Optimal. Leaf size=67 \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^4 x^4}+1}\right )}{4 c^4 x \sqrt{\frac{1}{c^4 x^4}+1} \sqrt{\text{sech}(2 \log (c x))}}+\frac{x^3}{4 \sqrt{\text{sech}(2 \log (c x))}} \]
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Rubi [A] time = 0.0549709, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5551, 5549, 266, 47, 63, 207} \[ \frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^4 x^4}+1}\right )}{4 c^4 x \sqrt{\frac{1}{c^4 x^4}+1} \sqrt{\text{sech}(2 \log (c x))}}+\frac{x^3}{4 \sqrt{\text{sech}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 5551
Rule 5549
Rule 266
Rule 47
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{\text{sech}(2 \log (c x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{\text{sech}(2 \log (x))}} \, dx,x,c x\right )}{c^3}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{1+\frac{1}{x^4}} x^3 \, dx,x,c x\right )}{c^4 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x^2} \, dx,x,\frac{1}{c^4 x^4}\right )}{4 c^4 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{x^3}{4 \sqrt{\text{sech}(2 \log (c x))}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\frac{1}{c^4 x^4}\right )}{8 c^4 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{x^3}{4 \sqrt{\text{sech}(2 \log (c x))}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\frac{1}{c^4 x^4}}\right )}{4 c^4 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{x^3}{4 \sqrt{\text{sech}(2 \log (c x))}}+\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{c^4 x^4}}\right )}{4 c^4 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ \end{align*}
Mathematica [A] time = 0.137535, size = 77, normalized size = 1.15 \[ \frac{x \left (c^2 x^2 \sqrt{c^4 x^4+1}+\sinh ^{-1}\left (c^2 x^2\right )\right )}{4 \sqrt{2} c^2 \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \sqrt{c^4 x^4+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 97, normalized size = 1.5 \begin{align*}{\frac{{x}^{3}\sqrt{2}}{8}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}}+{\frac{\sqrt{2}x}{8}\ln \left ({{c}^{4}{x}^{2}{\frac{1}{\sqrt{{c}^{4}}}}}+\sqrt{{c}^{4}{x}^{4}+1} \right ){\frac{1}{\sqrt{{c}^{4}}}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}{\frac{1}{\sqrt{{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^{2}}{\sqrt{\operatorname{sech}\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 [A] time = 3.01036, size = 194, normalized size = 2.9 \begin{align*} \frac{2 \, \sqrt{2}{\left (c^{5} x^{5} + c x\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}} + \sqrt{2} \log \left (-2 \, c^{4} x^{4} - 2 \,{\left (c^{5} x^{5} + c x\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}} - 1\right )}{16 \, c^{3}} \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^{2}}{\sqrt{\operatorname{sech}{\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^{2}}{\sqrt{\operatorname{sech}\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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