Optimal. Leaf size=15 \[ \frac{1}{2} \log \left (x^2+3\right )+3 \tan ^{-1}(x) \]
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Rubi [A] time = 0.106478, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {6725, 203, 260} \[ \frac{1}{2} \log \left (x^2+3\right )+3 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6725
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{9+x+3 x^2+x^3}{\left (1+x^2\right ) \left (3+x^2\right )} \, dx &=\int \left (\frac{3}{1+x^2}+\frac{x}{3+x^2}\right ) \, dx\\ &=3 \int \frac{1}{1+x^2} \, dx+\int \frac{x}{3+x^2} \, dx\\ &=3 \tan ^{-1}(x)+\frac{1}{2} \log \left (3+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0077366, size = 15, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+3\right )+3 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 14, normalized size = 0.9 \begin{align*} 3\,\arctan \left ( x \right ) +{\frac{\ln \left ({x}^{2}+3 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.9195, size = 18, normalized size = 1.2 \begin{align*} 3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47228, size = 43, normalized size = 2.87 \begin{align*} 3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.109246, size = 12, normalized size = 0.8 \begin{align*} \frac{\log{\left (x^{2} + 3 \right )}}{2} + 3 \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14973, size = 18, normalized size = 1.2 \begin{align*} 3 \, \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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