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

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

[Out]

I*(sin(1/4*Pi+I*ln(c*x))^2)^(1/2)/sin(1/4*Pi+I*ln(c*x))*EllipticF(cos(1/4*Pi+I*ln(c*x)),2^(1/2))*csch(2*ln(c*x
))^(1/2)*(I*sinh(2*ln(c*x)))^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.07, size = 43, normalized size = 0.93 \[ (i \sinh (2 \log (c x)))^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x)) F\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right ) \]

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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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x}, x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.29, size = 90, normalized size = 1.96 \[ \frac {i \sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \EllipticF \left (\sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}} \]

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*(sinh(2*ln(c*x))+I))^(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*(sinh(2*ln(c*x))+I))^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sinh(2*log(c*x)))^(1/2)/x,x)

[Out]

int((1/sinh(2*log(c*x)))^(1/2)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x}\, dx \]

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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