Optimal. Leaf size=16 \[ \frac {\sqrt {x^6-1}}{3 x^3} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} \frac {\sqrt {x^6-1}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx &=\frac {\sqrt {-1+x^6}}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-1}}{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 {\sqrt {-1+x^6}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^{3} + \sqrt {x^{6} - 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 20, normalized size = 1.25 \begin {gather*} \frac {\sqrt {-\frac {1}{x^{6}} + 1}}{3 \, \mathrm {sgn}\relax (x)} - \frac {1}{3} \, \mathrm {sgn}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 13, normalized size = 0.81
method | result | size |
trager | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {\sqrt {x^{6}-1}}{3 x^{3}}\) | \(13\) |
gosper | \(\frac {\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{3 x^{3} \sqrt {x^{6}-1}}\) | \(33\) |
meijerg | \(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{3}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x^6-1}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 29, normalized size = 1.81 \begin {gather*} \begin {cases} \frac {i \sqrt {-1 + \frac {1}{x^{6}}}}{3} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {\sqrt {1 - \frac {1}{x^{6}}}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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