Integrand size = 41, antiderivative size = 99 \[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=\sqrt {\frac {1}{5}-\frac {3 i}{5}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}}{\sqrt {-1-3 i}}\right )+\sqrt {\frac {1}{5}+\frac {3 i}{5}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}}{\sqrt {-1+3 i}}\right ) \]
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\[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=\int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-2-x+2 x^2} \int \frac {\left (1+x^2\right ) \sqrt {-1+x+x^2}}{\sqrt {-2-x+2 x^2} \left (1-x^2+x^4\right )} \, dx}{\sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}} \\ & = \frac {\sqrt {-2-x+2 x^2} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {-1+x+x^2}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {-2-x+2 x^2}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {-1+x+x^2}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {-2-x+2 x^2}}\right ) \, dx}{\sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \frac {\sqrt {-1+x+x^2}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {-2-x+2 x^2}} \, dx}{\sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \frac {\sqrt {-1+x+x^2}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {-2-x+2 x^2}} \, dx}{\sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}} \sqrt {-1+x+x^2}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}}+\frac {\sqrt {1+i \sqrt {3}} \sqrt {-1+x+x^2}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}}\right ) \, dx}{\sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}} \sqrt {-1+x+x^2}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}}+\frac {\sqrt {1-i \sqrt {3}} \sqrt {-1+x+x^2}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}}\right ) \, dx}{\sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}} \\ & = -\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \frac {\sqrt {-1+x+x^2}}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}} \, dx}{2 \sqrt {1+i \sqrt {3}} \sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \frac {\sqrt {-1+x+x^2}}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}} \, dx}{2 \sqrt {1+i \sqrt {3}} \sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \frac {\sqrt {-1+x+x^2}}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}} \, dx}{2 \sqrt {1-i \sqrt {3}} \sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {-2-x+2 x^2}\right ) \int \frac {\sqrt {-1+x+x^2}}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {-2-x+2 x^2}} \, dx}{2 \sqrt {1-i \sqrt {3}} \sqrt {-1+x+x^2} \sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}}} \\ \end{align*}
Time = 3.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.38 \[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=\frac {\left (2+x-2 x^2\right )^{3/2} \left (\sqrt {-1+3 i} \arctan \left (\frac {\sqrt {\frac {1}{5}-\frac {3 i}{5}} \sqrt {2+x-2 x^2}}{\sqrt {-1+x+x^2}}\right )+\sqrt {-1-3 i} \arctan \left (\frac {\sqrt {\frac {1}{5}+\frac {3 i}{5}} \sqrt {2+x-2 x^2}}{\sqrt {-1+x+x^2}}\right )\right )}{\sqrt {5} \left (-\frac {2+x-2 x^2}{-1+x+x^2}\right )^{3/2} \left (-1+x+x^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(675\) vs. \(2(77)=154\).
Time = 30.75 (sec) , antiderivative size = 676, normalized size of antiderivative = 6.83
method | result | size |
default | \(-\frac {\left (2 x^{2}-x -2\right ) \left (\sqrt {10 \sqrt {5}\, \sqrt {2}+26}\, \sqrt {5}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {5}\, \sqrt {\frac {2 x^{4}+x^{3}-5 x^{2}-x +2}{\left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right )^{2}}}\, \sqrt {10 \sqrt {5}\, \sqrt {2}+26}\, \left (2 \sqrt {5}\, \sqrt {2}+11\right ) \left (-\sqrt {2}\, x^{2}+2 x \sqrt {5}+3 x \sqrt {2}+\sqrt {2}\right ) \left (13 \sqrt {5}\, \sqrt {2}-40\right ) \left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right )}{1620 \left (x^{2}+x -1\right ) \left (2 x^{2}-x -2\right )}\right )-10 \,\operatorname {arctanh}\left (\frac {2 \sqrt {\frac {2 x^{4}+x^{3}-5 x^{2}-x +2}{\left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right )^{2}}}}{\sqrt {\sqrt {5}\, \sqrt {2}+3}}\right ) \sqrt {5}\, \sqrt {\sqrt {5}\, \sqrt {2}+3}+14 \,\operatorname {arctanh}\left (\frac {2 \sqrt {\frac {2 x^{4}+x^{3}-5 x^{2}-x +2}{\left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right )^{2}}}}{\sqrt {\sqrt {5}\, \sqrt {2}+3}}\right ) \sqrt {2}\, \sqrt {\sqrt {5}\, \sqrt {2}+3}-2 \arctan \left (\frac {\sqrt {5}\, \sqrt {\frac {2 x^{4}+x^{3}-5 x^{2}-x +2}{\left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right )^{2}}}\, \sqrt {10 \sqrt {5}\, \sqrt {2}+26}\, \left (2 \sqrt {5}\, \sqrt {2}+11\right ) \left (-\sqrt {2}\, x^{2}+2 x \sqrt {5}+3 x \sqrt {2}+\sqrt {2}\right ) \left (13 \sqrt {5}\, \sqrt {2}-40\right ) \left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right )}{1620 \left (x^{2}+x -1\right ) \left (2 x^{2}-x -2\right )}\right ) \sqrt {10 \sqrt {5}\, \sqrt {2}+26}\right ) \left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right ) \sqrt {\frac {2 x^{4}+x^{3}-5 x^{2}-x +2}{\left (\sqrt {2}\, x^{2}+2 x \sqrt {5}-3 x \sqrt {2}-\sqrt {2}\right )^{2}}}}{2 \sqrt {\frac {2 x^{2}-x -2}{x^{2}+x -1}}\, \sqrt {\left (x^{2}+x -1\right ) \left (2 x^{2}-x -2\right )}\, \sqrt {\frac {2 x^{4}+x^{3}-5 x^{2}-x +2}{x^{2}}}\, \left (13 \sqrt {5}\, \sqrt {2}-40\right ) \left (\sqrt {5}\, \sqrt {2}+3\right ) x}\) | \(676\) |
trager | \(\text {Expression too large to display}\) | \(840\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (69) = 138\).
