Optimal. Leaf size=378 \[ -\frac {\sqrt {2+x^6}}{4 x^2}+\frac {\sqrt {2+x^6}}{4 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}+\frac {\left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]
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Rubi [A]
time = 0.15, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 331, 309,
224, 1891} \begin {gather*} \frac {\left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} F\left (\text {ArcSin}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\text {ArcSin}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {\sqrt {x^6+2}}{4 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )}-\frac {\sqrt {x^6+2}}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 224
Rule 281
Rule 309
Rule 331
Rule 1891
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {2+x^6}}{4 x^2}+\frac {1}{8} \text {Subst}\left (\int \frac {x}{\sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {2+x^6}}{4 x^2}+\frac {1}{8} \text {Subst}\left (\int \frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x}{\sqrt {2+x^3}} \, dx,x,x^2\right )+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right )}{4 \sqrt [6]{2} \sqrt {2+\sqrt {3}}}\\ &=-\frac {\sqrt {2+x^6}}{4 x^2}+\frac {\sqrt {2+x^6}}{4 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \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}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}+\frac {\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 (\sin ^{-1}\left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\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 = 10.02, size = 29, normalized size = 0.08 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};-\frac {x^6}{2}\right )}{2 \sqrt {2} x^2} \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 = 0.19, size = 20, normalized size = 0.05
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {2}{3}\right ], -\frac {x^{6}}{2}\right )}{4 x^{2}}\) | \(20\) |
risch | \(-\frac {\sqrt {x^{6}+2}}{4 x^{2}}+\frac {\sqrt {2}\, x^{4} \hypergeom \left (\left [\frac {1}{2}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {x^{6}}{2}\right )}{32}\) | \(33\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.07, size = 26, normalized size = 0.07 \begin {gather*} -\frac {x^{2} {\rm weierstrassZeta}\left (0, -8, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right )\right ) + \sqrt {x^{6} + 2}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 39, normalized size = 0.10 \begin {gather*} \frac {\sqrt {2} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{12 x^{2} \Gamma \left (\frac {2}{3}\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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,\sqrt {x^6+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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