Optimal. Leaf size=29 \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )+x+\log (1-x)-\tan ^{-1}(x) \]
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Rubi [A] time = 0.03, 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 = {2074, 635, 203, 260} \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )+x+\log (1-x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 2074
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 \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )+x+\log (1-x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 27, normalized size = 0.93 \[ -\frac {1}{2} \log \left (x^2+1\right )+\frac {1}{2} x (x+2)+\log (x-1)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 23, normalized size = 0.79 \[ \frac {1}{2} \, x^{2} + x - \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 24, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} + x - \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 24, normalized size = 0.83
method | result | size |
default | \(x +\frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}-\arctan \relax (x )+\ln \left (-1+x \right )\) | \(24\) |
risch | \(x +\frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2}-\arctan \relax (x )+\ln \left (-1+x \right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 23, normalized size = 0.79 \[ \frac {1}{2} \, x^{2} + x - \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \log \left (x - 1\right ) \]
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 \[ x+\ln \left (x-1\right )+\frac {x^2}{2}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 22, normalized size = 0.76 \[ \frac {x^{2}}{2} + x + \log {\left (x - 1 \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} - \operatorname {atan}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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