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

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

[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)

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Rubi [A]  time = 0.03, antiderivative size = 36, 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} \[ -i \sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} F(i \log (c x)|2) \]

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 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 {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*}

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Mathematica [A]  time = 0.06, size = 36, normalized size = 1.00 \[ -i \sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} F(i \log (c x)|2) \]

Antiderivative was successfully verified.

[In]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x}, x\right ) \]

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

[Out]

Timed out

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maple [B]  time = 0.49, size = 167, normalized size = 4.64 \[ \frac {\sqrt {\left (2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \sqrt {-2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}+1}\, \EllipticF \left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right )}{\sqrt {2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{4}+\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1/cosh(2*log(c*x)))^(1/2)/x, x)

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

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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