Optimal. Leaf size=16 \[ \frac {2 \left (x^6+1\right )^{3/4}}{3 x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {449} \begin {gather*} \frac {2 \left (x^6+1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int \frac {-2+x^6}{x^4 \sqrt [4]{1+x^6}} \, dx &=\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^6+1\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+x^6\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \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*} \int \frac {x^{6} - 2}{{\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 13, normalized size = 0.81
method | result | size |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(13\) |
gosper | \(\frac {2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}{3 x^{3} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(28\) |
meijerg | \(\frac {2 \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], -x^{6}\right )}{3 x^{3}}+\frac {\hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{6}\right ) x^{3}}{3}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 29, normalized size = 1.81 \begin {gather*} \frac {2 \, {\left (x^{6} + 1\right )}}{3 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (x^6+1\right )}^{3/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.59, size = 42, normalized size = 2.62 \begin {gather*} \frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3} + \frac {2 {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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