Optimal. Leaf size=203 \[ -\frac{\sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}(c x),\frac{1}{2}\right )}{5 c^3 x \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))}}-\frac{2}{5 c^4 x^2 \left (c^2+\frac{1}{x^2}\right ) \sqrt{\text{sech}(2 \log (c x))}}+\frac{2 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 c^3 x \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))}}+\frac{2}{5 c^4 \sqrt{\text{sech}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{sech}(2 \log (c x))}} \]
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Rubi [A] time = 0.129896, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {5551, 5549, 335, 277, 325, 305, 220, 1196} \[ -\frac{2}{5 c^4 x^2 \left (c^2+\frac{1}{x^2}\right ) \sqrt{\text{sech}(2 \log (c x))}}-\frac{\sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 c^3 x \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))}}+\frac{2 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 c^3 x \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))}}+\frac{2}{5 c^4 \sqrt{\text{sech}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{sech}(2 \log (c x))}} \]
Antiderivative was successfully verified.
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Rule 5551
Rule 5549
Rule 335
Rule 277
Rule 325
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{\text{sech}(2 \log (c x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\sqrt{\text{sech}(2 \log (x))}} \, dx,x,c x\right )}{c^4}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{1+\frac{1}{x^4}} x^4 \, dx,x,c x\right )}{c^5 \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^4}}{x^6} \, dx,x,\frac{1}{c x}\right )}{c^5 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{x^4}{5 \sqrt{\text{sech}(2 \log (c x))}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{2}{5 c^4 \sqrt{\text{sech}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{sech}(2 \log (c x))}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{2}{5 c^4 \sqrt{\text{sech}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{sech}(2 \log (c x))}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}+\frac{2 \operatorname{Subst}\left (\int \frac{1-x^2}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}}\\ &=\frac{2}{5 c^4 \sqrt{\text{sech}(2 \log (c x))}}-\frac{2}{5 c^4 \left (c^2+\frac{1}{x^2}\right ) x^2 \sqrt{\text{sech}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{sech}(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 ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{5 c^3 \left (c^4+\frac{1}{x^4}\right ) x \sqrt{\text{sech}(2 \log (c x))}}-\frac{\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 )}{5 c^3 \left (c^4+\frac{1}{x^4}\right ) x \sqrt{\text{sech}(2 \log (c x))}}\\ \end{align*}
Mathematica [C] time = 0.112481, size = 65, normalized size = 0.32 \[ \frac{\left (\frac{c^2 x^2}{c^4 x^4+1}\right )^{3/2} \left (c^4 x^4+1\right )^{3/2} \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};-c^4 x^4\right )}{3 \sqrt{2} c^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.041, size = 134, normalized size = 0.7 \begin{align*}{\frac{{x}^{4}\sqrt{2}}{10}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}}+{\frac{{\frac{i}{5}}\sqrt{2}x}{ \left ({c}^{4}{x}^{4}+1 \right ){c}^{2}}\sqrt{1-i{c}^{2}{x}^{2}}\sqrt{1+i{c}^{2}{x}^{2}} \left ({\it EllipticF} \left ( x\sqrt{i{c}^{2}},i \right ) -{\it EllipticE} \left ( x\sqrt{i{c}^{2}},i \right ) \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^{3}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\sqrt{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )}}, x\right ) \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^{3}}{\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^{3}}{\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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