Integrand size = 23, antiderivative size = 117 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=-\sqrt {\frac {1}{2} \left (2+\sqrt {5}\right )} \arctan \left (\frac {5+2 \sqrt {5}-\sqrt {5} x}{\sqrt {10 \left (2+\sqrt {5}\right )} \sqrt {-1+x+x^2}}\right )+\sqrt {\frac {1}{2} \left (-2+\sqrt {5}\right )} \text {arctanh}\left (\frac {5-2 \sqrt {5}+\sqrt {5} x}{\sqrt {10 \left (-2+\sqrt {5}\right )} \sqrt {-1+x+x^2}}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1050, 1044, 213, 209} \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=\sqrt {\frac {1}{2} \left (\sqrt {5}-2\right )} \text {arctanh}\left (\frac {\sqrt {5} x-2 \sqrt {5}+5}{\sqrt {10 \left (\sqrt {5}-2\right )} \sqrt {x^2+x-1}}\right )-\sqrt {\frac {1}{2} \left (2+\sqrt {5}\right )} \arctan \left (\frac {-\sqrt {5} x+2 \sqrt {5}+5}{\sqrt {10 \left (2+\sqrt {5}\right )} \sqrt {x^2+x-1}}\right ) \]
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Rule 209
Rule 213
Rule 1044
Rule 1050
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {-\sqrt {5}+\left (-5-2 \sqrt {5}\right ) x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx}{2 \sqrt {5}}+\frac {\int \frac {\sqrt {5}+\left (-5+2 \sqrt {5}\right ) x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx}{2 \sqrt {5}} \\ & = -\left (\left (-5+2 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{10 \left (2-\sqrt {5}\right )+x^2} \, dx,x,\frac {-5+2 \sqrt {5}-\sqrt {5} x}{\sqrt {-1+x+x^2}}\right )\right )+\left (5+2 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{10 \left (2+\sqrt {5}\right )+x^2} \, dx,x,\frac {-5-2 \sqrt {5}+\sqrt {5} x}{\sqrt {-1+x+x^2}}\right ) \\ & = -\sqrt {\frac {1}{2} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {5+2 \sqrt {5}-\sqrt {5} x}{\sqrt {10 \left (2+\sqrt {5}\right )} \sqrt {-1+x+x^2}}\right )+\sqrt {\frac {1}{2} \left (-2+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {5-2 \sqrt {5}+\sqrt {5} x}{\sqrt {10 \left (-2+\sqrt {5}\right )} \sqrt {-1+x+x^2}}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=\frac {1}{2} \text {RootSum}\left [2-4 \text {$\#$1}+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {3 \log \left (-x+\sqrt {-1+x+x^2}-\text {$\#$1}\right )-2 \log \left (-x+\sqrt {-1+x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (-x+\sqrt {-1+x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-1+3 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.08 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.10
method | result | size |
trager | \(\operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{2}+\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{2}+\textit {\_Z}^{2}+1\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{2} x +2 \operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{2}+\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{2}+x -1}+\operatorname {RootOf}\left (\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{2}+\textit {\_Z}^{2}+1\right )}{4 x \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{2}+2 x -1}\right )-\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{3} x +2 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right ) x +\sqrt {x^{2}+x -1}-\operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )}{4 x \operatorname {RootOf}\left (16 \textit {\_Z}^{4}+16 \textit {\_Z}^{2}+5\right )^{2}+2 x +1}\right )\) | \(246\) |
default | \(\frac {\sqrt {\frac {10 \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}-\frac {5 \sqrt {5}\, \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+10+5 \sqrt {5}}\, \sqrt {5}\, \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {\left (-2+\sqrt {5}\right ) \left (-\frac {\left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+4 \sqrt {5}+9\right )}\, \sqrt {20+10 \sqrt {5}}\, \left (\frac {\sqrt {5}\, \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+\frac {2 \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}-\sqrt {5}+2\right ) \left (-\sqrt {5}-2+x \right ) \left (-2+\sqrt {5}\right )}{5 \left (-\sqrt {5}+2-x \right ) \left (\frac {\left (-\sqrt {5}-2+x \right )^{4}}{\left (-\sqrt {5}+2-x \right )^{4}}-\frac {18 \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+1\right )}\right ) \sqrt {5}+\operatorname {arctanh}\left (\frac {\sqrt {\frac {10 \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}-\frac {5 \sqrt {5}\, \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+10+5 \sqrt {5}}}{\sqrt {20+10 \sqrt {5}}}\right )+2 \arctan \left (\frac {\sqrt {5}\, \sqrt {\left (-2+\sqrt {5}\right ) \left (-\frac {\left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+4 \sqrt {5}+9\right )}\, \sqrt {20+10 \sqrt {5}}\, \left (\frac {\sqrt {5}\, \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+\frac {2 \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}-\sqrt {5}+2\right ) \left (-\sqrt {5}-2+x \right ) \left (-2+\sqrt {5}\right )}{5 \left (-\sqrt {5}+2-x \right ) \left (\frac {\left (-\sqrt {5}-2+x \right )^{4}}{\left (-\sqrt {5}+2-x \right )^{4}}-\frac {18 \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}+1\right )}\right )\right )}{\sqrt {-\frac {5 \left (\frac {\sqrt {5}\, \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}-\frac {2 \left (-\sqrt {5}-2+x \right )^{2}}{\left (-\sqrt {5}+2-x \right )^{2}}-\sqrt {5}-2\right )}{\left (1+\frac {-\sqrt {5}-2+x}{-\sqrt {5}+2-x}\right )^{2}}}\, \left (1+\frac {-\sqrt {5}-2+x}{-\sqrt {5}+2-x}\right ) \sqrt {20+10 \sqrt {5}}}\) | \(637\) |
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Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=\frac {1}{2} \, \sqrt {i - 2} \log \left (-x + \sqrt {i - 2} + \sqrt {x^{2} + x - 1} + i\right ) - \frac {1}{2} \, \sqrt {i - 2} \log \left (-x - \sqrt {i - 2} + \sqrt {x^{2} + x - 1} + i\right ) + \frac {1}{2} \, \sqrt {-i - 2} \log \left (-x + \sqrt {-i - 2} + \sqrt {x^{2} + x - 1} - i\right ) - \frac {1}{2} \, \sqrt {-i - 2} \log \left (-x - \sqrt {-i - 2} + \sqrt {x^{2} + x - 1} - i\right ) \]
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\[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=\int \frac {2 x + 1}{\left (x^{2} + 1\right ) \sqrt {x^{2} + x - 1}}\, dx \]
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\[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=\int { \frac {2 \, x + 1}{\sqrt {x^{2} + x - 1} {\left (x^{2} + 1\right )}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (86) = 172\).
Time = 0.34 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.91 \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=\frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 4} \log \left (16 \, {\left (15 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 33 \, x + 5 \, \sqrt {5} - 33 \, \sqrt {x^{2} + x - 1} + 2 \, \sqrt {5 \, \sqrt {5} + 11} + 11\right )}^{2} + 16 \, {\left (5 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 11 \, x - 5 \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} - 15 \, \sqrt {5} - 11 \, \sqrt {x^{2} + x - 1} - 11 \, \sqrt {5 \, \sqrt {5} + 11} - 33\right )}^{2}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 4} \log \left (16 \, {\left (15 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 33 \, x + 5 \, \sqrt {5} - 33 \, \sqrt {x^{2} + x - 1} - 2 \, \sqrt {5 \, \sqrt {5} + 11} + 11\right )}^{2} + 16 \, {\left (5 \, \sqrt {5} {\left (x - \sqrt {x^{2} + x - 1}\right )} + 11 \, x + 5 \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} - 15 \, \sqrt {5} - 11 \, \sqrt {x^{2} + x - 1} + 11 \, \sqrt {5 \, \sqrt {5} + 11} - 33\right )}^{2}\right ) + \frac {\sqrt {2 \, \sqrt {5} - 4} {\left (\arctan \left (3\right ) + \arctan \left (\frac {1}{10} \, {\left (x - \sqrt {x^{2} + x - 1}\right )} {\left (\sqrt {5} \sqrt {5 \, \sqrt {5} + 11} + 4 \, \sqrt {5} - 5 \, \sqrt {5 \, \sqrt {5} + 11}\right )} - \frac {7}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} + \frac {1}{5} \, \sqrt {5} + \frac {3}{2} \, \sqrt {5 \, \sqrt {5} + 11}\right )\right )}}{2 \, {\left (\sqrt {5} - 2\right )}} - \frac {\sqrt {2 \, \sqrt {5} - 4} {\left (\arctan \left (3\right ) + \arctan \left (-\frac {1}{10} \, {\left (x - \sqrt {x^{2} + x - 1}\right )} {\left (\sqrt {5} \sqrt {5 \, \sqrt {5} + 11} - 4 \, \sqrt {5} - 5 \, \sqrt {5 \, \sqrt {5} + 11}\right )} + \frac {7}{10} \, \sqrt {5} \sqrt {5 \, \sqrt {5} + 11} + \frac {1}{5} \, \sqrt {5} - \frac {3}{2} \, \sqrt {5 \, \sqrt {5} + 11}\right )\right )}}{2 \, {\left (\sqrt {5} - 2\right )}} \]
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Timed out. \[ \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {-1+x+x^2}} \, dx=\int \frac {2\,x+1}{\left (x^2+1\right )\,\sqrt {x^2+x-1}} \,d x \]
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