Optimal. Leaf size=103 \[ -\frac {3 (1-x)}{8 \left (x^2+1\right )}+\frac {3 x}{16 \left (x^2+1\right )}+\frac {x+1}{8 \left (x^2+1\right )^2}+\frac {15}{16} \log \left (x^2+1\right )-\frac {1}{2} \log \left (x^2+x+1\right )+\frac {1}{8} \log (1-x)-\log (x)+\frac {7}{16} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.54, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6728, 639, 199, 203, 635, 260, 634, 618, 204, 628} \[ -\frac {3 (1-x)}{8 \left (x^2+1\right )}+\frac {3 x}{16 \left (x^2+1\right )}+\frac {x+1}{8 \left (x^2+1\right )^2}+\frac {15}{16} \log \left (x^2+1\right )-\frac {1}{2} \log \left (x^2+x+1\right )+\frac {1}{8} \log (1-x)-\log (x)+\frac {7}{16} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 204
Rule 260
Rule 618
Rule 628
Rule 634
Rule 635
Rule 639
Rule 6728
Rubi steps
\begin {align*} \int \frac {1+x^2+x^3}{(-1+x) x \left (1+x^2\right )^3 \left (1+x+x^2\right )} \, dx &=\int \left (\frac {1}{8 (-1+x)}-\frac {1}{x}+\frac {1-x}{2 \left (1+x^2\right )^3}+\frac {3 (1+x)}{4 \left (1+x^2\right )^2}+\frac {-1+15 x}{8 \left (1+x^2\right )}+\frac {-1-x}{1+x+x^2}\right ) \, dx\\ &=\frac {1}{8} \log (1-x)-\log (x)+\frac {1}{8} \int \frac {-1+15 x}{1+x^2} \, dx+\frac {1}{2} \int \frac {1-x}{\left (1+x^2\right )^3} \, dx+\frac {3}{4} \int \frac {1+x}{\left (1+x^2\right )^2} \, dx+\int \frac {-1-x}{1+x+x^2} \, dx\\ &=\frac {1+x}{8 \left (1+x^2\right )^2}-\frac {3 (1-x)}{8 \left (1+x^2\right )}+\frac {1}{8} \log (1-x)-\log (x)-\frac {1}{8} \int \frac {1}{1+x^2} \, dx+\frac {3}{8} \int \frac {1}{\left (1+x^2\right )^2} \, dx+\frac {3}{8} \int \frac {1}{1+x^2} \, dx-\frac {1}{2} \int \frac {1}{1+x+x^2} \, dx-\frac {1}{2} \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {15}{8} \int \frac {x}{1+x^2} \, dx\\ &=\frac {1+x}{8 \left (1+x^2\right )^2}-\frac {3 (1-x)}{8 \left (1+x^2\right )}+\frac {3 x}{16 \left (1+x^2\right )}+\frac {1}{4} \tan ^{-1}(x)+\frac {1}{8} \log (1-x)-\log (x)+\frac {15}{16} \log \left (1+x^2\right )-\frac {1}{2} \log \left (1+x+x^2\right )+\frac {3}{16} \int \frac {1}{1+x^2} \, dx+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {1+x}{8 \left (1+x^2\right )^2}-\frac {3 (1-x)}{8 \left (1+x^2\right )}+\frac {3 x}{16 \left (1+x^2\right )}+\frac {7}{16} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{8} \log (1-x)-\log (x)+\frac {15}{16} \log \left (1+x^2\right )-\frac {1}{2} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 93, normalized size = 0.90 \[ \frac {1}{48} \left (-14 \log \left (1-x^3\right )+\frac {6 (x+1)}{\left (x^2+1\right )^2}+\frac {9 (3 x-2)}{x^2+1}+45 \log \left (x^2+1\right )-10 \log \left (x^2+x+1\right )+20 \log (1-x)-48 \log (x)+21 \tan ^{-1}(x)-16 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 136, normalized size = 1.32 \[ \frac {27 \, x^{3} - 16 \, \sqrt {3} {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 18 \, x^{2} + 21 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \relax (x) - 24 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (x^{2} + x + 1\right ) + 45 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 6 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (x - 1\right ) - 48 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \relax (x) + 33 \, x - 12}{48 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 74, normalized size = 0.72 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {9 \, x^{3} - 6 \, x^{2} + 11 \, x - 4}{16 \, {\left (x^{2} + 1\right )}^{2}} + \frac {7}{16} \, \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) + \frac {15}{16} \, \log \left (x^{2} + 1\right ) + \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 0.71 \[ \frac {7 \arctan \relax (x )}{16}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}-\ln \relax (x )+\frac {\ln \left (x -1\right )}{8}+\frac {15 \ln \left (x^{2}+1\right )}{16}-\frac {\ln \left (x^{2}+x +1\right )}{2}+\frac {\frac {9}{2} x^{3}-3 x^{2}+\frac {11}{2} x -2}{8 \left (x^{2}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 77, normalized size = 0.75 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {9 \, x^{3} - 6 \, x^{2} + 11 \, x - 4}{16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} + \frac {7}{16} \, \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) + \frac {15}{16} \, \log \left (x^{2} + 1\right ) + \frac {1}{8} \, \log \left (x - 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 96, normalized size = 0.93 \[ \frac {\ln \left (x-1\right )}{8}-\ln \relax (x)+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {\frac {9\,x^3}{16}-\frac {3\,x^2}{8}+\frac {11\,x}{16}-\frac {1}{4}}{x^4+2\,x^2+1}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {15}{16}-\frac {7}{32}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {15}{16}+\frac {7}{32}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 88, normalized size = 0.85 \[ - \log {\relax (x )} + \frac {\log {\left (x - 1 \right )}}{8} + \frac {15 \log {\left (x^{2} + 1 \right )}}{16} - \frac {\log {\left (x^{2} + x + 1 \right )}}{2} + \frac {7 \operatorname {atan}{\relax (x )}}{16} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{3} + \frac {9 x^{3} - 6 x^{2} + 11 x - 4}{16 x^{4} + 32 x^{2} + 16} \]
Verification of antiderivative is not currently implemented for this CAS.
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