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