Optimal. Leaf size=38 \[ \frac {1}{2} \tan ^{-1}(x)+\frac {1}{6} \log (1+x)+\frac {1}{4} \log \left (1+x^2\right )-\frac {1}{3} \log \left (1-x+x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {2083, 649, 209,
266, 642} \begin {gather*} \frac {\text {ArcTan}(x)}{2}+\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 642
Rule 649
Rule 2083
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*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 31, normalized size = 0.82
method | result | size |
default | \(\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}+1\right )}{4}-\frac {\ln \left (x^{2}-x +1\right )}{3}\) | \(31\) |
risch | \(\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}+1\right )}{4}-\frac {\ln \left (x^{2}-x +1\right )}{3}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 30, normalized size = 0.79 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 30, normalized size = 0.79 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 0.76 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.39, size = 31, normalized size = 0.82 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.16, size = 36, normalized size = 0.95 \begin {gather*} \frac {\ln \left (x+1\right )}{6}-\frac {\ln \left (x^2-x+1\right )}{3}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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