Optimal. Leaf size=239 \[ \frac {\text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3} x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2} \sqrt [4]{3}}+\frac {\sqrt [4]{3} x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2} \sqrt [4]{3}} \]
[Out]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.82, antiderivative size = 853, normalized size of antiderivative = 3.57, number of steps
used = 153, number of rules used = 22, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.880, Rules used = {1600, 6860,
1743, 1223, 1202, 228, 1199, 1229, 1471, 554, 259, 552, 551, 1262, 749, 858, 223, 212, 739, 210,
415, 418} \begin {gather*} -\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \text {ArcTan}\left (\frac {\left (1-i \sqrt {3}\right ) x^2+2}{\sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \text {ArcTan}\left (\frac {4-\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )+\frac {\left (i+\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {x^4-1}}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {x^4-1}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (-\frac {4}{\left (i-\sqrt {3}\right )^2};\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {2}{1-i \sqrt {3}};\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {2}{1+i \sqrt {3}};\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (-\frac {4}{\left (i+\sqrt {3}\right )^2};\text {ArcSin}(x)|-1\right )}{4 \sqrt {x^4-1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 210
Rule 212
Rule 223
Rule 228
Rule 259
Rule 415
Rule 418
Rule 551
Rule 552
Rule 554
Rule 739
Rule 749
Rule 858
Rule 1199
Rule 1202
Rule 1223
Rule 1229
Rule 1262
Rule 1471
Rule 1600
Rule 1743
Rule 6860
Rubi steps
\begin {align*} \int \frac {-1+x^{16}}{\sqrt {-1+x^4} \left (1+x^8+x^{16}\right )} \, dx &=\int \frac {\sqrt {-1+x^4} \left (1+x^4+x^8+x^{12}\right )}{1+x^8+x^{16}} \, dx\\ &=\int \left (\frac {(1-x) \sqrt {-1+x^4}}{8 \left (1-x+x^2\right )}+\frac {(1+x) \sqrt {-1+x^4}}{8 \left (1+x+x^2\right )}+\frac {\sqrt {-1+x^4}}{4 \left (1-x^2+x^4\right )}+\frac {\sqrt {-1+x^4} \left (1+x^4\right )}{2 \left (1-x^4+x^8\right )}\right ) \, dx\\ &=\frac {1}{8} \int \frac {(1-x) \sqrt {-1+x^4}}{1-x+x^2} \, dx+\frac {1}{8} \int \frac {(1+x) \sqrt {-1+x^4}}{1+x+x^2} \, dx+\frac {1}{4} \int \frac {\sqrt {-1+x^4}}{1-x^2+x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx\\ &=\frac {1}{8} \int \left (\frac {\left (-1-\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x}+\frac {\left (-1+\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{8} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{4} \int \left (\frac {2 i \sqrt {-1+x^4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^2\right )}+\frac {2 i \sqrt {-1+x^4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^2\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x^4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x^4}\right ) \, dx\\ &=\frac {i \int \frac {\sqrt {-1+x^4}}{1+i \sqrt {3}-2 x^2} \, dx}{2 \sqrt {3}}+\frac {i \int \frac {\sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x^2} \, dx}{2 \sqrt {3}}+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x^4} \, dx+\frac {1}{24} \left (3-i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{24} \left (-3+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x} \, dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x^4} \, dx-\frac {1}{24} \left (3+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x} \, dx+\frac {1}{24} \left (3+i \sqrt {3}\right ) \int \frac {\sqrt {-1+x^4}}{1+i \sqrt {3}+2 x} \, dx\\ &=-\frac {i \int \frac {-1+i \sqrt {3}-2 x^2}{\sqrt {-1+x^4}} \, dx}{8 \sqrt {3}}-\frac {i \int \frac {1+i \sqrt {3}+2 x^2}{\sqrt {-1+x^4}} \, dx}{8 \sqrt {3}}+\frac {i \int \frac {\sqrt {-1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}-\frac {i \int \frac {\sqrt {-1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}-\frac {i \int \frac {\sqrt {-1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}+\frac {i \int \frac {\sqrt {-1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx}{2 \sqrt {3}}+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{4} \left (1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx+\frac {1}{12} \left (3-i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (-1+i \sqrt {3}\right )^2-4 x^2} \, dx+\frac {1}{12} \left (-3+i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{4} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4} \left (-1-i \sqrt {3}+2 x^4\right )} \, dx+\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (-1-i \sqrt {3}\right )^2-4 x^2} \, dx-\frac {1}{12} \left (3+i \sqrt {3}\right ) \int \frac {x \sqrt {-1+x^4}}{\left (1+i \sqrt {3}\right )^2-4 x^2} \, dx\\ &=\frac {\left (1-i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {-1+x^4}}-\frac {1}{4} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx-\frac {i \int \frac {\left (-1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}+\frac {i \int \frac {\left (1-i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}+\frac {i \int \frac {\left (-1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}-\frac {i \int \frac {\left (1+i \sqrt {3}\right )^2+4 x^2}{\sqrt {-1+x^4}} \, dx}{32 \sqrt {3}}-\frac {1}{8} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx-\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{24} \left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (-1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {1}{24} \left (-3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{8} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx-\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{24} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (-1-i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )-\frac {1}{24} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\left (1+i \sqrt {3}\right )^2-4 x} \, dx,x,x^2\right )+\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \left (i+\sqrt {3}\right )}-\frac {\left (i-\sqrt {3}\right ) \int \frac {1-x^2}{\left (1+i \sqrt {3}-2 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \left (i-\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1-x^2}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (i-\sqrt {3}\right )}\\ &=\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (i-\sqrt {3}\right ) \sqrt {-1+x^4}}-2 \frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (3 i-\sqrt {3}\right )}+\frac {\left (i-\sqrt {3}\right ) \int \frac {1-x^2}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (3 i-\sqrt {3}\right )}+\frac {\left (i-\sqrt {3}\right ) \int \frac {1-x^2}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (3 i-\sqrt {3}\right )}+\frac {1}{96} \left (3-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {4+2 \left (1+i \sqrt {3}\right ) x}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )+\frac {1}{96} \left (-3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {4+2 \left (1+i \sqrt {3}\right ) x}{\left (\left (-1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )-2 \left (\frac {1}{48} \left (-3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx\right )-\frac {1}{96} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {4+2 \left (1-i \sqrt {3}\right ) x}{\left (\left (-1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )+\frac {1}{96} \left (3+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {4-\left (1+i \sqrt {3}\right )^2 x}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )+2 \left (\frac {1}{48} \left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx\right )+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{2 \left ((-2-i)+\sqrt {3}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{2 \left ((-2+i)+\sqrt {3}\right )}-2 \frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (3 i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1-x^2}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (3 i+\sqrt {3}\right )}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1-x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1+x^4}} \, dx}{2 \left (3 i+\sqrt {3}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )}-\frac {\int \frac {1-x^2}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx}{2 \sqrt {2} \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right )}-\frac {\int \frac {1-x^2}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx}{2 \sqrt {2} \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )}-\frac {\int \frac {1-x^2}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx}{2 \sqrt {2} \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right )}-\frac {\int \frac {1-x^2}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}} \, dx}{2 \sqrt {2} \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right )}-\frac {\left (\left (i-\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\left (1+i \sqrt {3}-2 x^2\right ) \sqrt {-1-x^2}} \, dx}{2 \left (i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx}{2 \left (i-\sqrt {3}\right ) \sqrt {-1+x^4}}\\ &=-\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2-i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2+i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (i-\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}+\frac {1}{48} \left (3-i \sqrt {3}\right )^2 \text {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )+\frac {1}{8} \left (-1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )-\frac {1}{8} \left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )+\frac {1}{8} \left (1+i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {-1+x^2}} \, dx,x,x^2\right )+\frac {\left (\left (i-\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2}} \, dx}{2 \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (i-\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2}} \, dx}{2 \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (i-\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \left (i+\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x^2\right ) \sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \left (i+\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \left (i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx}{4 \left (i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2}} \, dx}{2 \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2}} \, dx}{2 \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx}{2 \sqrt {2} \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx}{2 \sqrt {2} \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx}{2 \sqrt {2} \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx}{2 \sqrt {2} \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}\\ &=-\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2-i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2+i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (i-\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}-\frac {1}{8} \left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-16+\left (-1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {4-\left (-1-i \sqrt {3}\right )^2 x^2}{\sqrt {-1+x^4}}\right )+\frac {1}{8} \left (-1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-16+\left (1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {4-\left (1+i \sqrt {3}\right )^2 x^2}{\sqrt {-1+x^4}}\right )+\frac {1}{8} \left (1-i \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-16+\left (-1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {4-\left (-1+i \sqrt {3}\right )^2 x^2}{\sqrt {-1+x^4}}\right )-\frac {1}{48} \left (3-i \sqrt {3}\right )^2 \text {Subst}\left (\int \frac {1}{-16+\left (1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {4-\left (1-i \sqrt {3}\right )^2 x^2}{\sqrt {-1+x^4}}\right )-\frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \left (i+\sqrt {3}\right )}-\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \left (i-\sqrt {3}\right )}+2 \frac {\left (\left (i-\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{8 \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (i-\sqrt {3}\right ) \left (3+i \sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (i-\sqrt {3}\right ) \left (3+i \sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+2 \frac {\left (\left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{8 \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (3-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (3-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\sqrt {2} \left (i+\sqrt {3}\right ) \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}+\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\sqrt {2} \left (i+\sqrt {3}\right ) \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}+\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\sqrt {2} \left (i-\sqrt {3}\right ) \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\sqrt {2} \left (i-\sqrt {3}\right ) \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (-2+\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (2+\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (-2+\sqrt {2+2 i \sqrt {3}}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (2+\sqrt {2+2 i \sqrt {3}}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x^2\right ) \sqrt {1-x^2} \sqrt {1+x^2}} \, dx}{4 \left (i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx}{4 \left (i-\sqrt {3}\right ) \sqrt {-1+x^4}}\\ &=-\frac {\left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {2+\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{16 \sqrt {2 \left (3+i \sqrt {3}\right )}}+\frac {\left (3-i \sqrt {3}\right )^{3/2} \tan ^{-1}\left (\frac {2+\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (3-i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{96 \sqrt {2}}-\frac {\left (1-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {2+\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (3-i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{16 \sqrt {2 \left (3-i \sqrt {3}\right )}}+\frac {\left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {4-\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{16 \sqrt {2 \left (3+i \sqrt {3}\right )}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2-i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2+i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {2}{1-i \sqrt {3}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {-1+x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {2}{1+i \sqrt {3}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {-1+x^4}}+2 \frac {\left (i-\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (3 i-\sqrt {3}\right )}+2 \frac {\left (i+\sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (3 i+\sqrt {3}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{\sqrt {2} \left (i+\sqrt {3}\right ) \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{\sqrt {2} \left (i+\sqrt {3}\right ) \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{\sqrt {2} \left (i-\sqrt {3}\right ) \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{\sqrt {2} \left (i-\sqrt {3}\right ) \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right )}+\frac {\left (\left (i-\sqrt {3}\right ) \left (3+i \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {1+x^2}} \, dx}{4 \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (i-\sqrt {3}\right ) \left (3+i \sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\left (\left (-1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {1+x^2}} \, dx}{4 \left (3 i-\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (3-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\left (\left (-1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {1+x^2}} \, dx}{4 \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (3-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1-x^2} \sqrt {1+x^2}} \, dx}{4 \left (3 i+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (-2+\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (2+\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (-2+\sqrt {2+2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (\left (2+\sqrt {2+2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx}{4 \sqrt {2} \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}\\ &=-\frac {\left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {2+\left (1-i \sqrt {3}\right ) x^2}{\sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{16 \sqrt {2 \left (3+i \sqrt {3}\right )}}+\frac {\left (3-i \sqrt {3}\right )^{3/2} \tan ^{-1}\left (\frac {2+\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (3-i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{96 \sqrt {2}}-\frac {\left (1-i \sqrt {3}\right ) \tan ^{-1}\left (\frac {2+\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (3-i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{16 \sqrt {2 \left (3-i \sqrt {3}\right )}}+\frac {\left (1+i \sqrt {3}\right ) \tan ^{-1}\left (\frac {4-\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {-1+x^4}}\right )}{16 \sqrt {2 \left (3+i \sqrt {3}\right )}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2-i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {2} \left ((-2+i)+\sqrt {3}\right ) \sqrt {-1+x^4}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \left (i+\sqrt {3}\right ) \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \left (i+\sqrt {3}\right ) \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \left (1+\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \left (i-\sqrt {3}\right ) \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \left (i-\sqrt {3}\right ) \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (2+\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {2} \left (\sqrt {2}+\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (-\frac {4}{\left (i-\sqrt {3}\right )^2};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {-1+x^4}}-\frac {\left (2+\sqrt {2+2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {2} \left (\sqrt {2}+\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\left (2-\sqrt {2-2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {2} \left (\sqrt {2}-\sqrt {1-i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {2}{1-i \sqrt {3}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {-1+x^4}}-\frac {\left (2-\sqrt {2+2 i \sqrt {3}}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {2} \left (\sqrt {2}-\sqrt {1+i \sqrt {3}}\right ) \sqrt {-1+x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\frac {2}{1+i \sqrt {3}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {-1+x^4}}-\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (-\frac {4}{\left (i+\sqrt {3}\right )^2};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {-1+x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.59, size = 188, normalized size = 0.79 \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {-1-x^2+x^4}{\sqrt {2} x \sqrt {-1+x^4}}\right )+3^{3/4} \text {ArcTan}\left (\frac {-1-\left (-2 i+\sqrt {-1+4 i \sqrt {3}}\right ) x^2+x^4}{\sqrt {2} \sqrt [4]{3} x \sqrt {-1+x^4}}\right )-3 \tanh ^{-1}\left (\frac {-1+x^2+x^4}{\sqrt {2} x \sqrt {-1+x^4}}\right )-3^{3/4} \tanh ^{-1}\left (\frac {-1+\left (2 i+\sqrt {-1-4 i \sqrt {3}}\right ) x^2+x^4}{\sqrt {2} \sqrt [4]{3} x \sqrt {-1+x^4}}\right )}{12 \sqrt {2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.94, size = 685, normalized size = 2.87
method | result | size |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {x^{4}-1}{x^{2}}-\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}+1\right )}{8}+\frac {\arctan \left (-1+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )}{4}-\frac {\ln \left (\frac {x^{4}-1}{x^{2}}+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}+1\right )}{8}+\frac {\arctan \left (1+\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}\right )}{4}+\frac {3^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{3 x}+1\right )}{12}+\frac {3^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{3 x}-1\right )}{12}+\frac {3^{\frac {3}{4}} \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {3^{\frac {1}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}}{2}}{\frac {x^{4}-1}{2 x^{2}}+\frac {3^{\frac {1}{4}} \sqrt {x^{4}-1}\, \sqrt {2}}{2 x}+\frac {\sqrt {3}}{2}}\right )}{24}\right ) \sqrt {2}}{2}\) | \(231\) |
