Optimal. Leaf size=122 \[ \frac{x^5}{16 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{32 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^4 x^4}+1}\right )}{32 c^{12} x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.0752549, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {5551, 5549, 266, 47, 51, 63, 207} \[ \frac{x^5}{16 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{32 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{c^4 x^4}+1}\right )}{32 c^{12} x^3 \left (\frac{1}{c^4 x^4}+1\right )^{3/2} \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 5551
Rule 5549
Rule 266
Rule 47
Rule 51
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{x^8}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^8}{\text{sech}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^9}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}\right )^{3/2} x^{11} \, dx,x,c x\right )}{c^{12} \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^4} \, dx,x,\frac{1}{c^4 x^4}\right )}{4 c^{12} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^9}{12 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x^3} \, dx,x,\frac{1}{c^4 x^4}\right )}{8 c^{12} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^5}{16 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,\frac{1}{c^4 x^4}\right )}{32 c^{12} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{32 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{16 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\frac{1}{c^4 x^4}\right )}{64 c^{12} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{32 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{16 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{sech}^{\frac{3}{2}}(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 )}{32 c^{12} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{32 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^5}{16 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{c^4 x^4}}\right )}{32 c^{12} \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 = 0.185238, size = 98, normalized size = 0.8 \[ \frac{c^3 x^3 \sqrt{c^4 x^4+1} \left (8 c^8 x^8+14 c^4 x^4+3\right )-3 c x \sinh ^{-1}\left (c^2 x^2\right )}{192 \sqrt{2} c^9 \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \sqrt{c^4 x^4+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 121, normalized size = 1. \begin{align*}{\frac{{x}^{3} \left ( 8\,{c}^{8}{x}^{8}+14\,{c}^{4}{x}^{4}+3 \right ) \sqrt{2}}{384\,{c}^{6}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}}-{\frac{\sqrt{2}x}{128\,{c}^{6}}\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{x^{8}}{\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 = 3.25135, size = 242, normalized size = 1.98 \begin{align*} \frac{2 \, \sqrt{2}{\left (8 \, c^{13} x^{13} + 22 \, c^{9} x^{9} + 17 \, c^{5} x^{5} + 3 \, c x\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}} + 3 \, \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 )}{768 \, c^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{8}}{\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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