Optimal. Leaf size=223 \[ \frac {x^3 \sqrt {\frac {\left (x^2+1\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} x^4-2 x^2+\sqrt {2}}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \left (x^2+1\right ) \sqrt {x^8+1}}-\frac {x^3 \sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2} x^4+2 x^2+\sqrt {2}}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {x^8+1}} \]
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Rubi [A] time = 0.05, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {226, 1883} \[ \frac {x^3 \sqrt {\frac {\left (x^2+1\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} x^4-2 x^2+\sqrt {2}}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \left (x^2+1\right ) \sqrt {x^8+1}}-\frac {x^3 \sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {x^8+1}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2} x^4+2 x^2+\sqrt {2}}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {x^8+1}} \]
Antiderivative was successfully verified.
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Rule 226
Rule 1883
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x^8}} \, dx &=\frac {1}{2} \int \frac {1-x^2}{\sqrt {1+x^8}} \, dx+\frac {1}{2} \int \frac {1+x^2}{\sqrt {1+x^8}} \, dx\\ &=\frac {x^3 \sqrt {\frac {\left (1+x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x^2+\sqrt {2} x^4}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \left (1+x^2\right ) \sqrt {1+x^8}}-\frac {x^3 \sqrt {-\frac {\left (1-x^2\right )^2}{x^2}} \sqrt {-\frac {1+x^8}{x^4}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x^2+\sqrt {2} x^4}{x^2}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \left (1-x^2\right ) \sqrt {1+x^8}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 17, normalized size = 0.08 \[ x \, _2F_1\left (\frac {1}{8},\frac {1}{2};\frac {9}{8};-x^8\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {x^{8} + 1}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 14, normalized size = 0.06 \[ x \hypergeom \left (\left [\frac {1}{8}, \frac {1}{2}\right ], \left [\frac {9}{8}\right ], -x^{8}\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 12, normalized size = 0.05 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{8},\frac {1}{2};\ \frac {9}{8};\ -x^8\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.67, size = 27, normalized size = 0.12 \[ \frac {x \Gamma \left (\frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{8}, \frac {1}{2} \\ \frac {9}{8} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {9}{8}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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