Integrand size = 13, antiderivative size = 378 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=-\frac {\sqrt {2+x^6}}{4 x^2}+\frac {\sqrt {2+x^6}}{4 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}+\frac {\left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]
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Time = 0.18 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {281, 331, 309, 224, 1891} \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\frac {\left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{4\ 2^{5/6} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {\sqrt {x^6+2}}{4 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )}-\frac {\sqrt {x^6+2}}{4 x^2} \]
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Rule 224
Rule 281
Rule 309
Rule 331
Rule 1891
Rubi steps \begin{align*} \text {integral}& = \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 {\left (1-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right )}{4\ 2^{2/3}} \\ & = -\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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.08 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{2},\frac {2}{3},-\frac {x^6}{2}\right )}{2 \sqrt {2} x^2} \]
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Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 5.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05
method | result | size |
meijerg | \(-\frac {\sqrt {2}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};-\frac {x^{6}}{2}\right )}{4 x^{2}}\) | \(20\) |
risch | \(-\frac {\sqrt {x^{6}+2}}{4 x^{2}}+\frac {\sqrt {2}\, x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {x^{6}}{2}\right )}{32}\) | \(33\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.07 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=-\frac {x^{2} {\rm weierstrassZeta}\left (0, -8, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right )\right ) + \sqrt {x^{6} + 2}}{4 \, x^{2}} \]
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Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\frac {\sqrt {2} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{12 x^{2} \Gamma \left (\frac {2}{3}\right )} \]
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\[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\int { \frac {1}{\sqrt {x^{6} + 2} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\int { \frac {1}{\sqrt {x^{6} + 2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \sqrt {2+x^6}} \, dx=\int \frac {1}{x^3\,\sqrt {x^6+2}} \,d x \]
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