Optimal. Leaf size=26 \[ \frac {x^4}{4}+\frac {1}{4} \log \left (1+x^4\right )-\log \left (2+x^4\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1371, 717, 646,
31} \begin {gather*} \frac {x^4}{4}+\frac {1}{4} \log \left (x^4+1\right )-\log \left (x^4+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 717
Rule 1371
Rubi steps
\begin {align*} \int \frac {x^{11}}{2+3 x^4+x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{2+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac {x^4}{4}+\frac {1}{4} \text {Subst}\left (\int \frac {-2-3 x}{2+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac {x^4}{4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^4\right )-\text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^4\right )\\ &=\frac {x^4}{4}+\frac {1}{4} \log \left (1+x^4\right )-\log \left (2+x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^4}{4}+\frac {1}{4} \log \left (1+x^4\right )-\log \left (2+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 23, normalized size = 0.88
method | result | size |
default | \(\frac {x^{4}}{4}+\frac {\ln \left (x^{4}+1\right )}{4}-\ln \left (x^{4}+2\right )\) | \(23\) |
norman | \(\frac {x^{4}}{4}+\frac {\ln \left (x^{4}+1\right )}{4}-\ln \left (x^{4}+2\right )\) | \(23\) |
risch | \(\frac {x^{4}}{4}+\frac {\ln \left (x^{4}+1\right )}{4}-\ln \left (x^{4}+2\right )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{4}}{4} + \frac {\log {\left (x^{4} + 1 \right )}}{4} - \log {\left (x^{4} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.94, size = 22, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, x^{4} - \log \left (x^{4} + 2\right ) + \frac {1}{4} \, \log \left (x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 22, normalized size = 0.85 \begin {gather*} \frac {\ln \left (x^4+1\right )}{4}-\ln \left (x^4+2\right )+\frac {x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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