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

Optimal. Leaf size=41 \[ -\frac{1}{2} c^2 x \sqrt{1-\frac{1}{c^4 x^4}} \csc ^{-1}\left (c^2 x^2\right ) \sqrt{\text{csch}(2 \log (c x))} \]

[Out]

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

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Rubi [A]  time = 0.0460243, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5552, 5550, 335, 275, 216} \[ -\frac{1}{2} c^2 x \sqrt{1-\frac{1}{c^4 x^4}} \csc ^{-1}\left (c^2 x^2\right ) \sqrt{\text{csch}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5552

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

Rule 5550

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), 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[(e*x)^m/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x]
/; FreeQ[{a, b, d, e, m, 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 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{\sqrt{\text{csch}(2 \log (c x))}}{x^2} \, dx &=c \operatorname{Subst}\left (\int \frac{\sqrt{\text{csch}(2 \log (x))}}{x^2} \, dx,x,c x\right )\\ &=\left (c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{1}{x^4}} x^3} \, dx,x,c x\right )\\ &=-\left (\left (c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )\right )\\ &=-\left (\frac{1}{2} \left (c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\frac{1}{c^2 x^2}\right )\right )\\ &=-\frac{1}{2} c^2 \sqrt{1-\frac{1}{c^4 x^4}} x \csc ^{-1}\left (c^2 x^2\right ) \sqrt{\text{csch}(2 \log (c x))}\\ \end{align*}

Mathematica [A]  time = 0.110806, size = 54, normalized size = 1.32 \[ \frac{\sqrt{c^4 x^4-1} \sqrt{\frac{c^2 x^2}{2 c^4 x^4-2}} \tan ^{-1}\left (\sqrt{c^4 x^4-1}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\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(csch(2*ln(c*x))^(1/2)/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57141, size = 96, normalized size = 2.34 \begin{align*} \frac{1}{2} \, \sqrt{2} c \arctan \left (\frac{{\left (c^{4} x^{4} - 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out