Optimal. Leaf size=18 \[ -\frac {\sqrt {1-x^4}}{2 x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270}
\begin {gather*} -\frac {\sqrt {1-x^4}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x^4}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 25, normalized size = 1.39
method | result | size |
trager | \(-\frac {\sqrt {-x^{4}+1}}{2 x^{2}}\) | \(15\) |
meijerg | \(-\frac {\sqrt {-x^{4}+1}}{2 x^{2}}\) | \(15\) |
risch | \(\frac {x^{4}-1}{2 x^{2} \sqrt {-x^{4}+1}}\) | \(20\) |
default | \(\frac {\left (x^{2}+1\right ) \left (x^{2}-1\right )}{2 x^{2} \sqrt {-x^{4}+1}}\) | \(25\) |
elliptic | \(\frac {\left (x^{2}+1\right ) \left (x^{2}-1\right )}{2 x^{2} \sqrt {-x^{4}+1}}\) | \(25\) |
gosper | \(\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right )}{2 x^{2} \sqrt {-x^{4}+1}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} \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 = 0.31, size = 34, normalized size = 1.89 \begin {gather*} \begin {cases} - \frac {i \sqrt {x^{4} - 1}}{2 x^{2}} & \text {for}\: \left |{x^{4}}\right | > 1 \\- \frac {\sqrt {1 - x^{4}}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (14) = 28\).
time = 1.89, size = 35, normalized size = 1.94 \begin {gather*} \frac {x^{2}}{4 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}} - \frac {\sqrt {-x^{4} + 1} - 1}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-x^4}}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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