Optimal. Leaf size=26 \[ 2 x+\frac {x^2}{2}+\frac {x^3}{3}+2 \log (1-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 786}
\begin {gather*} \frac {x^3}{3}+\frac {x^2}{2}+2 x+2 \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 786
Rule 1607
Rubi steps
\begin {align*} \int \frac {x+x^3}{-1+x} \, dx &=\int \frac {x \left (1+x^2\right )}{-1+x} \, dx\\ &=\int \left (2+\frac {2}{-1+x}+x+x^2\right ) \, dx\\ &=2 x+\frac {x^2}{2}+\frac {x^3}{3}+2 \log (1-x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 0.96 \begin {gather*} \frac {1}{6} \left (-17+12 x+3 x^2+2 x^3+12 \log (-1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.77, size = 20, normalized size = 0.77 \begin {gather*} 2 x+\frac {x^2}{2}+\frac {x^3}{3}+2 \text {Log}\left [-1+x\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 21, normalized size = 0.81
method | result | size |
default | \(\frac {x^{3}}{3}+\frac {x^{2}}{2}+2 x +2 \ln \left (-1+x \right )\) | \(21\) |
norman | \(\frac {x^{3}}{3}+\frac {x^{2}}{2}+2 x +2 \ln \left (-1+x \right )\) | \(21\) |
risch | \(\frac {x^{3}}{3}+\frac {x^{2}}{2}+2 x +2 \ln \left (-1+x \right )\) | \(21\) |
meijerg | \(\frac {x \left (4 x^{2}+6 x +12\right )}{12}+2 \ln \left (1-x \right )+x\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 20, normalized size = 0.77 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} + 2 \, x + 2 \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 0.77 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} + 2 \, x + 2 \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{3}}{3} + \frac {x^{2}}{2} + 2 x + 2 \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 25, normalized size = 0.96 \begin {gather*} \frac {1}{3} x^{3}+\frac {1}{2} x^{2}+2 x+2 \ln \left |x-1\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 20, normalized size = 0.77 \begin {gather*} 2\,x+2\,\ln \left (x-1\right )+\frac {x^2}{2}+\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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