Optimal. Leaf size=87 \[ \frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{3 c x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}} \]
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Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5551, 5549, 335, 277, 220} \[ \frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{3 c x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 335
Rule 5549
Rule 5551
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {\text {sech}(2 \log (c x))}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c^2}\\ &=\frac {\operatorname {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x^2 \, dx,x,c x\right )}{c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^4} \, dx,x,\frac {1}{c x}\right )}{c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{3 c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}}\\ &=\frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )}{3 c \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 58, normalized size = 0.67 \[ \frac {\sqrt {c^4 x^4+1} \sqrt {\frac {c^2 x^2}{2 c^4 x^4+2}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-c^4 x^4\right )}{c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 114, normalized size = 1.31 \[ \frac {x^{2} \sqrt {2}}{6 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {\sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \EllipticF \left (x \sqrt {i c^{2}}, i\right ) \sqrt {2}\, x}{3 \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}} \]
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 \[ \int \frac {x}{\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}} \,d x \]
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 \[ \int \frac {x}{\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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