Optimal. Leaf size=13 \[ \log \left (x^2+x+1\right )-\frac{1}{x} \]
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Rubi [A] time = 0.0490023, antiderivative size = 13, 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 = {1594, 1628, 628} \[ \log \left (x^2+x+1\right )-\frac{1}{x} \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1628
Rule 628
Rubi steps
\begin{align*} \int \frac{1+x+2 x^2+2 x^3}{x^2+x^3+x^4} \, dx &=\int \frac{1+x+2 x^2+2 x^3}{x^2 \left (1+x+x^2\right )} \, dx\\ &=\int \left (\frac{1}{x^2}+\frac{1+2 x}{1+x+x^2}\right ) \, dx\\ &=-\frac{1}{x}+\int \frac{1+2 x}{1+x+x^2} \, dx\\ &=-\frac{1}{x}+\log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.005564, size = 13, normalized size = 1. \[ \log \left (x^2+x+1\right )-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 14, normalized size = 1.1 \begin{align*} -{x}^{-1}+\ln \left ({x}^{2}+x+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03172, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{x} + \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43135, size = 38, normalized size = 2.92 \begin{align*} \frac{x \log \left (x^{2} + x + 1\right ) - 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.092617, size = 10, normalized size = 0.77 \begin{align*} \log{\left (x^{2} + x + 1 \right )} - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13992, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{x} + \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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