\(\int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x} \, dx\) [164]

Optimal result

Integrand size = 15, antiderivative size = 36 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x} \, dx=-i \sqrt {\cosh (2 \log (c x))} \operatorname {EllipticF}(i \log (c x),2) \sqrt {\text {sech}(2 \log (c x))} \]

[Out]

-I*((1/2*c*x+1/2/c/x)^2)^(1/2)/(1/2*c*x+1/2/c/x)*EllipticF(I*(1/2*c*x-1/2/c/x),2^(1/2))*cosh(2*ln(c*x))^(1/2)*
sech(2*ln(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, 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} \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x} \, dx=-i \sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} \operatorname {EllipticF}(i \log (c x),2) \]

[In]

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

[Out]

(-I)*Sqrt[Cosh[2*Log[c*x]]]*EllipticF[I*Log[c*x], 2]*Sqrt[Sech[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 {sech}(2 x)} \, dx,x,\log (c x)\right ) \\ & = \left (\sqrt {\cosh (2 \log (c x))} \sqrt {\text {sech}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\cosh (2 x)}} \, dx,x,\log (c x)\right ) \\ & = -i \sqrt {\cosh (2 \log (c x))} \operatorname {EllipticF}(i \log (c x),2) \sqrt {\text {sech}(2 \log (c x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x} \, dx=-i \sqrt {\cosh (2 \log (c x))} \operatorname {EllipticF}(i \log (c x),2) \sqrt {\text {sech}(2 \log (c x))} \]

[In]

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

[Out]

(-I)*Sqrt[Cosh[2*Log[c*x]]]*EllipticF[I*Log[c*x], 2]*Sqrt[Sech[2*Log[c*x]]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(73)=146\).

Time = 0.67 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.64

method	result	size
derivativedivides	\(\frac {\sqrt {\left (2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \sqrt {-2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}+1}\, \operatorname {EllipticF}\left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right )}{\sqrt {2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{4}+\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}}\)	\(167\)
default	\(\frac {\sqrt {\left (2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \sqrt {-2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}+1}\, \operatorname {EllipticF}\left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right )}{\sqrt {2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{4}+\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}}\)	\(167\)

[In]

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

[Out]

((2*(1/2*c*x+1/2/c/x)^2-1)*(1/2*c*x-1/2/c/x)^2)^(1/2)*(-(1/2*c*x-1/2/c/x)^2)^(1/2)*(-2*(1/2*c*x+1/2/c/x)^2+1)^
(1/2)/(2*(1/2*c*x-1/2/c/x)^4+(1/2*c*x-1/2/c/x)^2)^(1/2)*EllipticF(1/2*c*x+1/2/c/x,2^(1/2))/(1/2*c*x-1/2/c/x)/(
2*(1/2*c*x+1/2/c/x)^2-1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x} \, dx=-\frac {\sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} F(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1)}{c^{3}} \]

[In]

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

[Out]

-sqrt(2)*(-c^4)^(3/4)*elliptic_f(arcsin((-c^4)^(1/4)*x), -1)/c^3

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(sech(2*log(c*x)))/x, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

int((1/cosh(2*log(c*x)))^(1/2)/x, x)