Optimal. Leaf size=239 \[ \frac {\tan ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\frac {x^4}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {1}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^4}{\sqrt {2} \sqrt [4]{3}}+\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {1}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2} \sqrt [4]{3}} \]
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Rubi [C] time = 4.44, 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 = {1586, 6728, 1729, 1209, 1188, 222, 1185, 1215, 1457, 540, 253, 538, 537, 1248, 735, 844, 217, 206, 725, 204, 406, 409} \begin {gather*} -\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \tan ^{-1}\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 ) \tan ^{-1}\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 (\sin ^{-1}\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 (\sin ^{-1}\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 (\sin ^{-1}\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 (\sin ^{-1}\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 );\left .\sin ^{-1}(x)\right |-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};\left .\sin ^{-1}(x)\right |-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 );\left .\sin ^{-1}(x)\right |-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 )}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {2}{1-i \sqrt {3}};\left .\sin ^{-1}(x)\right |-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 )}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {2}{1+i \sqrt {3}};\left .\sin ^{-1}(x)\right |-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};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 204
Rule 206
Rule 217
Rule 222
Rule 253
Rule 406
Rule 409
Rule 537
Rule 538
Rule 540
Rule 725
Rule 735
Rule 844
Rule 1185
Rule 1188
Rule 1209
Rule 1215
Rule 1248
Rule 1457
Rule 1586
Rule 1729
Rule 6728
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 \operatorname {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] time = 2.75, size = 195, normalized size = 0.82 \begin {gather*} \frac {\sqrt {1-x^4} \left (4 F\left (\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2}-\frac {i \sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2}+\frac {i \sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2} i \left (i+\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )\right )}{4 \sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 2.29, size = 278, normalized size = 1.16 \begin {gather*} \frac {\tan ^{-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 {\tan ^{-1}\left (\frac {-\frac {1}{\sqrt {2} \sqrt [4]{3}}+\frac {\left (2 i-\sqrt {-1+4 i \sqrt {3}}\right ) x^2}{\sqrt {2} \sqrt [4]{3}}+\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 {\left (2 i+\sqrt {-1-4 i \sqrt {3}}\right ) x^2}{\sqrt {2} \sqrt [4]{3}}+\frac {x^4}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {-1+x^4}}\right )}{4 \sqrt {2} \sqrt [4]{3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 1173, normalized size = 4.91
result too large to display
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*} \int \frac {x^{16} - 1}{{\left (x^{16} + x^{8} + 1\right )} \sqrt {x^{4} - 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.14, size = 231, normalized size = 0.97
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 | \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}-1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}-2}\, \sqrt {x^{4}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}-2}}+\frac {2 i \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{2}-1, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{16}+\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{2}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-1}}\right )}{2 \sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \frac {1}{2}+\frac {i \sqrt {3}}{2}, i\right )}{\sqrt {x^{4}-1}}\right )}{8}+\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{2}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-1}}\right )}{2 \sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}+\frac {i \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \frac {1}{2}-\frac {i \sqrt {3}}{2}, i\right )}{\sqrt {x^{4}-1}}\right )}{8}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6}-\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}\, \sqrt {x^{4}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}}+\frac {2 i \left (-\underline {\hspace {1.25 ex}}\alpha ^{7}+\underline {\hspace {1.25 ex}}\alpha ^{3}\right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{6}-\underline {\hspace {1.25 ex}}\alpha ^{2}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{16}+\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x^{2}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-1}}\right )}{2 \sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}+\frac {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \frac {1}{2}-\frac {i \sqrt {3}}{2}, i\right )}{\sqrt {x^{4}-1}}\right )}{8}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x^{2}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}-1}}\right )}{2 \sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}+\frac {i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \frac {1}{2}+\frac {i \sqrt {3}}{2}, i\right )}{\sqrt {x^{4}-1}}\right )}{8}\) | \(685\) |
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*} \int \frac {x^{16} - 1}{{\left (x^{16} + x^{8} + 1\right )} \sqrt {x^{4} - 1}}\,{d x} \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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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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