3.137 \(\int \frac{1}{\sqrt{\text{csch}(2 \log (c x))}} \, dx\)

Optimal. Leaf size=60 \[ \frac{\csc ^{-1}\left (c^2 x^2\right )}{2 c^2 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}+\frac{x}{2 \sqrt{\text{csch}(2 \log (c x))}} \]

[Out]

x/(2*Sqrt[Csch[2*Log[c*x]]]) + ArcCsc[c^2*x^2]/(2*c^2*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]])

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Rubi [A]  time = 0.0353633, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {5546, 5544, 335, 275, 277, 216} \[ \frac{\csc ^{-1}\left (c^2 x^2\right )}{2 c^2 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}+\frac{x}{2 \sqrt{\text{csch}(2 \log (c x))}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Csch[2*Log[c*x]]],x]

[Out]

x/(2*Sqrt[Csch[2*Log[c*x]]]) + ArcCsc[c^2*x^2]/(2*c^2*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]])

Rule 5546

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5544

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(Csch[d*(a + b*Log[x])]^p*(1 - 1/(E^(2*a*d)*x
^(2*b*d)))^p)/x^(-(b*d*p)), Int[1/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, p}, x
] &&  !IntegerQ[p]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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 \frac{1}{\sqrt{\text{csch}(2 \log (c x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\text{csch}(2 \log (x))}} \, dx,x,c x\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{1-\frac{1}{x^4}} x \, dx,x,c x\right )}{c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-x^4}}{x^3} \, dx,x,\frac{1}{c x}\right )}{c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-x^2}}{x^2} \, dx,x,\frac{1}{c^2 x^2}\right )}{2 c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=\frac{x}{2 \sqrt{\text{csch}(2 \log (c x))}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\frac{1}{c^2 x^2}\right )}{2 c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=\frac{x}{2 \sqrt{\text{csch}(2 \log (c x))}}+\frac{\csc ^{-1}\left (c^2 x^2\right )}{2 c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ \end{align*}

Mathematica [A]  time = 0.0876883, size = 77, normalized size = 1.28 \[ \frac{x \left (2 \sqrt{c^4 x^4-1}-2 \tan ^{-1}\left (\sqrt{c^4 x^4-1}\right )\right )}{4 \sqrt{2} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}} \sqrt{c^4 x^4-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Csch[2*Log[c*x]]],x]

[Out]

(x*(2*Sqrt[-1 + c^4*x^4] - 2*ArcTan[Sqrt[-1 + c^4*x^4]]))/(4*Sqrt[2]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)]*Sqrt[-1 +
c^4*x^4])

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{{\rm csch} \left (2\,\ln \left ( cx \right ) \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/csch(2*ln(c*x))^(1/2),x)

[Out]

int(1/csch(2*ln(c*x))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csch(2*log(c*x))), x)

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Fricas [A]  time = 1.53477, size = 184, normalized size = 3.07 \begin{align*} -\frac{\sqrt{2} c x \arctan \left (\frac{{\left (c^{4} x^{4} - 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) - \sqrt{2}{\left (c^{4} x^{4} - 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}}}{4 \, c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(2)*c*x*arctan((c^4*x^4 - 1)*sqrt(c^2*x^2/(c^4*x^4 - 1))/(c*x)) - sqrt(2)*(c^4*x^4 - 1)*sqrt(c^2*x^2
/(c^4*x^4 - 1)))/(c^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/csch(2*ln(c*x))**(1/2),x)

[Out]

Integral(1/sqrt(csch(2*log(c*x))), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/csch(2*log(c*x))^(1/2),x, algorithm="giac")

[Out]

Timed out