frac{{{\rm e}^{2\,x}}}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{1}{2\,{{\rm e}^{2\,x}}-2}{\frac{1}{\sqrt{{\frac{a{{\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/(a*csch(x)^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.73158, size = 23, normalized size = 1.77 \begin{align*} -\frac{e^{\left (-x\right )}}{2 \, \sqrt{a}} - \frac{e^{x}}{2 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-x)/sqrt(a) - 1/2*e^x/sqrt(a)

________________________________________________________________________________________

Fricas [B]  time = 1.88567, size = 247, normalized size = 19. \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \,{\left (a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((e^(2*x) - 1)*sinh(x)^2 - cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) - cosh(x))*sinh(x) - 1
)*sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*e^x/(a*cosh(x)*e^x + a*e^x*sinh(x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*csch(x)**2), x)

________________________________________________________________________________________

Giac [B]  time = 1.14493, size = 32, normalized size = 2.46 \begin{align*} \frac{e^{\left (-x\right )} + e^{x}}{2 \, \sqrt{a} \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/(a*csch(x)^2)^(1/2),x, algorithm="giac")

[Out]

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