Optimal. Leaf size=19 \[ \frac {1}{4} \log (1-x)-\frac {1}{4} \log (3+x) \]
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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 = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {630, 31}
\begin {gather*} \frac {1}{4} \log (1-x)-\frac {1}{4} \log (x+3) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 630
Rubi steps
\begin {align*} \int \frac {1}{-3+2 x+x^2} \, dx &=\frac {1}{4} \int \frac {1}{-1+x} \, dx-\frac {1}{4} \int \frac {1}{3+x} \, dx\\ &=\frac {1}{4} \log (1-x)-\frac {1}{4} \log (3+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log (1-x)-\frac {1}{4} \log (3+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 14, normalized size = 0.74
method | result | size |
default | \(\frac {\ln \left (-1+x \right )}{4}-\frac {\ln \left (3+x \right )}{4}\) | \(14\) |
norman | \(\frac {\ln \left (-1+x \right )}{4}-\frac {\ln \left (3+x \right )}{4}\) | \(14\) |
risch | \(\frac {\ln \left (-1+x \right )}{4}-\frac {\ln \left (3+x \right )}{4}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.21, size = 13, normalized size = 0.68 \begin {gather*} -\frac {1}{4} \, \log \left (x + 3\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.41, size = 13, normalized size = 0.68 \begin {gather*} -\frac {1}{4} \, \log \left (x + 3\right ) + \frac {1}{4} \, \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 )}}{4} - \frac {\log {\left (x + 3 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 15, normalized size = 0.79 \begin {gather*} -\frac {1}{4} \, \log \left ({\left | x + 3 \right |}\right ) + \frac {1}{4} \, \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 = 0.08, size = 8, normalized size = 0.42 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {x}{2}+\frac {1}{2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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