Optimal. Leaf size=12 \[ \frac {1}{3} \log \left (3 x+x^3\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 12, 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 = {1601}
\begin {gather*} \frac {1}{3} \log \left (x^3+3 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 1601
Rubi steps
\begin {align*} \int \frac {1+x^2}{3 x+x^3} \, dx &=\frac {1}{3} \log \left (3 x+x^3\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.42 \begin {gather*} \frac {\log (x)}{3}+\frac {1}{3} \log \left (3+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 11, normalized size = 0.92
method | result | size |
default | \(\frac {\ln \left (x \left (x^{2}+3\right )\right )}{3}\) | \(11\) |
risch | \(\frac {\ln \left (x^{3}+3 x \right )}{3}\) | \(11\) |
norman | \(\frac {\ln \left (x \right )}{3}+\frac {\ln \left (x^{2}+3\right )}{3}\) | \(14\) |
meijerg | \(\frac {\ln \left (1+\frac {x^{2}}{3}\right )}{3}+\frac {\ln \left (x \right )}{3}-\frac {\ln \left (3\right )}{6}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, \log \left (x^{3} + 3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, \log \left (x^{3} + 3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log {\left (x^{3} + 3 x \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.21, size = 13, normalized size = 1.08 \begin {gather*} \frac {1}{3} \, \log \left (3 \, {\left | \frac {1}{3} \, x^{3} + x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.27, size = 10, normalized size = 0.83 \begin {gather*} \frac {\ln \left (x^3+3\,x\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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