Optimal. Leaf size=16 \[ \frac {2 \left (-1+x^6\right )^{3/4}}{3 x^3} \]
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Rubi [A]
time = 0.00, 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 = {460}
\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 460
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 = 0.11, 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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Maple [A]
time = 0.28, 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 | \(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{3 x^{3} \left (x^{6}-1\right )^{\frac {1}{4}}}\) | \(33\) |
meijerg | \(-\frac {2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], x^{6}\right )}{3 \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{4}} x^{3}}+\frac {\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], x^{6}\right )}{3 \mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{4}}}\) | \(66\) |
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. 38 vs.
\(2 (12) = 24\).
time = 0.50, size = 38, normalized size = 2.38 \begin {gather*} \frac {2 \, {\left (x^{6} - 1\right )}}{3 \, {\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^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, 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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Sympy [C] Result contains complex when optimal does not.
time = 1.74, size = 48, normalized size = 3.00 \begin {gather*} \frac {x^{3} e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{6}} \right )}}{3} + \frac {2 e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6}} \right )}}{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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, 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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