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

Optimal. Leaf size=46 \[ i \sqrt{i \sinh (2 \log (c x))} \sqrt{\text{csch}(2 \log (c x))} \text{EllipticF}\left (\frac{\pi }{4}-i \log (c x),2\right ) \]

[Out]

I*Sqrt[Csch[2*Log[c*x]]]*EllipticF[Pi/4 - I*Log[c*x], 2]*Sqrt[I*Sinh[2*Log[c*x]]]

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Rubi [A]  time = 0.0321486, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3771, 2641} \[ i \sqrt{i \sinh (2 \log (c x))} \sqrt{\text{csch}(2 \log (c x))} F\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

I*Sqrt[Csch[2*Log[c*x]]]*EllipticF[Pi/4 - I*Log[c*x], 2]*Sqrt[I*Sinh[2*Log[c*x]]]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{\text{csch}(2 \log (c x))}}{x} \, dx &=\operatorname{Subst}\left (\int \sqrt{\text{csch}(2 x)} \, dx,x,\log (c x)\right )\\ &=\left (\sqrt{\text{csch}(2 \log (c x))} \sqrt{i \sinh (2 \log (c x))}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{i \sinh (2 x)}} \, dx,x,\log (c x)\right )\\ &=i \sqrt{\text{csch}(2 \log (c x))} F\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right ) \sqrt{i \sinh (2 \log (c x))}\\ \end{align*}

Mathematica [A]  time = 0.0680655, size = 43, normalized size = 0.93 \[ (i \sinh (2 \log (c x)))^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x)) \text{EllipticF}\left (\frac{\pi }{4}-i \log (c x),2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Csch[2*Log[c*x]]^(3/2)*EllipticF[Pi/4 - I*Log[c*x], 2]*(I*Sinh[2*Log[c*x]])^(3/2)

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Maple [A]  time = 0.168, size = 90, normalized size = 2. \begin{align*}{\frac{{\frac{i}{2}}\sqrt{2}}{\cosh \left ( 2\,\ln \left ( cx \right ) \right ) }\sqrt{-i \left ( i+\sinh \left ( 2\,\ln \left ( cx \right ) \right ) \right ) }\sqrt{-i \left ( -\sinh \left ( 2\,\ln \left ( cx \right ) \right ) +i \right ) }\sqrt{i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }{\it EllipticF} \left ( \sqrt{-i \left ( i+\sinh \left ( 2\,\ln \left ( cx \right ) \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*(-I*(I+sinh(2*ln(c*x))))^(1/2)*2^(1/2)*(-I*(-sinh(2*ln(c*x))+I))^(1/2)*(I*sinh(2*ln(c*x)))^(1/2)*Ellipti
cF((-I*(I+sinh(2*ln(c*x))))^(1/2),1/2*2^(1/2))/cosh(2*ln(c*x))/sinh(2*ln(c*x))^(1/2)

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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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(csch(2*log(c*x)))/x, 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}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(csch(2*log(c*x)))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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