Optimal. Leaf size=6 \[ -\tanh ^{-1}(x+2) \]
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Rubi [B] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 2.83, 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 = {616, 31} \[ \frac {1}{2} \log (x+1)-\frac {1}{2} \log (x+3) \]
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rubi steps
\begin {align*} \int \frac {1}{3+4 x+x^2} \, dx &=\frac {1}{2} \int \frac {1}{1+x} \, dx-\frac {1}{2} \int \frac {1}{3+x} \, dx\\ &=\frac {1}{2} \log (1+x)-\frac {1}{2} \log (3+x)\\ \end {align*}
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Mathematica [B] time = 0.00, size = 17, normalized size = 2.83 \[ \frac {1}{2} \log (x+1)-\frac {1}{2} \log (x+3) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 13, normalized size = 2.17 \[ -\frac {1}{2} \, \log \left (x + 3\right ) + \frac {1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 15, normalized size = 2.50 \[ -\frac {1}{2} \, \log \left ({\left | x + 3 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 14, normalized size = 2.33 \[ -\frac {\ln \left (x +3\right )}{2}+\frac {\ln \left (x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.29, size = 13, normalized size = 2.17 \[ -\frac {1}{2} \, \log \left (x + 3\right ) + \frac {1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 6, normalized size = 1.00 \[ -\mathrm {atanh}\left (x+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 12, normalized size = 2.00 \[ \frac {\log {\left (x + 1 \right )}}{2} - \frac {\log {\left (x + 3 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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