Optimal. Leaf size=14 \[ \frac {2 \sqrt [4]{-1+x^6}}{x} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, 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 = {460}
\begin {gather*} \frac {2 \sqrt [4]{x^6-1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rubi steps
\begin {align*} \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx &=\frac {2 \sqrt [4]{-1+x^6}}{x}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{-1+x^6}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 13, normalized size = 0.93
method | result | size |
trager | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
risch | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
gosper | \(\frac {2 \left (x^{2}-x +1\right ) \left (x^{2}+x +1\right ) \left (1+x \right ) \left (-1+x \right )}{x \left (x^{6}-1\right )^{\frac {3}{4}}}\) | \(33\) |
meijerg | \(\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} x^{5} \hypergeom \left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right )}{5 \mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}-\frac {2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{\mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs.
\(2 (12) = 24\).
time = 0.51, size = 33, normalized size = 2.36 \begin {gather*} \frac {2 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.12, size = 66, normalized size = 4.71 \begin {gather*} \frac {x^{5} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x \Gamma \left (\frac {5}{6}\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 [B]
time = 0.14, size = 12, normalized size = 0.86 \begin {gather*} \frac {2\,{\left (x^6-1\right )}^{1/4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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