Optimal. Leaf size=29 \[ x+\frac {x^2}{2}-\tan ^{-1}(x)+\log (1-x)-\frac {1}{2} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2099, 649, 209,
266} \begin {gather*} -\text {ArcTan}(x)+\frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )+x+\log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 2099
Rubi steps
\begin {align*} \int \frac {1+x^4}{-1+x-x^2+x^3} \, dx &=\int \left (1+\frac {1}{-1+x}+x+\frac {-1-x}{1+x^2}\right ) \, dx\\ &=x+\frac {x^2}{2}+\log (1-x)+\int \frac {-1-x}{1+x^2} \, dx\\ &=x+\frac {x^2}{2}+\log (1-x)-\int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=x+\frac {x^2}{2}-\tan ^{-1}(x)+\log (1-x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} x+\frac {x^2}{2}-\tan ^{-1}(x)+\log (1-x)-\frac {1}{2} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 24, normalized size = 0.83
method | result | size |
default | \(x +\frac {x^{2}}{2}+\ln \left (-1+x \right )-\frac {\ln \left (x^{2}+1\right )}{2}-\arctan \left (x \right )\) | \(24\) |
risch | \(x +\frac {x^{2}}{2}+\ln \left (-1+x \right )-\frac {\ln \left (x^{2}+1\right )}{2}-\arctan \left (x \right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.32, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, x^{2} + x - \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, x^{2} + x - \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 0.76 \begin {gather*} \frac {x^{2}}{2} + x + \log {\left (x - 1 \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} - \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 24, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, x^{2} + x - \arctan \left (x\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 29, normalized size = 1.00 \begin {gather*} x+\ln \left (x-1\right )+\frac {x^2}{2}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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