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

Optimal. Leaf size=36 \[ -i \sqrt{\text{sech}(2 \log (c x))} \sqrt{\cosh (2 \log (c x))} \text{EllipticF}(i \log (c x),2) \]

[Out]

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

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Rubi [A]  time = 0.0286682, antiderivative size = 36, 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} \[ -i \sqrt{\text{sech}(2 \log (c x))} \sqrt{\cosh (2 \log (c x))} F(i \log (c x)|2) \]

Antiderivative was successfully verified.

[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 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{sech}(2 \log (c x))}}{x} \, dx &=\operatorname{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 ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{\cosh (2 x)}} \, dx,x,\log (c x)\right )\\ &=-i \sqrt{\cosh (2 \log (c x))} F(i \log (c x)|2) \sqrt{\text{sech}(2 \log (c x))}\\ \end{align*}

Mathematica [A]  time = 0.0551606, size = 36, normalized size = 1. \[ -i \sqrt{\text{sech}(2 \log (c x))} \sqrt{\cosh (2 \log (c x))} \text{EllipticF}(i \log (c x),2) \]

Antiderivative was successfully verified.

[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]]]

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Maple [B]  time = 0.417, size = 167, normalized size = 4.6 \begin{align*}{\sqrt{ \left ( 2\, \left ( 1/2\,cx+1/2\,{\frac{1}{cx}} \right ) ^{2}-1 \right ) \left ({\frac{cx}{2}}-{\frac{1}{2\,cx}} \right ) ^{2}}\sqrt{- \left ({\frac{cx}{2}}-{\frac{1}{2\,cx}} \right ) ^{2}}\sqrt{-2\, \left ( 1/2\,cx+1/2\,{\frac{1}{cx}} \right ) ^{2}+1}{\it EllipticF} \left ({\frac{cx}{2}}+{\frac{1}{2\,cx}},\sqrt{2} \right ){\frac{1}{\sqrt{2\, \left ( 1/2\,cx-1/2\,{\frac{1}{cx}} \right ) ^{4}+ \left ({\frac{cx}{2}}-{\frac{1}{2\,cx}} \right ) ^{2}}}} \left ({\frac{cx}{2}}-{\frac{1}{2\,cx}} \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( 1/2\,cx+1/2\,{\frac{1}{cx}} \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{sech}\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(sech(2*log(c*x))^(1/2)/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(sqrt(sech(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{sech}{\left (2 \log{\left (c x \right )} \right )}}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sech(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{sech}\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(sech(2*log(c*x))^(1/2)/x,x, algorithm="giac")

[Out]

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