Integrand size = 22, antiderivative size = 45 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Result contains complex when optimal does not.
Time = 5.43 (sec) , antiderivative size = 2243, normalized size of antiderivative = 49.84, number of steps used = 227, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {1600, 6857, 1743, 1223, 1212, 226, 1210, 1231, 1721, 1262, 749, 858, 221, 739, 212, 210} \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\frac {1}{32} \sqrt [16]{-1} \left ((1+i)+\sqrt {2}\right ) \sqrt {-4+(2+2 i) \sqrt {2}} \arctan \left (\frac {(-1)^{3/16} \sqrt {-1+\sqrt [4]{-1}} x}{\sqrt {x^4+1}}\right )-\frac {1}{16} (-1)^{13/16} \sqrt {-1+\sqrt [4]{-1}} \left ((1+i)+i \sqrt {2}\right ) \arctan \left (\frac {(-1)^{3/16} \sqrt {-1+\sqrt [4]{-1}} x}{\sqrt {x^4+1}}\right )-\frac {(-1)^{5/8} \sqrt {(-1)^{3/8}-(-1)^{5/8}} \left (1+(-1)^{5/8}\right ) \left ((1-i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {x^4+1}}\right )}{16 \left (i-\sqrt [8]{-1}\right )}-\frac {\arctan \left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {x^4+1}}\right )}{8 \sqrt {(-1)^{3/8}-(-1)^{5/8}}}-\frac {1}{16} (-1)^{11/16} \sqrt {-1+(-1)^{3/4}} \left ((-1+i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt [16]{-1} \sqrt {-1+(-1)^{3/4}} x}{\sqrt {x^4+1}}\right )+\frac {(-1)^{15/16} \arctan \left (\frac {\sqrt [16]{-1} \sqrt {-1+(-1)^{3/4}} x}{\sqrt {x^4+1}}\right )}{4 \sqrt {-4-(2-2 i) \sqrt {2}}}+\frac {1}{16} (-1)^{5/8} \sqrt {\sqrt [8]{-1}-(-1)^{7/8}} \left ((-1+i)+\sqrt {2}\right ) \arctan \left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {x^4+1}}\right )-\frac {\arctan \left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {x^4+1}}\right )}{8 \sqrt {\sqrt [8]{-1}-(-1)^{7/8}}}+\frac {\sqrt [4]{-1} \left (1+(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}+\frac {\sqrt [4]{-1} \left (1-(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {\left (1+(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {x^4+1}}-\frac {\left (1-(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {x^4+1}}+\frac {(-1)^{5/8} \left (i-\sqrt [8]{-1}\right ) \left ((1-i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {(-1)^{5/8} \left (1+\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}+\frac {(-1)^{5/8} \left (1-\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}+\frac {\sqrt [4]{-1} \left (1+\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}+\frac {\sqrt [4]{-1} \left (1-\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {\left (1+\sqrt [8]{-1}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {2} \left ((-1-i)+\sqrt {2}\right ) \sqrt {x^4+1}}-\frac {\left (1-\sqrt [8]{-1}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {2} \left ((-1-i)+\sqrt {2}\right ) \sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \left (i+\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{16 \sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (1+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (i+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {\sqrt [8]{-1} \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (1-\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {\sqrt [8]{-1} \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \left (i-\sqrt [8]{-1}\right ) \sqrt {x^4+1}}+\frac {\left ((-1+i)+(2+2 i) \sqrt [8]{-1}+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{3/8} \left (i-\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{16 \left ((1-i)+\sqrt {2}\right ) \sqrt {x^4+1}}-\frac {\left ((-1+i)+(2+2 i) (-1)^{5/8}-\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{7/8} \left (1-\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{16 \left ((-1+i)+\sqrt {2}\right ) \sqrt {x^4+1}}-\frac {\sqrt [8]{-1} \left (1-(-1)^{3/8}\right ) \left (1+(-1)^{5/8}\right ) \left ((-1+i)+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{3/8} \left (i+\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{32 \sqrt {x^4+1}}+\frac {\left (\frac {1}{32}+\frac {i}{32}\right ) \left ((-1+i)+\sqrt [8]{-1}-(-1)^{5/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{7/8} \left (1+\sqrt [8]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{\left (1+\sqrt [8]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (i-\sqrt [4]{-1}-2 (-1)^{7/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+(-1)^{3/8}-(-1)^{5/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{16 \left (i+\sqrt [4]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (i-\sqrt [4]{-1}+2 (-1)^{7/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-(-1)^{3/8}+(-1)^{5/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{16 \left (i+\sqrt [4]{-1}\right ) \sqrt {x^4+1}}+\frac {\left ((1+i)-(2-2 i) (-1)^{3/8}+\sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+\sqrt [8]{-1}-(-1)^{7/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{16 \left ((-1-i)+\sqrt {2}\right ) \sqrt {x^4+1}}-\frac {\left (i+\sqrt [4]{-1}+2 (-1)^{3/8}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2-\sqrt [8]{-1}+(-1)^{7/8}\right ),2 \arctan (x),\frac {1}{2}\right )}{16 \left (i-\sqrt [4]{-1}\right ) \sqrt {x^4+1}} \]
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Rule 210
Rule 212
Rule 221
