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 = 0.0282889, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, 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 2074
Rule 635
Rule 203
Rule 260
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*}
Mathematica [A] time = 0.0072064, size = 29, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2+1\right )+x+\log (1-x)-\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 24, normalized size = 0.8 \begin{align*} x+{\frac{{x}^{2}}{2}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-\arctan \left ( x \right ) +\ln \left ( -1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3999, size = 31, normalized size = 1.07 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08234, size = 77, normalized size = 2.66 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.107935, size = 22, normalized size = 0.76 \begin{align*} \frac{x^{2}}{2} + x + \log{\left (x - 1 \right )} - \frac{\log{\left (x^{2} + 1 \right )}}{2} - \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0573, size = 32, normalized size = 1.1 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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