Optimal. Leaf size=88 \[ \frac{1}{2 x \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\text{csch}^{-1}\left (c^2 x^2\right )}{2 c^6 x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^3}{6 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.0670216, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5551, 5549, 335, 275, 277, 215} \[ \frac{1}{2 x \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\text{csch}^{-1}\left (c^2 x^2\right )}{2 c^6 x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^3}{6 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 5551
Rule 5549
Rule 335
Rule 275
Rule 277
Rule 215
Rubi steps
\begin{align*} \int \frac{x^2}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\text{sech}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}\right )^{3/2} x^5 \, dx,x,c x\right )}{c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^4\right )^{3/2}}{x^7} \, dx,x,\frac{1}{c x}\right )}{c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^{3/2}}{x^4} \, dx,x,\frac{1}{c^2 x^2}\right )}{2 c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^3}{6 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{x^2} \, dx,x,\frac{1}{c^2 x^2}\right )}{2 c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{1}{2 \left (c^4+\frac{1}{x^4}\right ) x \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^3}{6 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\frac{1}{c^2 x^2}\right )}{2 c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{1}{2 \left (c^4+\frac{1}{x^4}\right ) x \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^3}{6 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\text{csch}^{-1}\left (c^2 x^2\right )}{2 c^6 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ \end{align*}
Mathematica [A] time = 0.168229, size = 88, normalized size = 1. \[ \frac{x \left (\sqrt{c^4 x^4+1} \left (c^4 x^4+4\right )-3 \tanh ^{-1}\left (\sqrt{c^4 x^4+1}\right )\right )}{12 \sqrt{2} c^2 \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \sqrt{c^4 x^4+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\rm sech} \left (2\,\ln \left ( cx \right ) \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \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}}{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.1002, size = 234, normalized size = 2.66 \begin{align*} \frac{3 \, \sqrt{2} c x \log \left (\frac{c^{5} x^{5} + 2 \, c x - 2 \,{\left (c^{4} x^{4} + 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) + 2 \, \sqrt{2}{\left (c^{8} x^{8} + 5 \, c^{4} x^{4} + 4\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}}}{48 \, c^{4} 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^{2}}{\operatorname{sech}^{\frac{3}{2}}{\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}}{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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