Optimal. Leaf size=25 \[ \frac {x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1593, 1802} \[ \frac {x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1802
Rubi steps
\begin {align*} \int \frac {1+2 x^2+x^5}{-x+x^3} \, dx &=\int \frac {1+2 x^2+x^5}{x \left (-1+x^2\right )} \, dx\\ &=\int \left (1+\frac {2}{-1+x}-\frac {1}{x}+x^2+\frac {1}{1+x}\right ) \, dx\\ &=x+\frac {x^3}{3}+2 \log (1-x)-\log (x)+\log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 21, normalized size = 0.84 \[ \frac {1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 24, normalized size = 0.96 \[ \frac {1}{3} \, x^{3} + x + \log \left ({\left | x + 1 \right |}\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.88 \[ \frac {x^{3}}{3}+x -\ln \relax (x )+2 \ln \left (x -1\right )+\ln \left (x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 21, normalized size = 0.84 \[ \frac {1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 30, normalized size = 1.20 \[ x+2\,\ln \left (x-1\right )+\frac {x^3}{3}+\mathrm {atan}\left (\frac {48{}\mathrm {i}}{11\,\left (22\,x-2\right )}+\frac {13}{11}{}\mathrm {i}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 0.80 \[ \frac {x^{3}}{3} + x - \log {\relax (x )} + 2 \log {\left (x - 1 \right )} + \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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