Optimal. Leaf size=36 \[ -i \sqrt{\text{sech}(2 \log (c x))} \sqrt{\cosh (2 \log (c x))} \text{EllipticF}(i \log (c x),2) \]
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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.
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Rule 3771
Rule 2641
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.
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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.
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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.
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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.
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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.
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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.
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