Optimal. Leaf size=45 \[ \frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{4 \sqrt {x^8+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {275, 220} \[ \frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{4 \sqrt {x^8+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1+x^8}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=\frac {\left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{4 \sqrt {1+x^8}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 22, normalized size = 0.49 \[ \frac {1}{2} x^2 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^8\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{\sqrt {x^{8} + 1}}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 17, normalized size = 0.38 \[ \frac {x^{2} \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -x^{8}\right )}{2} \]
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 \[ \int \frac {x}{\sqrt {x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x}{\sqrt {x^8+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.64, size = 29, normalized size = 0.64 \[ \frac {x^{2} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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