Optimal. Leaf size=179 \[ \frac {x}{6 \sqrt {2+x^6}}+\frac {x \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}+\left (1-\sqrt {3}\right ) x^2}{\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2 \left (\sqrt [3]{2}+x^2\right )}{\left (\sqrt [3]{2}+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {2+x^6}} \]
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Rubi [A]
time = 0.02, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {205, 231}
\begin {gather*} \frac {\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}} x 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 )}{6 \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 {x}{6 \sqrt {x^6+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 231
Rubi steps
\begin {align*} \int \frac {1}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {x}{6 \sqrt {2+x^6}}+\frac {1}{3} \int \frac {1}{\sqrt {2+x^6}} \, dx\\ &=\frac {x}{6 \sqrt {2+x^6}}+\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 )}{6 \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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 38, normalized size = 0.21 \begin {gather*} \frac {1}{6} x \left (\frac {1}{\sqrt {2+x^6}}+\sqrt {2} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\frac {x^6}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.17, size = 18, normalized size = 0.10
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{6}, \frac {3}{2}\right ], \left [\frac {7}{6}\right ], -\frac {x^{6}}{2}\right )}{4}\) | \(18\) |
risch | \(\frac {x}{6 \sqrt {x^{6}+2}}+\frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{6}, \frac {1}{2}\right ], \left [\frac {7}{6}\right ], -\frac {x^{6}}{2}\right )}{6}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F]
time = 0.07, size = 22, normalized size = 0.12 \begin {gather*} {\rm integral}\left (\frac {\sqrt {x^{6} + 2}}{x^{12} + 4 \, x^{6} + 4}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.34, size = 34, normalized size = 0.19 \begin {gather*} \frac {\sqrt {2} x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{2} \\ \frac {7}{6} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac {7}{6}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [B]
time = 0.08, size = 16, normalized size = 0.09 \begin {gather*} \frac {\sqrt {2}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{6},\frac {3}{2};\ \frac {7}{6};\ -\frac {x^6}{2}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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