Optimal. Leaf size=141 \[ \frac{2 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}(c x),\frac{1}{2}\right )}{77 c^5 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{6 x^4}{77 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4}{77 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^8}{11 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
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Rubi [A] time = 0.0979203, antiderivative size = 141, 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, 277, 325, 220} \[ \frac{6 x^4}{77 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4}{77 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{2 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{77 c^5 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^8}{11 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
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Rule 5551
Rule 5549
Rule 335
Rule 277
Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{x^7}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^7}{\text{sech}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^8}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}\right )^{3/2} x^{10} \, dx,x,c x\right )}{c^{11} \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^{12}} \, dx,x,\frac{1}{c x}\right )}{c^{11} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^8}{11 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{6 \operatorname{Subst}\left (\int \frac{\sqrt{1+x^4}}{x^8} \, dx,x,\frac{1}{c x}\right )}{11 c^{11} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{6 x^4}{77 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^8}{11 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{77 c^{11} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{77 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{6 x^4}{77 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^8}{11 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{77 c^{11} \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{77 c^4 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{6 x^4}{77 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^8}{11 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{2 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{77 c^5 \left (c^4+\frac{1}{x^4}\right )^2 x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ \end{align*}
Mathematica [C] time = 0.177372, size = 77, normalized size = 0.55 \[ \frac{\sqrt{c^4 x^4+1} \sqrt{\frac{c^2 x^2}{2 c^4 x^4+2}} \left (\left (c^4 x^4+1\right )^{5/2}-\, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-c^4 x^4\right )\right )}{22 c^8} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.036, size = 138, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( 7\,{c}^{8}{x}^{8}+13\,{c}^{4}{x}^{4}+4 \right ) \sqrt{2}}{308\,{c}^{6}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}}-{\frac{\sqrt{2}x}{77\,{c}^{6} \left ({c}^{4}{x}^{4}+1 \right ) }\sqrt{1-i{c}^{2}{x}^{2}}\sqrt{1+i{c}^{2}{x}^{2}}{\it EllipticF} \left ( x\sqrt{i{c}^{2}},i \right ){\frac{1}{\sqrt{i{c}^{2}}}}{\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^{7}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{7}}{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}, x\right ) \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^{7}}{\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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