Integrand size = 15, antiderivative size = 80 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=-\frac {1}{3} \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}+\frac {1}{6} c^3 \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) x \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5670, 5668, 342, 327, 226} \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=\frac {1}{6} c^3 x \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))} \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )-\frac {1}{3} \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))} \]
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Rule 226
Rule 327
Rule 342
Rule 5668
Rule 5670
Rubi steps \begin{align*} \text {integral}& = c^4 \text {Subst}\left (\int \frac {\sqrt {\text {sech}(2 \log (x))}}{x^5} \, dx,x,c x\right ) \\ & = \left (c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {1}{x^4}} x^6} \, dx,x,c x\right ) \\ & = -\left (\left (c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )\right ) \\ & = -\frac {1}{3} \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}+\frac {1}{3} \left (c^5 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right ) \\ & = -\frac {1}{3} \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}+\frac {1}{6} c^3 \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 \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=-\frac {\sqrt {2} \sqrt {\frac {c^2 x^2}{1+c^4 x^4}} \sqrt {1+c^4 x^4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-c^4 x^4\right )}{3 x^4} \]
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Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.46
method | result | size |
risch | \(-\frac {\left (c^{4} x^{4}+1\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}{3 x^{4}}-\frac {c^{4} \sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i c^{2}}, i\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}{3 \sqrt {i c^{2}}\, x}\) | \(117\) |
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none
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=\frac {\sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} c x^{4} F(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1) - \sqrt {2} {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{3 \, x^{4}} \]
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\[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=\int \frac {\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}{x^{5}}\, dx \]
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\[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=\int { \frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^5} \, dx=\int \frac {\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^5} \,d x \]
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