Optimal. Leaf size=86 \[ \frac {\sqrt {x^2-x+1} (x+1)}{x^2+x+1}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} (x+1)}{\sqrt {x^2-x+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} (1-x)}{\sqrt {x^2-x+1}}\right )}{\sqrt {6}} \]
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Rubi [A] time = 0.08, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1060, 1035, 1029, 206, 204} \[ \frac {\sqrt {x^2-x+1} (x+1)}{x^2+x+1}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} (x+1)}{\sqrt {x^2-x+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} (1-x)}{\sqrt {x^2-x+1}}\right )}{\sqrt {6}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 1029
Rule 1035
Rule 1060
Rubi steps
\begin {align*} \int \frac {1-x+3 x^2}{\sqrt {1-x+x^2} \left (1+x+x^2\right )^2} \, dx &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+\frac {1}{12} \int \frac {18-6 x}{\sqrt {1-x+x^2} \left (1+x+x^2\right )} \, dx\\ &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+\frac {1}{48} \int \frac {24+24 x}{\sqrt {1-x+x^2} \left (1+x+x^2\right )} \, dx-\frac {1}{48} \int \frac {-48+48 x}{\sqrt {1-x+x^2} \left (1+x+x^2\right )} \, dx\\ &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+24 \operatorname {Subst}\left (\int \frac {1}{1728-2 x^2} \, dx,x,\frac {-24+24 x}{\sqrt {1-x+x^2}}\right )+288 \operatorname {Subst}\left (\int \frac {1}{-20736-2 x^2} \, dx,x,\frac {-144-144 x}{\sqrt {1-x+x^2}}\right )\\ &=\frac {(1+x) \sqrt {1-x+x^2}}{1+x+x^2}+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} (1+x)}{\sqrt {1-x+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}} (1-x)}{\sqrt {1-x+x^2}}\right )}{\sqrt {6}}\\ \end {align*}
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Mathematica [C] time = 2.52, size = 961, normalized size = 11.17 \[ \frac {\sqrt {x^2-x+1} (x+1)}{x^2+x+1}+\frac {\left (7-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {3 \left (\left (-21-4 i \sqrt {3}\right ) x^4+14 \left (7-2 i \sqrt {3}\right ) x^3+\left (-103-36 i \sqrt {3}\right ) x^2+\left (94+32 i \sqrt {3}\right ) x-64 i \sqrt {3}-17\right )}{\left (84 i-113 \sqrt {3}\right ) x^4+2 \left (52 \sqrt {3-3 i \sqrt {3}} \sqrt {x^2-x+1}+21 \sqrt {3}+138 i\right ) x^3+\left (52 \sqrt {3-3 i \sqrt {3}} \sqrt {x^2-x+1}-59 \sqrt {3}-180 i\right ) x^2+2 \left (26 \sqrt {3-3 i \sqrt {3}} \sqrt {x^2-x+1}-69 \sqrt {3}+132 i\right ) x-52 \sqrt {3-3 i \sqrt {3}} \sqrt {x^2-x+1}+67 \sqrt {3}+96 i}\right )}{4 \sqrt {3-3 i \sqrt {3}}}-\frac {i \left (-7 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {3 \left (\left (-21+4 i \sqrt {3}\right ) x^4+14 \left (7+2 i \sqrt {3}\right ) x^3+\left (-103+36 i \sqrt {3}\right ) x^2+\left (94-32 i \sqrt {3}\right ) x+64 i \sqrt {3}-17\right )}{\left (84 i+113 \sqrt {3}\right ) x^4-2 \left (52 \sqrt {3+3 i \sqrt {3}} \sqrt {x^2-x+1}+21 \sqrt {3}-138 i\right ) x^3+\left (-52 \sqrt {3+3 i \sqrt {3}} \sqrt {x^2-x+1}+59 \sqrt {3}-180 i\right ) x^2+\left (-52 \sqrt {3+3 i \sqrt {3}} \sqrt {x^2-x+1}+138 \sqrt {3}+264 i\right ) x+52 \sqrt {3+3 i \sqrt {3}} \sqrt {x^2-x+1}-67 \sqrt {3}+96 i}\right )}{4 \sqrt {3+3 i \sqrt {3}}}-\frac {\left (7 i+\sqrt {3}\right ) \log \left (16 \left (x^2+x+1\right )^2\right )}{8 \sqrt {3-3 i \sqrt {3}}}-\frac {\left (-7 i+\sqrt {3}\right ) \log \left (16 \left (x^2+x+1\right )^2\right )}{8 \sqrt {3+3 i \sqrt {3}}}+\frac {\left (7 i+\sqrt {3}\right ) \log \left (\left (x^2+x+1\right ) \left (\left (11 i+4 \sqrt {3}\right ) x^2-\left (8 i \sqrt {1-i \sqrt {3}} \sqrt {x^2-x+1}+4 \sqrt {3}+17 i\right ) x+10 i \sqrt {1-i \sqrt {3}} \sqrt {x^2-x+1}+4 \sqrt {3}+11 i\right )\right )}{8 \sqrt {3-3 i \sqrt {3}}}+\frac {\left (-7 i+\sqrt {3}\right ) \log \left (\left (x^2+x+1\right ) \left (\left (-11 i+4 \sqrt {3}\right ) x^2+\left (8 i \sqrt {1+i \sqrt {3}} \sqrt {x^2-x+1}-4 \sqrt {3}+17 i\right ) x-10 i \sqrt {1+i \sqrt {3}} \sqrt {x^2-x+1}+4 \sqrt {3}-11 i\right )\right )}{8 \sqrt {3+3 i \sqrt {3}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 358, normalized size = 4.16 \[ -\frac {8 \, \sqrt {6} \sqrt {3} {\left (x^{2} + x + 1\right )} \arctan \left (\frac {2}{3} \, \sqrt {6} \sqrt {3} {\left (x - 1\right )} + \frac {2}{3} \, \sqrt {2 \, x^{2} - \sqrt {x^{2} - x + 1} {\left (2 \, x - \sqrt {6} + 1\right )} - \sqrt {6} {\left (x + 1\right )} + 4} {\left (\sqrt {6} \sqrt {3} + 3 \, \sqrt {3}\right )} - \frac {2}{3} \, \sqrt {x^{2} - x + 1} {\left (\sqrt {6} \sqrt {3} + 3 \, \sqrt {3}\right )} + \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 8 \, \sqrt {6} \sqrt {3} {\left (x^{2} + x + 1\right )} \arctan \left (\frac {2}{3} \, \sqrt {6} \sqrt {3} {\left (x - 1\right )} + \frac {2}{3} \, \sqrt {2 \, x^{2} - \sqrt {x^{2} - x + 1} {\left (2 \, x + \sqrt {6} + 1\right )} + \sqrt {6} {\left (x + 1\right )} + 4} {\left (\sqrt {6} \sqrt {3} - 3 \, \sqrt {3}\right )} - \frac {2}{3} \, \sqrt {x^{2} - x + 1} {\left (\sqrt {6} \sqrt {3} - 3 \, \sqrt {3}\right )} - \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \sqrt {6} {\left (x^{2} + x + 1\right )} \log \left (12168 \, x^{2} - 6084 \, \sqrt {x^{2} - x + 1} {\left (2 \, x + \sqrt {6} + 1\right )} + 6084 \, \sqrt {6} {\left (x + 1\right )} + 24336\right ) + \sqrt {6} {\left (x^{2} + x + 1\right )} \log \left (12168 \, x^{2} - 6084 \, \sqrt {x^{2} - x + 1} {\left (2 \, x - \sqrt {6} + 1\right )} - 6084 \, \sqrt {6} {\left (x + 1\right )} + 24336\right ) - 12 \, x^{2} - 12 \, \sqrt {x^{2} - x + 1} {\left (x + 1\right )} - 12 \, x - 12}{12 \, {\left (x^{2} + x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 304, normalized size = 3.53 \[ -\frac {1}{3} \, \sqrt {6} \sqrt {3} \arctan \left (-\frac {2 \, x + \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1}{\sqrt {3} + \sqrt {2}}\right ) + \frac {1}{3} \, \sqrt {6} \sqrt {3} \arctan \left (-\frac {2 \, x - \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1}{\sqrt {3} - \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \log \left (4 \, {\left (\sqrt {6} \sqrt {3} + 3 \, \sqrt {3}\right )}^{2} + 36 \, {\left (2 \, x + \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1\right )}^{2}\right ) - \frac {1}{12} \, \sqrt {6} \log \left (4 \, {\left (\sqrt {6} \sqrt {3} - 3 \, \sqrt {3}\right )}^{2} + 36 \, {\left (2 \, x - \sqrt {6} - 2 \, \sqrt {x^{2} - x + 1} + 1\right )}^{2}\right ) + \frac {{\left (x - \sqrt {x^{2} - x + 1}\right )}^{3} + 4 \, {\left (x - \sqrt {x^{2} - x + 1}\right )}^{2} - 10 \, x + 10 \, \sqrt {x^{2} - x + 1} + 5}{{\left (x - \sqrt {x^{2} - x + 1}\right )}^{4} + 2 \, {\left (x - \sqrt {x^{2} - x + 1}\right )}^{3} + {\left (x - \sqrt {x^{2} - x + 1}\right )}^{2} - 6 \, x + 6 \, \sqrt {x^{2} - x + 1} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 455, normalized size = 5.29 \[ -\frac {-\frac {6 \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}\, \left (x +1\right )^{2} \arctanh \left (\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}}{4}\right )}{\left (-x +1\right )^{2}}-2 \sqrt {6}\, \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \arctanh \left (\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}}{4}\right )+\frac {9 \sqrt {2}\, \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (x +1\right )^{2} \arctan \left (\frac {2 \sqrt {2}\, \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-x +1\right )}\right )}{\left (-x +1\right )^{2}}+3 \sqrt {2}\, \sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \arctan \left (\frac {2 \sqrt {2}\, \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-x +1\right )}\right )-\frac {12 \left (x +1\right )^{3}}{\left (-x +1\right )^{3}}-\frac {36 \left (x +1\right )}{-x +1}}{6 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}{\left (\frac {x +1}{-x +1}+1\right )^{2}}}\, \left (\frac {x +1}{-x +1}+1\right ) \left (\frac {3 \left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+1\right )}+\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-\sqrt {6}\, \arctanh \left (\frac {\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \sqrt {6}}{4}\right )+3 \sqrt {2}\, \arctan \left (\frac {2 \sqrt {2}\, \left (x +1\right )}{\sqrt {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}\, \left (-x +1\right )}\right )\right )}{2 \left (\frac {x +1}{-x +1}+1\right ) \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (-x +1\right )^{2}}+3}{\left (\frac {x +1}{-x +1}+1\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 \, x^{2} - x + 1}{{\left (x^{2} + x + 1\right )}^{2} \sqrt {x^{2} - x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {3\,x^2-x+1}{\sqrt {x^2-x+1}\,{\left (x^2+x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3 x^{2} - x + 1}{\sqrt {x^{2} - x + 1} \left (x^{2} + x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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