Optimal. Leaf size=19 \[ \frac {1}{8} \log (3 x+1)-\frac {1}{8} \log (x+3) \]
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Rubi [A] time = 0.06, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 31
Rule 616
Rule 6688
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*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ \frac {1}{8} \log (3 x+1)-\frac {1}{8} \log (x+3) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 15, normalized size = 0.79 \[ \frac {1}{8} \, \log \left (3 \, x + 1\right ) - \frac {1}{8} \, \log \left (x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 17, normalized size = 0.89 \[ \frac {1}{8} \, \log \left ({\left | 3 \, x + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x + 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 16, normalized size = 0.84 \[ \frac {\ln \left (3 x +1\right )}{8}-\frac {\ln \left (x +3\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 15, normalized size = 0.79 \[ \frac {1}{8} \, \log \left (3 \, x + 1\right ) - \frac {1}{8} \, \log \left (x + 3\right ) \]
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 \[ -\frac {\mathrm {atanh}\left (\frac {3\,x}{4}+\frac {5}{4}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 14, normalized size = 0.74 \[ \frac {\log {\left (x + \frac {1}{3} \right )}}{8} - \frac {\log {\left (x + 3 \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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