Integrand size = 40, antiderivative size = 81 \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=-\frac {x \left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6}-\frac {3}{2} \arctan \left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right )-\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right ) \]
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\[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \left (-2+x^3\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2}+\frac {2 \left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6}\right ) \, dx \\ & = 2 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6} \, dx+3 \int \frac {\left (-2+x^3\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2} \, dx \\ & = 2 \int \left (\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^3\right )}+\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx+3 \int \left (-\frac {2 \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2}+\frac {x^3 \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2}\right ) \, dx \\ & = 3 \int \frac {x^3 \left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2} \, dx-6 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1-x^3+x^6\right )^2} \, dx+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1+i \sqrt {3}-2 x^3} \, dx}{\sqrt {3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}} \\ & = 3 \int \left (-\frac {2 \left (1+i \sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}-2 x^3\right )^2}+\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x^3\right )}-\frac {2 \left (1-i \sqrt {3}\right ) \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}+2 x^3\right )^2}+\frac {2 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx-6 \int \left (-\frac {4 \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}-2 x^3\right )^2}+\frac {4 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x^3\right )}-\frac {4 \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}+2 x^3\right )^2}+\frac {4 i \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx+\frac {(4 i) \int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}}+\frac {(4 i) \int \left (\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{3 \left (1+i \sqrt {3}\right )^{2/3} \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx}{\sqrt {3}} \\ & = 8 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1+i \sqrt {3}-2 x^3\right )^2} \, dx+8 \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-1+i \sqrt {3}+2 x^3\right )^2} \, dx+\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1+i \sqrt {3}-2 x^3} \, dx}{\sqrt {3}}+\frac {(2 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}-\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{1+i \sqrt {3}-2 x^3} \, dx}{\sqrt {3}}-\frac {(8 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{-1+i \sqrt {3}+2 x^3} \, dx}{\sqrt {3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1-i \sqrt {3}\right )^{2/3}}-\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (-1+i \sqrt {3}+2 x^3\right )^2} \, dx+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}+\frac {(4 i) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx}{3 \sqrt {3} \left (1+i \sqrt {3}\right )^{2/3}}-\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4}}{\left (1+i \sqrt {3}-2 x^3\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=-\frac {x \left (1-x^3+x^4+x^6\right )^{3/4}}{1-x^3+x^6}-\frac {3}{2} \arctan \left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right )-\frac {3}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1-x^3+x^4+x^6}}\right ) \]
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Time = 10.62 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.73
method | result | size |
pseudoelliptic | \(\frac {\left (3 x^{6}-3 x^{3}+3\right ) \ln \left (\frac {\left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}}-x}{x}\right )+\left (-3 x^{6}+3 x^{3}-3\right ) \ln \left (\frac {\left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}}+x}{x}\right )+\left (6 x^{6}-6 x^{3}+6\right ) \arctan \left (\frac {\left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}}}{x}\right )-4 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x}{4 x^{6}-4 x^{3}+4}\) | \(140\) |
trager | \(-\frac {x \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}}}{x^{6}-x^{3}+1}-\frac {3 \ln \left (-\frac {x^{6}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x +2 \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}+2 x^{4}-x^{3}+1}{x^{6}-x^{3}+1}\right )}{4}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{3}+1}\right )}{4}\) | \(247\) |
risch | \(-\frac {x \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}}}{x^{6}-x^{3}+1}+\frac {3 \ln \left (\frac {-x^{6}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}-2 x^{4}+x^{3}-1}{x^{6}-x^{3}+1}\right )}{4}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}+x^{4}-x^{3}+1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{6}+x^{4}-x^{3}+1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{3}+1}\right )}{4}\) | \(248\) |
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Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (71) = 142\).
Time = 29.65 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.42 \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=-\frac {3 \, {\left (x^{6} - x^{3} + 1\right )} \arctan \left (\frac {2 \, {\left ({\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{6} - x^{3} + 1}\right ) - 3 \, {\left (x^{6} - x^{3} + 1\right )} \log \left (\frac {x^{6} + 2 \, x^{4} - 2 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {1}{4}} x^{3} - x^{3} + 2 \, \sqrt {x^{6} + x^{4} - x^{3} + 1} x^{2} - 2 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{6} - x^{3} + 1}\right ) + 4 \, {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}} x}{4 \, {\left (x^{6} - x^{3} + 1\right )}} \]
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Timed out. \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 4\right )} {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}}}{{\left (x^{6} - x^{3} + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 4\right )} {\left (x^{6} + x^{4} - x^{3} + 1\right )}^{\frac {3}{4}}}{{\left (x^{6} - x^{3} + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (1-x^3+x^4+x^6\right )^{3/4} \left (-4+x^3+2 x^6\right )}{\left (1-x^3+x^6\right )^2} \, dx=\int \frac {\left (2\,x^6+x^3-4\right )\,{\left (x^6+x^4-x^3+1\right )}^{3/4}}{{\left (x^6-x^3+1\right )}^2} \,d x \]
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