Optimal. Leaf size=10 \[ \frac {2}{x+1}+\log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1594, 27, 894} \begin {gather*} \frac {2}{x+1}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 894
Rule 1594
Rubi steps
\begin {align*} \int \frac {1+x^2}{x+2 x^2+x^3} \, dx &=\int \frac {1+x^2}{x \left (1+2 x+x^2\right )} \, dx\\ &=\int \frac {1+x^2}{x (1+x)^2} \, dx\\ &=\int \left (\frac {1}{x}-\frac {2}{(1+x)^2}\right ) \, dx\\ &=\frac {2}{1+x}+\log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {2}{x+1}+\log (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 {1+x^2}{x+2 x^2+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.03, size = 14, normalized size = 1.40 \begin {gather*} \frac {{\left (x + 1\right )} \log \relax (x) + 2}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 11, normalized size = 1.10 \begin {gather*} \frac {2}{x + 1} + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 11, normalized size = 1.10 \begin {gather*} \ln \relax (x )+\frac {2}{x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 10, normalized size = 1.00 \begin {gather*} \frac {2}{x + 1} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 10, normalized size = 1.00 \begin {gather*} \ln \relax (x)+\frac {2}{x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 7, normalized size = 0.70 \begin {gather*} \log {\relax (x )} + \frac {2}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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