Optimal. Leaf size=137 \[ -\frac {\left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}{c^2+\frac {1}{x^2}}-\frac {1}{2} c x \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))} F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )+c x \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))} E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5551, 5549, 335, 305, 220, 1196} \[ -\frac {\left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}{c^2+\frac {1}{x^2}}-\frac {1}{2} c x \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))} F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right )+c x \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))} E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 335
Rule 1196
Rule 5549
Rule 5551
Rubi steps
\begin {align*} \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^3} \, dx &=c^2 \operatorname {Subst}\left (\int \frac {\sqrt {\text {sech}(2 \log (x))}}{x^3} \, dx,x,c x\right )\\ &=\left (c^3 \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^4} \, dx,x,c x\right )\\ &=-\left (\left (c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )\right )\\ &=-\left (\left (c^3 \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^4}} \, dx,x,\frac {1}{c x}\right )\right )+\left (c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )\\ &=-\frac {\left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}{c^2+\frac {1}{x^2}}+c \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 ) x E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))}-\frac {1}{2} c \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 ) x F\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 59, normalized size = 0.43 \[ -\frac {c^2 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-c^4 x^4\right )}{\sqrt {c^4 x^4+1} \sqrt {\frac {c^2 x^2}{2 c^4 x^4+2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x^{3}}, x\right ) \]
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 [C] time = 0.20, size = 134, normalized size = 0.98 \[ -\frac {\left (c^{4} x^{4}+1\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}{x^{2}}+\frac {i c^{2} \sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \left (\EllipticF \left (x \sqrt {i c^{2}}, i\right )-\EllipticE \left (x \sqrt {i c^{2}}, i\right )\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}{\sqrt {i c^{2}}\, x} \]
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 \[ \int \frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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