Optimal. Leaf size=12 \[ -\frac {x}{\sqrt {-1+x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {28, 1418, 391}
\begin {gather*} -\frac {x}{\sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 391
Rule 1418
Rubi steps
\begin {align*} \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx &=\int \frac {-1+x^8}{\left (-1+x^4\right )^{5/2}} \, dx\\ &=\int \frac {1+x^4}{\left (-1+x^4\right )^{3/2}} \, dx\\ &=-\frac {x}{\sqrt {-1+x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 12, normalized size = 1.00 \begin {gather*} -\frac {x}{\sqrt {-1+x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 11, normalized size = 0.92
method | result | size |
gosper | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
default | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
trager | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
risch | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
elliptic | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
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. 27 vs.
\(2 (10) = 20\).
time = 0.52, size = 27, normalized size = 2.25 \begin {gather*} -\frac {\sqrt {x^{2} + 1} \sqrt {x + 1} \sqrt {x - 1} x}{x^{4} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 10, normalized size = 0.83 \begin {gather*} -\frac {x}{\sqrt {x^{4} - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 10, normalized size = 0.83 \begin {gather*} -\frac {x}{\sqrt {x^{4} - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 10, normalized size = 0.83 \begin {gather*} -\frac {x}{\sqrt {x^4-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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