Optimal. Leaf size=114 \[ \frac {\text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2}}-\sqrt {2} x-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{(1+x) \sqrt [4]{1+x^4}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {2}+\sqrt {2} x\right ) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 1.05, antiderivative size = 797, normalized size of antiderivative = 6.99, number of steps
used = 46, number of rules used = 19, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {6860, 2184,
1254, 385, 218, 214, 211, 524, 1262, 760, 408, 504, 1227, 551, 455, 65, 304, 209, 212}
\begin {gather*} -\frac {1}{6} \left (1+i \sqrt {3}\right ) F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right ) x^3-\frac {1}{6} \left (1-i \sqrt {3}\right ) F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right ) x^3-\frac {\text {ArcTan}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x^4+1}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1-i \sqrt {3}}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x^4+1}}{\sqrt [4]{1+i \sqrt {3}}}\right )+\frac {i \sqrt {-x^4} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\left .\text {ArcSin}\left (\sqrt [4]{x^4+1}\right )\right |-1\right )}{2 x^2}-\frac {i \sqrt {-x^4} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\left .\text {ArcSin}\left (\sqrt [4]{x^4+1}\right )\right |-1\right )}{2 x^2}-\frac {i \sqrt {-x^4} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\left .\text {ArcSin}\left (\sqrt [4]{x^4+1}\right )\right |-1\right )}{2 x^2}+\frac {i \sqrt {-x^4} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\left .\text {ArcSin}\left (\sqrt [4]{x^4+1}\right )\right |-1\right )}{2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 65
Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 304
Rule 385
Rule 408
Rule 455
Rule 504
Rule 524
Rule 551
Rule 760
Rule 1227
Rule 1254
Rule 1262
Rule 2184
Rule 6860
Rubi steps
\begin {align*} \int \frac {-1+x}{\left (1+x+x^2\right ) \sqrt [4]{1+x^4}} \, dx &=\int \left (\frac {1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}}+\frac {1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}}\right ) \, dx\\ &=\left (1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}} \, dx+\left (1+i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^4}} \, dx\\ &=\left (1-i \sqrt {3}\right ) \int \left (\frac {-i+\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^2\right ) \sqrt [4]{1+x^4}}+\frac {x}{\left (1-i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}}\right ) \, dx+\left (1+i \sqrt {3}\right ) \int \left (\frac {i+\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^2\right ) \sqrt [4]{1+x^4}}+\frac {x}{\left (1+i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}}\right ) \, dx\\ &=2 i \int \frac {1}{\left (-i+\sqrt {3}-2 i x^2\right ) \sqrt [4]{1+x^4}} \, dx-2 i \int \frac {1}{\left (i+\sqrt {3}+2 i x^2\right ) \sqrt [4]{1+x^4}} \, dx+\left (1-i \sqrt {3}\right ) \int \frac {x}{\left (1-i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}} \, dx+\left (1+i \sqrt {3}\right ) \int \frac {x}{\left (1+i \sqrt {3}+2 x^2\right ) \sqrt [4]{1+x^4}} \, dx\\ &=2 i \int \left (\frac {1+i \sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^4\right ) \sqrt [4]{1+x^4}}+\frac {i x^2}{\sqrt [4]{1+x^4} \left (1-i \sqrt {3}+2 x^4\right )}\right ) \, dx-2 i \int \left (\frac {1-i \sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^4\right ) \sqrt [4]{1+x^4}}-\frac {i x^2}{\sqrt [4]{1+x^4} \left (1+i \sqrt {3}+2 x^4\right )}\right ) \, dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )+\frac {1}{2} \left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )\\ &=-\left (2 \int \frac {x^2}{\sqrt [4]{1+x^4} \left (1-i \sqrt {3}+2 x^4\right )} \, dx\right )-2 \int \frac {x^2}{\sqrt [4]{1+x^4} \left (1+i \sqrt {3}+2 x^4\right )} \, dx+\left (i-\sqrt {3}\right ) \int \frac {1}{\left (i+\sqrt {3}+2 i x^4\right ) \sqrt [4]{1+x^4}} \, dx+\left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )+\left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )+\left (-1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {x}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )+\frac {1}{2} \left (1+i \sqrt {3}\right )^2 \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt [4]{1+x^2}} \, dx,x,x^2\right )-\left (i+\sqrt {3}\right ) \int \frac {1}{\left (-i+\sqrt {3}-2 i x^4\right ) \sqrt [4]{1+x^4}} \, dx\\ &=-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (1+i \sqrt {3}\right )}+\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{i+\sqrt {3}-\left (-i+\sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt [4]{1+x}} \, dx,x,x^4\right )+\frac {1}{2} \left (-1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt [4]{1+x}} \, dx,x,x^4\right )-\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-i+\sqrt {3}-\left (i+\sqrt {3}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {\left (2 \left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (4+\left (1-i \sqrt {3}\right )^2-4 