Optimal. Leaf size=38 \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0249394, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2058, 635, 203, 260, 628} \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 2058
Rule 635
Rule 203
Rule 260
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{1+x^2+x^3+x^5} \, dx &=\int \left (\frac{1}{6 (1+x)}+\frac{1+x}{2 \left (1+x^2\right )}+\frac{1-2 x}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac{1}{6} \log (1+x)+\frac{1}{3} \int \frac{1-2 x}{1-x+x^2} \, dx+\frac{1}{2} \int \frac{1+x}{1+x^2} \, dx\\ &=\frac{1}{6} \log (1+x)-\frac{1}{3} \log \left (1-x+x^2\right )+\frac{1}{2} \int \frac{1}{1+x^2} \, dx+\frac{1}{2} \int \frac{x}{1+x^2} \, dx\\ &=\frac{1}{2} \tan ^{-1}(x)+\frac{1}{6} \log (1+x)+\frac{1}{4} \log \left (1+x^2\right )-\frac{1}{3} \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0070307, size = 38, normalized size = 1. \[ \frac{1}{4} \log \left (x^2+1\right )-\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{6} \log (x+1)+\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 31, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ( x \right ) }{2}}+{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{4}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.70635, size = 41, normalized size = 1.08 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79973, size = 100, normalized size = 2.63 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.141915, size = 29, normalized size = 0.76 \begin{align*} \frac{\log{\left (x + 1 \right )}}{6} + \frac{\log{\left (x^{2} + 1 \right )}}{4} - \frac{\log{\left (x^{2} - x + 1 \right )}}{3} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14541, size = 42, normalized size = 1.11 \begin{align*} \frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right ) + \frac{1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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