Optimal. Leaf size=6 \[ -\tanh ^{-1}(x+2) \]
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Rubi [B] time = 0.0036109, antiderivative size = 17, normalized size of antiderivative = 2.83, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 616
Rule 31
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*}
Mathematica [B] time = 0.0025821, 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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Maple [B] time = 0.047, size = 14, normalized size = 2.3 \begin{align*}{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ( 3+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13718, size = 18, normalized size = 3. \begin{align*} -\frac{1}{2} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09493, size = 46, normalized size = 7.67 \begin{align*} -\frac{1}{2} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.094988, size = 12, normalized size = 2. \begin{align*} \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x + 3 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26072, size = 20, normalized size = 3.33 \begin{align*} -\frac{1}{2} \, \log \left ({\left | x + 3 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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