Optimal. Leaf size=60 \[ \frac{2 \text{EllipticF}\left (\csc ^{-1}(c x),-1\right )}{3 c^3 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^2}{3 \sqrt{\text{csch}(2 \log (c x))}} \]
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Rubi [A] time = 0.0405378, antiderivative size = 60, normalized size of antiderivative = 1., 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 = {5552, 5550, 335, 277, 221} \[ \frac{2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{3 c^3 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^2}{3 \sqrt{\text{csch}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 5552
Rule 5550
Rule 335
Rule 277
Rule 221
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{\text{csch}(2 \log (c x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{\text{csch}(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{csch}(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{csch}(2 \log (c x))}}\\ &=\frac{x^2}{3 \sqrt{\text{csch}(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{csch}(2 \log (c x))}}\\ &=\frac{x^2}{3 \sqrt{\text{csch}(2 \log (c x))}}+\frac{2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{3 c^3 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ \end{align*}
Mathematica [C] time = 0.103116, size = 57, normalized size = 0.95 \[ \frac{x^2 \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};c^4 x^4\right )}{\sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 109, normalized size = 1.8 \begin{align*}{\frac{{x}^{2}\sqrt{2}}{6}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}-{\frac{\sqrt{2}x}{3\,{c}^{4}{x}^{4}-3}\sqrt{{c}^{2}{x}^{2}+1}\sqrt{-{c}^{2}{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{-{c}^{2}},i \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}} \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{x}{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}\,{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{x}{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}, 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{x}{\sqrt{\operatorname{csch}{\left (2 \log{\left (c x \right )} \right )}}}\, 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{x}{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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