Optimal. Leaf size=11 \[ -\frac {1}{4 \left (1+x^4\right )} \]
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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 267}
\begin {gather*} -\frac {1}{4 \left (x^4+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 267
Rubi steps
\begin {align*} \int \frac {x^3}{1+2 x^4+x^8} \, dx &=\int \frac {x^3}{\left (1+x^4\right )^2} \, dx\\ &=-\frac {1}{4 \left (1+x^4\right )}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \left (1+x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 10, normalized size = 0.91
method | result | size |
gosper | \(-\frac {1}{4 \left (x^{4}+1\right )}\) | \(10\) |
default | \(-\frac {1}{4 \left (x^{4}+1\right )}\) | \(10\) |
norman | \(-\frac {1}{4 \left (x^{4}+1\right )}\) | \(10\) |
risch | \(-\frac {1}{4 \left (x^{4}+1\right )}\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 9, normalized size = 0.82 \begin {gather*} -\frac {1}{4 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 9, normalized size = 0.82 \begin {gather*} -\frac {1}{4 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.73 \begin {gather*} - \frac {1}{4 x^{4} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.34, size = 9, normalized size = 0.82 \begin {gather*} -\frac {1}{4 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} -\frac {1}{4\,\left (x^4+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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