Optimal. Leaf size=13 \[ \log \left (x^2+x+1\right )-\frac {1}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 13, 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 = {1594, 1628, 628} \begin {gather*} \log \left (x^2+x+1\right )-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1594
Rule 1628
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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \log \left (x^2+x+1\right )-\frac {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 {1+x+2 x^2+2 x^3}{x^2+x^3+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 2.06, size = 15, normalized size = 1.15 \begin {gather*} \frac {x \log \left (x^{2} + x + 1\right ) - 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{x} + \log \left (x^{2} + x + 1\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 = 1.08 \begin {gather*} \ln \left (x^{2}+x +1\right )-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{x} + \log \left (x^{2} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.14, size = 13, normalized size = 1.00 \begin {gather*} \ln \left (x^2+x+1\right )-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (x^{2} + x + 1 \right )} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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