x^4\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{x^2}+\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (4+\left (1+i \sqrt {3}\right )^2-4 x^4\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{x^2}\\ &=-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (1+i \sqrt {3}\right )}-\left (2 \left (1-i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {x^2}{4+\left (1-i \sqrt {3}\right )^2-4 x^4} \, dx,x,\sqrt [4]{1+x^4}\right )-\left (2 \left (1+i \sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {x^2}{4+\left (1+i \sqrt {3}\right )^2-4 x^4} \, dx,x,\sqrt [4]{1+x^4}\right )+\frac {\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {i+\sqrt {3}}}+\frac {\left (i-\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {i+\sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}-\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {-i+\sqrt {3}}}-\frac {\left (i+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}+\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {-i+\sqrt {3}}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}+\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2}-\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2}\\ &=-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (1+i \sqrt {3}\right )}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )-\frac {\left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}+\frac {\left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}-\frac {\left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}+\frac {\left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{2 \sqrt {2} x^2}+\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2}-\frac {\left (\left (1+i \sqrt {3}\right )^2 \sqrt {-x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x^2\right )} \, dx,x,\sqrt [4]{1+x^4}\right )}{4 \sqrt {2} x^2}\\ &=-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1-i \sqrt {3}}\right )}{3 \left (1-i \sqrt {3}\right )}-\frac {2 x^3 F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};-x^4,-\frac {2 x^4}{1+i \sqrt {3}}\right )}{3 \left (1+i \sqrt {3}\right )}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1-i \sqrt {3}}}\right )+\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1+i \sqrt {3}}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{1+x^4}}\right )}{2 \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4}}-\frac {1}{2} \left (-\frac {i-\sqrt {3}}{i+\sqrt {3}}\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1-i \sqrt {3}}}\right )-\frac {1}{2} \left (\frac {1}{2} \left (1+i \sqrt {3}\right )\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{1+x^4}}{\sqrt [4]{1+i \sqrt {3}}}\right )+\frac {\left (1+i \sqrt {3}\right ) \sqrt {-x^4} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\left .\sin ^{-1}\left (\sqrt [4]{1+x^4}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {1-i \sqrt {3}} x^2}-\frac {\left (1+i \sqrt {3}\right )^{3/2} \sqrt {-x^4} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\left .\sin ^{-1}\left (\sqrt [4]{1+x^4}\right )\right |-1\right )}{4 \sqrt {2} x^2}-\frac {\left (1+i \sqrt {3}\right ) \sqrt {-x^4} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\left .\sin ^{-1}\left (\sqrt [4]{1+x^4}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {1-i \sqrt {3}} x^2}+\frac {\left (1+i \sqrt {3}\right )^{3/2} \sqrt {-x^4} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\left .\sin ^{-1}\left (\sqrt [4]{1+x^4}\right )\right |-1\right )}{4 \sqrt {2} x^2}\\ \end {align*}
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Mathematica [A]
time = 1.39, size = 87, normalized size = 0.76 \begin {gather*} \frac {\text {ArcTan}\left (\frac {-1-2 x-x^2+\sqrt {1+x^4}}{\sqrt {2} (1+x) \sqrt [4]{1+x^4}}\right )-\tanh ^{-1}\left (\frac {\sqrt {2} (1+x) \sqrt [4]{1+x^4}}{1+2 x+x^2+\sqrt {1+x^4}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 5.52, size = 396, normalized size = 3.47
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-\left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-3 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -3 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x +2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x -\left (x^{4}+1\right )^{\frac {3}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )-\left (x^{4}+1\right )^{\frac {3}{4}}}{\left (x^{2}+x +1\right )^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -\left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\sqrt {x^{4}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-3 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-3 \left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x +2 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+\left (x^{4}+1\right )^{\frac {3}{4}} x -\left (x^{4}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+3 \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+\left (x^{4}+1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) x}{\left (x^{2}+x +1\right )^{2}}\right )}{2}\) | \(396\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1098 vs.