Time = 0.37 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.77 \[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {-3 i - 1} \log \left (\frac {\sqrt {5} \sqrt {-3 i - 1} {\left (\left (459 i + 187\right ) \, x^{4} - \left (232 i - 724\right ) \, x^{3} - \left (1183 i + 419\right ) \, x^{2} + \left (232 i - 724\right ) \, x + 459 i + 187\right )} - 20 \, {\left (\left (13 i + 84\right ) \, x^{4} - \left (71 i - 97\right ) \, x^{3} - \left (110 i + 155\right ) \, x^{2} + \left (71 i - 97\right ) \, x + 13 i + 84\right )} \sqrt {\frac {2 \, x^{2} - x - 2}{x^{2} + x - 1}}}{x^{4} - x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {3 i - 1} \log \left (\frac {\sqrt {5} \sqrt {3 i - 1} {\left (-\left (459 i - 187\right ) \, x^{4} + \left (232 i + 724\right ) \, x^{3} + \left (1183 i - 419\right ) \, x^{2} - \left (232 i + 724\right ) \, x - 459 i + 187\right )} - 20 \, {\left (-\left (13 i - 84\right ) \, x^{4} + \left (71 i + 97\right ) \, x^{3} + \left (110 i - 155\right ) \, x^{2} - \left (71 i + 97\right ) \, x - 13 i + 84\right )} \sqrt {\frac {2 \, x^{2} - x - 2}{x^{2} + x - 1}}}{x^{4} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {3 i - 1} \log \left (\frac {\sqrt {5} \sqrt {3 i - 1} {\left (\left (459 i - 187\right ) \, x^{4} - \left (232 i + 724\right ) \, x^{3} - \left (1183 i - 419\right ) \, x^{2} + \left (232 i + 724\right ) \, x + 459 i - 187\right )} - 20 \, {\left (-\left (13 i - 84\right ) \, x^{4} + \left (71 i + 97\right ) \, x^{3} + \left (110 i - 155\right ) \, x^{2} - \left (71 i + 97\right ) \, x - 13 i + 84\right )} \sqrt {\frac {2 \, x^{2} - x - 2}{x^{2} + x - 1}}}{x^{4} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-3 i - 1} \log \left (\frac {\sqrt {5} \sqrt {-3 i - 1} {\left (-\left (459 i + 187\right ) \, x^{4} + \left (232 i - 724\right ) \, x^{3} + \left (1183 i + 419\right ) \, x^{2} - \left (232 i - 724\right ) \, x - 459 i - 187\right )} - 20 \, {\left (\left (13 i + 84\right ) \, x^{4} - \left (71 i - 97\right ) \, x^{3} - \left (110 i + 155\right ) \, x^{2} + \left (71 i - 97\right ) \, x + 13 i + 84\right )} \sqrt {\frac {2 \, x^{2} - x - 2}{x^{2} + x - 1}}}{x^{4} - x^{2} + 1}\right ) \]
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Timed out. \[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} - x^{2} + 1\right )} \sqrt {\frac {2 \, x^{2} - x - 2}{x^{2} + x - 1}}} \,d x } \]
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\[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=\int { \frac {x^{2} + 1}{{\left (x^{4} - x^{2} + 1\right )} \sqrt {\frac {2 \, x^{2} - x - 2}{x^{2} + x - 1}}} \,d x } \]
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Timed out. \[ \int \frac {1+x^2}{\sqrt {\frac {-2-x+2 x^2}{-1+x+x^2}} \left (1-x^2+x^4\right )} \, dx=\int \frac {x^2+1}{\sqrt {-\frac {-2\,x^2+x+2}{x^2+x-1}}\,\left (x^4-x^2+1\right )} \,d x \]
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