Optimal. Leaf size=80 \[ \frac{1}{6} c^3 x \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \sqrt{\text{sech}(2 \log (c x))} \text{EllipticF}\left (2 \cot ^{-1}(c x),\frac{1}{2}\right )-\frac{1}{3} \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))} \]
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Rubi [A] time = 0.0696465, antiderivative size = 80, normalized size of antiderivative = 1., 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, 321, 220} \[ \frac{1}{6} c^3 x \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \sqrt{\text{sech}(2 \log (c x))} F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )-\frac{1}{3} \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 5551
Rule 5549
Rule 335
Rule 321
Rule 220
Rubi steps
\begin{align*} \int \frac{\sqrt{\text{sech}(2 \log (c x))}}{x^5} \, dx &=c^4 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right ) \sqrt{\text{sech}(2 \log (c x))}\\ \end{align*}
Mathematica [C] time = 0.0982415, size = 65, normalized size = 0.81 \[ -\frac{\sqrt{2} \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \sqrt{c^4 x^4+1} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};-c^4 x^4\right )}{3 x^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.042, size = 117, normalized size = 1.5 \begin{align*} -{\frac{ \left ({c}^{4}{x}^{4}+1 \right ) \sqrt{2}}{3\,{x}^{4}}\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}-{\frac{{c}^{4}\sqrt{2}}{3\,x}\sqrt{1-i{c}^{2}{x}^{2}}\sqrt{1+i{c}^{2}{x}^{2}}{\it EllipticF} \left ( x\sqrt{i{c}^{2}},i \right ) \sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}{\frac{1}{\sqrt{i{c}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )}}{x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{sech}{\left (2 \log{\left (c x \right )} \right )}}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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