Optimal. Leaf size=92 \[ -\frac{3}{4 x^3 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^4 x^4}+1}\right )}{4 c^4 x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.0392663, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {5545, 5543, 266, 47, 50, 63, 207} \[ -\frac{3}{4 x^3 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \tanh ^{-1}\left (\sqrt{\frac{1}{c^4 x^4}+1}\right )}{4 c^4 x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 5545
Rule 5543
Rule 266
Rule 47
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\text{sech}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}\right )^{3/2} x^3 \, dx,x,c x\right )}{c^4 \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{(1+x)^{3/2}}{x^2} \, dx,x,\frac{1}{c^4 x^4}\right )}{4 c^4 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{4 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x} \, dx,x,\frac{1}{c^4 x^4}\right )}{8 c^4 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{3}{4 \left (c^4+\frac{1}{x^4}\right ) x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\frac{1}{c^4 x^4}\right )}{8 c^4 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{3}{4 \left (c^4+\frac{1}{x^4}\right ) x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\frac{1}{c^4 x^4}}\right )}{4 c^4 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{3}{4 \left (c^4+\frac{1}{x^4}\right ) x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^4 x^4}}\right )}{4 c^4 \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 [C] time = 0.0857202, size = 64, normalized size = 0.7 \[ -\frac{\sqrt{c^4 x^4+1} \sqrt{\frac{c^2 x^2}{2 c^4 x^4+2}} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-c^4 x^4\right )}{4 c^4 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 131, normalized size = 1.4 \begin{align*}{\frac{ \left ({c}^{8}{x}^{8}-{c}^{4}{x}^{4}-2 \right ) \sqrt{2}}{16\,x \left ({c}^{4}{x}^{4}+1 \right ){c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}}+{\frac{3\,{c}^{2}\sqrt{2}x}{16}\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{{c}^{4}{x}^{4}+1}}}{\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{1}{\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 = 3.32069, size = 227, normalized size = 2.47 \begin{align*} \frac{3 \, \sqrt{2} c^{3} x^{3} \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 ) + 2 \, \sqrt{2}{\left (c^{8} x^{8} - c^{4} x^{4} - 2\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}}}{32 \, c^{4} x^{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{1}{\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{1}{\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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