Optimal. Leaf size=40 \[ -\frac {1}{2} c^2 x \sqrt {\frac {1}{c^4 x^4}+1} \text {csch}^{-1}\left (c^2 x^2\right ) \sqrt {\text {sech}(2 \log (c x))} \]
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Rubi [A] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5551, 5549, 335, 275, 215} \[ -\frac {1}{2} c^2 x \sqrt {\frac {1}{c^4 x^4}+1} \text {csch}^{-1}\left (c^2 x^2\right ) \sqrt {\text {sech}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rule 335
Rule 5549
Rule 5551
Rubi steps
\begin {align*} \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx &=c \operatorname {Subst}\left (\int \frac {\sqrt {\text {sech}(2 \log (x))}}{x^2} \, dx,x,c x\right )\\ &=\left (c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {1}{x^4}} x^3} \, dx,x,c x\right )\\ &=-\left (\left (c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )\right )\\ &=-\left (\frac {1}{2} \left (c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\frac {1}{c^2 x^2}\right )\right )\\ &=-\frac {1}{2} c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \text {csch}^{-1}\left (c^2 x^2\right ) \sqrt {\text {sech}(2 \log (c x))}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 55, normalized size = 1.38 \[ -\frac {\sqrt {c^4 x^4+1} \sqrt {\frac {c^2 x^2}{2 c^4 x^4+2}} \tanh ^{-1}\left (\sqrt {c^4 x^4+1}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 57, normalized size = 1.42 \[ \frac {1}{4} \, \sqrt {2} c \log \left (\frac {c^{5} x^{5} + 2 \, c x - 2 \, {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) \]
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 [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\mathrm {sech}\left (2 \ln \left (c x \right )\right )}}{x^{2}}\, dx \]
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 \[ \int \frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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