\(\int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx\) [138]

Optimal result

Integrand size = 15, antiderivative size = 46 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=i \sqrt {\text {csch}(2 \log (c x))} \operatorname {EllipticF}\left (\frac {\pi }{4}-i \log (c x),2\right ) \sqrt {i \sinh (2 \log (c x))} \]

[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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3856, 2720} \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=i \sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))} \operatorname {EllipticF}\left (\frac {\pi }{4}-i \log (c x),2\right ) \]

[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 2720

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

Rule 3856

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=\text {csch}^{\frac {3}{2}}(2 \log (c x)) \operatorname {EllipticF}\left (\frac {\pi }{4}-i \log (c x),2\right ) (i \sinh (2 \log (c x)))^{3/2} \]

[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)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.96

method	result	size
derivativedivides	\(\frac {i \sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}}\)	\(90\)
default	\(\frac {i \sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (2 \ln \left (c x \right )\right )+i\right )}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (2 \ln \left (c x \right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}}\)	\(90\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=-i \, \sqrt {2} F(\arcsin \left (c x\right )\,|\,-1) \]

[In]

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

[Out]

-I*sqrt(2)*elliptic_f(arcsin(c*x), -1)

Sympy [F]

\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=\int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x}\, dx \]

[In]

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

[Out]

Integral(sqrt(csch(2*log(c*x)))/x, x)

Maxima [F]

\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=\int { \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x} \, dx=\int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x} \,d x \]

[In]

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

[Out]

int((1/sinh(2*log(c*x)))^(1/2)/x, x)