Optimal. Leaf size=19 \[ \frac {1}{3} \log (1-x)-\frac {1}{3} \log (2+x) \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {36, 31}
\begin {gather*} \frac {1}{3} \log (1-x)-\frac {1}{3} \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {1}{(-1+x) (2+x)} \, dx &=\frac {1}{3} \int \frac {1}{-1+x} \, dx-\frac {1}{3} \int \frac {1}{2+x} \, dx\\ &=\frac {1}{3} \log (1-x)-\frac {1}{3} \log (2+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{3} \log (1-x)-\frac {1}{3} \log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 14, normalized size = 0.74
method | result | size |
default | \(\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (2+x \right )}{3}\) | \(14\) |
norman | \(\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (2+x \right )}{3}\) | \(14\) |
risch | \(\frac {\ln \left (-1+x \right )}{3}-\frac {\ln \left (2+x \right )}{3}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.94, size = 13, normalized size = 0.68 \begin {gather*} -\frac {1}{3} \, \log \left (x + 2\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 13, normalized size = 0.68 \begin {gather*} -\frac {1}{3} \, \log \left (x + 2\right ) + \frac {1}{3} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 12, normalized size = 0.63 \begin {gather*} \frac {\log {\left (x - 1 \right )}}{3} - \frac {\log {\left (x + 2 \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 15, normalized size = 0.79 \begin {gather*} -\frac {1}{3} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 12, normalized size = 0.63 \begin {gather*} \frac {\ln \left (\frac {x-1}{x+2}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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