Optimal. Leaf size=21 \[ \frac {1}{4} \log \left (1+x^4\right )-\frac {1}{4} \log \left (2+x^4\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 630, 31}
\begin {gather*} \frac {1}{4} \log \left (x^4+1\right )-\frac {1}{4} \log \left (x^4+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 630
Rule 1366
Rubi steps
\begin {align*} \int \frac {x^3}{2+3 x^4+x^8} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{2+3 x+x^2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^4\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \log \left (1+x^4\right )-\frac {1}{4} \log \left (2+x^4\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log \left (1+x^4\right )-\frac {1}{4} \log \left (2+x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 18, normalized size = 0.86
method | result | size |
default | \(\frac {\ln \left (x^{4}+1\right )}{4}-\frac {\ln \left (x^{4}+2\right )}{4}\) | \(18\) |
norman | \(\frac {\ln \left (x^{4}+1\right )}{4}-\frac {\ln \left (x^{4}+2\right )}{4}\) | \(18\) |
risch | \(\frac {\ln \left (x^{4}+1\right )}{4}-\frac {\ln \left (x^{4}+2\right )}{4}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{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 = 0.33, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{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 = 15, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x^{4} + 1 \right )}}{4} - \frac {\log {\left (x^{4} + 2 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.93, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{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 = 0.06, size = 16, normalized size = 0.76 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {256}{9\,\left (144\,x^4+160\right )}-\frac {7}{9}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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