Optimal. Leaf size=15 \[ \frac {1}{2} \log \left (x^2+3\right )+3 \tan ^{-1}(x) \]
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Rubi [A] time = 0.11, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{2} \log \left (x^2+3\right )+3 \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 6725
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*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (x^2+3\right )+3 \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {9+x+3 x^2+x^3}{\left (1+x^2\right ) \left (3+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.06, size = 13, normalized size = 0.87 \begin {gather*} 3 \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} 3 \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} 3 \arctan \relax (x )+\frac {\ln \left (x^{2}+3\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 13, normalized size = 0.87 \begin {gather*} 3 \, \arctan \relax (x) + \frac {1}{2} \, \log \left (x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (x^2+3\right )}{2}+3\,\mathrm {atan}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.80 \begin {gather*} \frac {\log {\left (x^{2} + 3 \right )}}{2} + 3 \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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