Optimal. Leaf size=181 \[ -\frac {\sqrt {2+x^6}}{10 x^5}-\frac {x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{10 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 181, normalized size of antiderivative = 1.00, 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 = {331, 231}
\begin {gather*} -\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 (\text {ArcCos}\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 231
Rule 331
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 29, normalized size = 0.16 \begin {gather*} -\frac {\, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};-\frac {x^6}{2}\right )}{5 \sqrt {2} x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.18, size = 20, normalized size = 0.11
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \hypergeom \left (\left [-\frac {5}{6}, \frac {1}{2}\right ], \left [\frac {1}{6}\right ], -\frac {x^{6}}{2}\right )}{10 x^{5}}\) | \(20\) |
risch | \(-\frac {\sqrt {x^{6}+2}}{10 x^{5}}-\frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {7}{6}\right ], -\frac {x^{6}}{2}\right )}{10}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.07, size = 21, normalized size = 0.12 \begin {gather*} {\rm integral}\left (\frac {\sqrt {x^{6} + 2}}{x^{12} + 2 \, x^{6}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.43, size = 39, normalized size = 0.22 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^6\,\sqrt {x^6+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________