\(2 (90) = 180\).
time = 4.68, size = 1098, normalized size = 9.63 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, x^{7} + 10 \, x^{6} + 16 \, x^{5} + 19 \, x^{4} + 16 \, x^{3} + \sqrt {2} {\left (x^{5} + 7 \, x^{4} + 15 \, x^{3} + 15 \, x^{2} + 7 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 10 \, x^{2} - \sqrt {2} {\left (x^{7} + x^{6} - 6 \, x^{5} - 16 \, x^{4} - 16 \, x^{3} - 6 \, x^{2} + x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 2 \, {\left (x^{6} + 4 \, x^{5} + 8 \, x^{4} + 10 \, x^{3} + 8 \, x^{2} + 4 \, x + 1\right )} \sqrt {x^{4} + 1} - {\left (\sqrt {2} {\left (x^{6} + 8 \, x^{5} + 22 \, x^{4} + 30 \, x^{3} + 22 \, x^{2} + 8 \, x + 1\right )} \sqrt {x^{4} + 1} + 4 \, {\left (x^{5} + 5 \, x^{4} + 10 \, x^{3} + 10 \, x^{2} + 5 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + \sqrt {2} {\left (2 \, x^{8} + 10 \, x^{7} + 19 \, x^{6} + 22 \, x^{5} + 21 \, x^{4} + 22 \, x^{3} + 19 \, x^{2} + 10 \, x + 2\right )} + 2 \, {\left (x^{7} + 5 \, x^{6} + 12 \, x^{5} + 18 \, x^{4} + 18 \, x^{3} + 12 \, x^{2} + 5 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, x^{3} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + 3 \, x^{2} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 2 \, x + 1}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}} + 4 \, x + 1}{3 \, x^{8} + 12 \, x^{7} + 14 \, x^{6} - 11 \, x^{4} + 14 \, x^{2} + 12 \, x + 3}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, x^{7} + 10 \, x^{6} + 16 \, x^{5} + 19 \, x^{4} + 16 \, x^{3} - \sqrt {2} {\left (x^{5} + 7 \, x^{4} + 15 \, x^{3} + 15 \, x^{2} + 7 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + 10 \, x^{2} + \sqrt {2} {\left (x^{7} + x^{6} - 6 \, x^{5} - 16 \, x^{4} - 16 \, x^{3} - 6 \, x^{2} + x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}} + 2 \, {\left (x^{6} + 4 \, x^{5} + 8 \, x^{4} + 10 \, x^{3} + 8 \, x^{2} + 4 \, x + 1\right )} \sqrt {x^{4} + 1} + {\left (\sqrt {2} {\left (x^{6} + 8 \, x^{5} + 22 \, x^{4} + 30 \, x^{3} + 22 \, x^{2} + 8 \, x + 1\right )} \sqrt {x^{4} + 1} - 4 \, {\left (x^{5} + 5 \, x^{4} + 10 \, x^{3} + 10 \, x^{2} + 5 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {3}{4}} + \sqrt {2} {\left (2 \, x^{8} + 10 \, x^{7} + 19 \, x^{6} + 22 \, x^{5} + 21 \, x^{4} + 22 \, x^{3} + 19 \, x^{2} + 10 \, x + 2\right )} - 2 \, {\left (x^{7} + 5 \, x^{6} + 12 \, x^{5} + 18 \, x^{4} + 18 \, x^{3} + 12 \, x^{2} + 5 \, x + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, x^{3} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + 3 \, x^{2} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 2 \, x + 1}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}} + 4 \, x + 1}{3 \, x^{8} + 12 \, x^{7} + 14 \, x^{6} - 11 \, x^{4} + 14 \, x^{2} + 12 \, x + 3}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} + 2 \, x^{3} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + 3 \, x^{2} + \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 2 \, x + 1\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} + 2 \, x^{3} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )} + 3 \, x^{2} - \sqrt {2} {\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (x^{2} + 2 \, x + 1\right )} + 2 \, x + 1\right )}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) \end {gather*}
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 \begin {gather*} \int \frac {x - 1}{\sqrt [4]{x^{4} + 1} \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x-1}{{\left (x^4+1\right )}^{1/4}\,\left (x^2+x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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