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 = 0.12, 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 = 0.44, 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 = 0.37, size = 18, normalized size = 1.12 \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 = 0.04, size = 13, normalized size = 0.81
method | result | size |
trager | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\) | \(13\) |
meijerg | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\) | \(13\) |
risch | \(-\frac {\sqrt {x^{6}+1}}{3 x^{3}}\) | \(13\) |
gosper | \(-\frac {\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}{3 x^{3} \sqrt {x^{6}+1}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, 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 = 0.22, 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 = 0.55, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\sqrt {1 + \frac {1}{x^{6}}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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