default | \(\text {Expression too large to display}\) | \(685\) |
trager | \(\text {Expression too large to display}\) | \(849\) |
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. 1173 vs.
\(2 (180) = 360\).
time = 0.68, size = 1173, normalized size = 4.91 \begin {gather*} -\frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {3 \, x^{16} + 6 \, x^{12} + 9 \, x^{8} + 6 \, x^{4} + \sqrt {3} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (x^{14} - 12 \, x^{10} + 12 \, x^{6} - x^{2}\right )} + 3^{\frac {1}{4}} \sqrt {2} {\left (x^{16} - 28 \, x^{12} + 45 \, x^{8} - 28 \, x^{4} + 1\right )} + 12 \, {\left (x^{11} + x^{7} + x^{3} + 4 \, \sqrt {3} {\left (x^{9} - x^{5}\right )}\right )} \sqrt {x^{4} - 1}\right )} \sqrt {\frac {12 \, x^{6} - 12 \, x^{2} + \sqrt {3} {\left (x^{8} + x^{4} + 1\right )} + 2 \, \sqrt {x^{4} - 1} {\left (3 \cdot 3^{\frac {1}{4}} \sqrt {2} x^{3} + 3^{\frac {3}{4}} \sqrt {2} {\left (x^{5} - x\right )}\right )}}{x^{8} + x^{4} + 1}} + 12 \, \sqrt {3} {\left (x^{14} - x^{2}\right )} + 6 \, \sqrt {x^{4} - 1} {\left (3 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (x^{11} - 3 \, x^{7} + x^{3}\right )} + 3^{\frac {1}{4}} \sqrt {2} {\left (x^{13} - 12 \, x^{9} + 12 \, x^{5} - x\right )}\right )} + 3}{3 \, {\left (x^{16} - 46 \, x^{12} + 99 \, x^{8} - 46 \, x^{4} + 1\right )}}\right ) + \frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {3 \, x^{16} + 6 \, x^{12} + 9 \, x^{8} + 6 \, x^{4} - \sqrt {3} {\left (2 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (x^{14} - 12 \, x^{10} + 12 \, x^{6} - x^{2}\right )} + 3^{\frac {1}{4}} \sqrt {2} {\left (x^{16} - 28 \, x^{12} + 45 \, x^{8} - 28 \, x^{4} + 1\right )} - 12 \, {\left (x^{11} + x^{7} + x^{3} + 4 \, \sqrt {3} {\left (x^{9} - x^{5}\right )}\right )} \sqrt {x^{4} - 1}\right )} \sqrt {\frac {12 \, x^{6} - 12 \, x^{2} + \sqrt {3} {\left (x^{8} + x^{4} + 1\right )} - 2 \, \sqrt {x^{4} - 1} {\left (3 \cdot 3^{\frac {1}{4}} \sqrt {2} x^{3} + 3^{\frac {3}{4}} \sqrt {2} {\left (x^{5} - x\right )}\right )}}{x^{8} + x^{4} + 1}} + 12 \, \sqrt {3} {\left (x^{14} - x^{2}\right )} - 6 \, \sqrt {x^{4} - 1} {\left (3 \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (x^{11} - 3 \, x^{7} + x^{3}\right )} + 3^{\frac {1}{4}} \sqrt {2} {\left (x^{13} - 12 \, x^{9} + 12 \, x^{5} - x\right )}\right )} + 3}{3 \, {\left (x^{16} - 46 \, x^{12} + 99 \, x^{8} - 46 \, x^{4} + 1\right )}}\right ) - \frac {1}{96} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {12 \, {\left (12 \, x^{6} - 12 \, x^{2} + \sqrt {3} {\left (x^{8} + x^{4} + 1\right )} + 2 \, \sqrt {x^{4} - 1} {\left (3 \cdot 3^{\frac {1}{4}} \sqrt {2} x^{3} + 3^{\frac {3}{4}} \sqrt {2} {\left (x^{5} - x\right )}\right )}\right )}}{x^{8} + x^{4} + 1}\right ) + \frac {1}{96} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (\frac {12 \, {\left (12 \, x^{6} - 12 \, x^{2} + \sqrt {3} {\left (x^{8} + x^{4} + 1\right )} - 2 \, \sqrt {x^{4} - 1} {\left (3 \cdot 3^{\frac {1}{4}} \sqrt {2} x^{3} + 3^{\frac {3}{4}} \sqrt {2} {\left (x^{5} - x\right )}\right )}\right )}}{x^{8} + x^{4} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {x^{8} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {x^{4} - 1} + {\left (4 \, \sqrt {x^{4} - 1} x^{3} + \sqrt {2} {\left (x^{8} - 2 \, x^{6} - 3 \, x^{4} + 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} + 4 \, x^{6} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {x^{4} - 1} - 4 \, x^{2} + 1}{x^{8} - x^{4} + 1}} + 1}{x^{8} - 4 \, x^{6} - x^{4} + 4 \, x^{2} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {x^{8} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {x^{4} - 1} + {\left (4 \, \sqrt {x^{4} - 1} x^{3} - \sqrt {2} {\left (x^{8} - 2 \, x^{6} - 3 \, x^{4} + 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} + 4 \, x^{6} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {x^{4} - 1} - 4 \, x^{2} + 1}{x^{8} - x^{4} + 1}} + 1}{x^{8} - 4 \, x^{6} - x^{4} + 4 \, x^{2} + 1}\right ) - \frac {1}{32} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} + 4 \, x^{6} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {x^{4} - 1} - 4 \, x^{2} + 1\right )}}{x^{8} - x^{4} + 1}\right ) + \frac {1}{32} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} + 4 \, x^{6} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {x^{4} - 1} - 4 \, x^{2} + 1\right )}}{x^{8} - x^{4} + 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 {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right ) \left (x^{8} + 1\right )}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right ) \left (x^{8} - x^{4} + 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 = -0.00 \begin {gather*} \int \frac {x^{16}-1}{\sqrt {x^4-1}\,\left (x^{16}+x^8+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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