Integrand size = 29, antiderivative size = 177 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {4 \left (-5+x+4 x^2\right ) \sqrt [4]{-x^3+x^4}}{45 x^3}+\sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.79 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.44, number of steps used = 36, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2081, 1600, 6865, 6874, 277, 270, 508, 304, 209, 212, 6857, 1543} \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {(1-i)^{5/4} \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1-i)^{3/4} \sqrt [4]{x-1} x^{3/4}}-\frac {(1+i)^{5/4} \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1+i)^{3/4} \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{2} \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {(1-i)^{5/4} \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1-i)^{3/4} \sqrt [4]{x-1} x^{3/4}}+\frac {(1+i)^{5/4} \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{(1+i)^{3/4} \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {16 \sqrt [4]{x^4-x^3}}{45 x}+\frac {4 \sqrt [4]{x^4-x^3}}{9 x^3}-\frac {4 \sqrt [4]{x^4-x^3}}{45 x^2} \]
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Rule 209
Rule 212
Rule 270
Rule 277
Rule 304
Rule 508
Rule 1543
Rule 1600
Rule 2081
Rule 6857
Rule 6865
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} \left (1+x^4\right )}{x^{13/4} \left (-1+x^4\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\sqrt [4]{-x^3+x^4} \int \frac {1+x^4}{(-1+x)^{3/4} x^{13/4} \left (1+x+x^2+x^3\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1+x^{16}}{x^{10} \left (-1+x^4\right )^{3/4} \left (1+x^4+x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{x^{10} \left (-1+x^4\right )^{3/4}}-\frac {1}{x^6 \left (-1+x^4\right )^{3/4}}-\frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^4\right )}+\frac {x^2 \left (1+x^4\right )}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (1+x^4\right )}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{5 x^2}-\frac {\left (16 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{5 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (32 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{9 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )}+\frac {x^6}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{5 x}+\frac {\left (128 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{45 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^6}{\left (-1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\sqrt {2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {i x^2}{2 \left (-1+x^4\right )^{3/4} \left (-i+x^4\right )}+\frac {i x^2}{2 \left (-1+x^4\right )^{3/4} \left (i+x^4\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (-i+x^4\right )}+\frac {x^2}{2 \left (-1+x^4\right )^{3/4} \left (i+x^4\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (i+x^4\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{i-(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-i-(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{i-(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-i-(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (i (1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (i (1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left ((1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((1-i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (i (1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (i (1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left ((1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left ((1+i)^{3/2} \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4}}{9 x^3}-\frac {4 \sqrt [4]{-x^3+x^4}}{45 x^2}-\frac {16 \sqrt [4]{-x^3+x^4}}{45 x}-\frac {\sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1-i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}-\frac {(1-i)^{5/4} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1+i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}-\frac {(1+i)^{5/4} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1-i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}+\frac {(1-i)^{5/4} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{(1+i)^{3/4} \sqrt [4]{-1+x} x^{3/4}}+\frac {(1+i)^{5/4} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{2} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.10 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\frac {(-1+x)^{3/4} \left (-8 \left (4 \sqrt [4]{-1+x} \left (-5+x+4 x^2\right )-45 \sqrt [4]{2} x^{9/4} \arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )+45 \sqrt [4]{2} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )+45 x^{9/4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{360 \left ((-1+x) x^3\right )^{3/4}} \]
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Time = 51.90 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {-45 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-1\right )}\right ) x^{3}-45 \,2^{\frac {1}{4}} x^{3} \ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right )-90 \,2^{\frac {1}{4}} x^{3} \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right )-32 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x +\frac {5}{4}\right ) \left (-1+x \right )}{90 x^{3}}\) | \(151\) |
trager | \(\text {Expression too large to display}\) | \(4008\) |
risch | \(\text {Expression too large to display}\) | \(7766\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.26 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.50 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=-\frac {45 \cdot 8^{\frac {3}{4}} x^{3} \log \left (\frac {8^{\frac {3}{4}} x + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 45 \cdot 8^{\frac {3}{4}} x^{3} \log \left (-\frac {8^{\frac {3}{4}} x - 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 45 i \cdot 8^{\frac {3}{4}} x^{3} \log \left (\frac {i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 45 i \cdot 8^{\frac {3}{4}} x^{3} \log \left (\frac {-i \cdot 8^{\frac {3}{4}} x + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 180 \, x^{3} \sqrt {-\sqrt {i + 1}} \log \left (\frac {x \sqrt {-\sqrt {i + 1}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 180 \, x^{3} \sqrt {-\sqrt {i + 1}} \log \left (-\frac {x \sqrt {-\sqrt {i + 1}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 180 \, x^{3} \sqrt {-\sqrt {-i + 1}} \log \left (\frac {x \sqrt {-\sqrt {-i + 1}} + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 180 \, x^{3} \sqrt {-\sqrt {-i + 1}} \log \left (-\frac {x \sqrt {-\sqrt {-i + 1}} - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 180 \, \left (i + 1\right )^{\frac {1}{4}} x^{3} \log \left (\frac {\left (i + 1\right )^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 180 \, \left (i + 1\right )^{\frac {1}{4}} x^{3} \log \left (-\frac {\left (i + 1\right )^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 180 \, \left (-i + 1\right )^{\frac {1}{4}} x^{3} \log \left (\frac {\left (-i + 1\right )^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 180 \, \left (-i + 1\right )^{\frac {1}{4}} x^{3} \log \left (-\frac {\left (-i + 1\right )^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 32 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x^{2} + x - 5\right )}}{360 \, x^{3}} \]
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Not integrable
Time = 1.89 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.18 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x^{4} + 1\right )}{x^{4} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.16 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{{\left (x^{4} - 1\right )} x^{4}} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.54 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\frac {4}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {4}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \left (-\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} - \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \left (-\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} + \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + i \, \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} - \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - i \, \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} + \left (65536 i - 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (-i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} + \left (65536 i + 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right )^{\frac {1}{4}} \log \left (-i \, \left (-73786976294838206464 i - 73786976294838206464\right )^{\frac {1}{4}} - \left (65536 i + 65536\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 2 i \, \left (-\frac {1}{256} i + \frac {1}{256}\right )^{\frac {1}{4}} \log \left (-i \, \left (85070591730234615865843651857942052864 i - 85070591730234615865843651857942052864\right )^{\frac {1}{4}} + \left (2147483648 i + 2147483648\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 2 i \, \left (-\frac {1}{256} i + \frac {1}{256}\right )^{\frac {1}{4}} \log \left (-i \, \left (85070591730234615865843651857942052864 i - 85070591730234615865843651857942052864\right )^{\frac {1}{4}} - \left (2147483648 i + 2147483648\right ) \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.16 \[ \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^3+x^4}}{x^4 \left (-1+x^4\right )} \, dx=\int \frac {\left (x^4+1\right )\,{\left (x^4-x^3\right )}^{1/4}}{x^4\,\left (x^4-1\right )} \,d x \]
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