Optimal. Leaf size=251 \[ -\frac{2 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}(c x),\frac{1}{2}\right )}{15 c^3 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{2 x^2}{15 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4}{15 c^4 x^2 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{4}{15 c^4 x^4 \left (c^4+\frac{1}{x^4}\right ) \left (c^2+\frac{1}{x^2}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{15 c^3 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14933, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 x^2}{15 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4}{15 c^4 x^2 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{4}{15 c^4 x^4 \left (c^4+\frac{1}{x^4}\right ) \left (c^2+\frac{1}{x^2}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{2 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{15 c^3 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{15 c^3 x^3 \left (c^4+\frac{1}{x^4}\right )^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{sech}^{\frac{3}{2}}(2 \log (c x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5551
Rule 5549
Rule 335
Rule 277
Rule 325
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^5}{\text{sech}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{\text{sech}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^4}\right )^{3/2} x^8 \, dx,x,c x\right )}{c^9 \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^{10}} \, dx,x,\frac{1}{c x}\right )}{c^9 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^6}{9 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{1+x^4}}{x^6} \, dx,x,\frac{1}{c x}\right )}{3 c^9 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{2 x^2}{15 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{15 c^4 \left (c^4+\frac{1}{x^4}\right ) x^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{2 x^2}{15 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{sech}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{15 c^4 \left (c^4+\frac{1}{x^4}\right ) x^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{2 x^2}{15 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \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 )}{15 c^9 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \operatorname{Subst}\left (\int \frac{1-x^2}{\sqrt{1+x^4}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1+\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{sech}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{4}{15 c^4 \left (c^4+\frac{1}{x^4}\right ) \left (c^2+\frac{1}{x^2}\right ) x^4 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4}{15 c^4 \left (c^4+\frac{1}{x^4}\right ) x^2 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{2 x^2}{15 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{sech}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \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 )}{15 c^3 \left (c^4+\frac{1}{x^4}\right )^2 x^3 \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 )}{15 c^3 \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 = 0.118474, size = 65, normalized size = 0.26 \[ \frac{\left (\frac{c^2 x^2}{c^4 x^4+1}\right )^{3/2} \left (c^4 x^4+1\right )^{3/2} \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-c^4 x^4\right )}{6 \sqrt{2} c^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.037, size = 147, normalized size = 0.6 \begin{align*}{\frac{{x}^{4} \left ( 5\,{c}^{4}{x}^{4}+11 \right ) \sqrt{2}}{180\,{c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}}}}+{\frac{{\frac{i}{15}}\sqrt{2}x}{ \left ({c}^{4}{x}^{4}+1 \right ){c}^{4}}\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5}}{\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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.
[In]
[Out]