Optimal. Leaf size=181 \[ -\frac{x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}}-\frac{\sqrt{x^6+2}}{10 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0302554, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {325, 225} \[ -\frac{\sqrt{x^6+2}}{10 x^5}-\frac{x \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x^2+\sqrt [3]{2}}{\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (x^2+\sqrt [3]{2}\right )}{\left (\left (1+\sqrt{3}\right ) x^2+\sqrt [3]{2}\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 225
Rubi steps
\begin{align*} \int \frac{1}{x^6 \sqrt{2+x^6}} \, dx &=-\frac{\sqrt{2+x^6}}{10 x^5}-\frac{1}{5} \int \frac{1}{\sqrt{2+x^6}} \, dx\\ &=-\frac{\sqrt{2+x^6}}{10 x^5}-\frac{x \left (\sqrt [3]{2}+x^2\right ) \sqrt{\frac{2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}+\left (1-\sqrt{3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt{3}\right ) x^2\right )^2}} \sqrt{2+x^6}}\\ \end{align*}
Mathematica [C] time = 0.0044754, size = 29, normalized size = 0.16 \[ -\frac{\, _2F_1\left (-\frac{5}{6},\frac{1}{2};\frac{1}{6};-\frac{x^6}{2}\right )}{5 \sqrt{2} x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.024, size = 31, normalized size = 0.2 \begin{align*} -{\frac{1}{10\,{x}^{5}}\sqrt{{x}^{6}+2}}-{\frac{x\sqrt{2}}{10}{\mbox{$_2$F$_1$}({\frac{1}{6}},{\frac{1}{2}};\,{\frac{7}{6}};\,-{\frac{{x}^{6}}{2}})}} \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{1}{\sqrt{x^{6} + 2} x^{6}}\,{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{\sqrt{x^{6} + 2}}{x^{12} + 2 \, x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.829708, size = 39, normalized size = 0.22 \begin{align*} \frac{\sqrt{2} \Gamma \left (- \frac{5}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{6}, \frac{1}{2} \\ \frac{1}{6} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{12 x^{5} \Gamma \left (\frac{1}{6}\right )} \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{1}{\sqrt{x^{6} + 2} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]