Optimal. Leaf size=30 \[ \frac{x^5 \left (c^4-\frac{1}{x^4}\right )}{6 c^4 \sqrt{\text{csch}(2 \log (c x))}} \]
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Rubi [A] time = 0.0452772, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5552, 5550, 264} \[ \frac{x^5 \left (c^4-\frac{1}{x^4}\right )}{6 c^4 \sqrt{\text{csch}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 5552
Rule 5550
Rule 264
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{\text{csch}(2 \log (c x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\sqrt{\text{csch}(2 \log (x))}} \, dx,x,c x\right )}{c^5}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{1-\frac{1}{x^4}} x^5 \, dx,x,c x\right )}{c^6 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=\frac{\left (c^4-\frac{1}{x^4}\right ) x^5}{6 c^4 \sqrt{\text{csch}(2 \log (c x))}}\\ \end{align*}
Mathematica [A] time = 0.0475923, size = 44, normalized size = 1.47 \[ \frac{\left (c^4 x^4-1\right )^2 \sqrt{\frac{c^2 x^2}{2 c^4 x^4-2}}}{6 c^6 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 39, normalized size = 1.3 \begin{align*}{\frac{\sqrt{2}x \left ({c}^{4}{x}^{4}-1 \right ) }{12\,{c}^{4}}{\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 [A] time = 1.77885, size = 62, normalized size = 2.07 \begin{align*} \frac{{\left (\sqrt{2} c^{4} x^{4} - \sqrt{2}\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{c x + 1} \sqrt{c x - 1}}{12 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61067, size = 103, normalized size = 3.43 \begin{align*} \frac{\sqrt{2}{\left (c^{8} x^{8} - 2 \, c^{4} x^{4} + 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}}}{12 \, c^{6} x} \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^{4}}{\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^{4}}{\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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