Optimal. Leaf size=19 \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]
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Rubi [A] time = 0.0618751, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6688, 616, 31} \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]
Antiderivative was successfully verified.
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Rule 6688
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\left (1+x^2\right ) \left (3+\frac{10 x}{1+x^2}\right )} \, dx &=\int \frac{1}{3+10 x+3 x^2} \, dx\\ &=\frac{3}{8} \int \frac{1}{1+3 x} \, dx-\frac{3}{8} \int \frac{1}{9+3 x} \, dx\\ &=-\frac{1}{8} \log (3+x)+\frac{1}{8} \log (1+3 x)\\ \end{align*}
Mathematica [A] time = 0.0031748, size = 19, normalized size = 1. \[ \frac{1}{8} \log (3 x+1)-\frac{1}{8} \log (x+3) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 16, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 3+x \right ) }{8}}+{\frac{\ln \left ( 1+3\,x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08613, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{8} \, \log \left (3 \, x + 1\right ) - \frac{1}{8} \, \log \left (x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23929, size = 47, normalized size = 2.47 \begin{align*} \frac{1}{8} \, \log \left (3 \, x + 1\right ) - \frac{1}{8} \, \log \left (x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098616, size = 14, normalized size = 0.74 \begin{align*} \frac{\log{\left (x + \frac{1}{3} \right )}}{8} - \frac{\log{\left (x + 3 \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15885, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{8} \, \log \left ({\left | 3 \, x + 1 \right |}\right ) - \frac{1}{8} \, \log \left ({\left | x + 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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