Rule 226
Rule 739
Rule 749
Rule 858
Rule 1210
Rule 1212
Rule 1223
Rule 1231
Rule 1262
Rule 1600
Rule 1721
Rule 1743
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+x^4} \left (-1+x^4-x^8+x^{12}\right )}{1+x^{16}} \, dx \\ & = \int \left (\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-i x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+i x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}-(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-\sqrt [8]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+\sqrt [8]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-\sqrt [4]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+\sqrt [4]{-1} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/8} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{3/8} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{5/8} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}+(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{5/8} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{3/4} x\right )}+\frac {\left (-\sqrt [16]{-1}-(-1)^{5/16}-(-1)^{9/16}-(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{3/4} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}-(-1)^{7/8} x\right )}+\frac {\left (-\sqrt [16]{-1}+(-1)^{5/16}+(-1)^{9/16}+(-1)^{13/16}\right ) \sqrt {1+x^4}}{16 \left (\sqrt [16]{-1}+(-1)^{7/8} x\right )}\right ) \, dx \\ & = -\left (\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-\sqrt [4]{-1} x} \, dx\right )-\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+\sqrt [4]{-1} x} \, dx-\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{3/4} x} \, dx-\frac {1}{16} \left (\sqrt [16]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{3/4} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-\sqrt [8]{-1} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+\sqrt [8]{-1} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{5/8} x} \, dx-\frac {1}{16} \left ((-1)^{9/16} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{5/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-i x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+i x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{3/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{3/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}-(-1)^{7/8} x} \, dx+\frac {1}{16} \left ((-1)^{9/16} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [16]{-1}+(-1)^{7/8} x} \, dx \\ & = -2 \left (\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-i x^2} \, dx\right )-2 \left (\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+i x^2} \, dx\right )-2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-\sqrt [4]{-1} x^2} \, dx\right )-2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+\sqrt [4]{-1} x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}-(-1)^{3/4} x^2} \, dx\right )+2 \left (\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt {1+x^4}}{\sqrt [8]{-1}+(-1)^{3/4} x^2} \, dx\right ) \\ & = -2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-i x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-i x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+i x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [8]{-1} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-\sqrt [4]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [8]{-1} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+\sqrt [4]{-1} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+x^2\right ) \sqrt {1+x^4}} \, dx\right )-\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-x^2\right ) \sqrt {1+x^4}} \, dx\right )-\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}+(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}-(-1)^{3/4} x^2}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\left (\frac {1}{8} \sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx\right )+\frac {1}{16} \left (\sqrt [8]{-1} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {\sqrt [8]{-1}+(-1)^{3/4} x^2}{\sqrt {1+x^4}} \, dx\right ) \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Time = 13.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {x^{4}+1}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(38\) |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {x^{4}+1}}{x}\right )}{\textit {\_R}}\right )}{8}\) | \(38\) |
elliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{8}-8 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}-\textit {\_R}^{2}\right ) \ln \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right ) \sqrt {2}}{16}\) | \(67\) |
trager | \(\text {Expression too large to display}\) | \(1528\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.45 (sec) , antiderivative size = 1957, normalized size of antiderivative = 43.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\int { \frac {x^{16} - 1}{{\left (x^{16} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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Not integrable
Time = 5.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49 \[ \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx=\int \frac {x^{16}-1}{\sqrt {x^4+1}\,\left (x^{16}+1\right )} \,d x \]
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