Optimal. Leaf size=45 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt {x^4+1}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [C] time = 7.41, antiderivative size = 2243, normalized size of antiderivative = 49.84, number of steps used = 227, number of rules used = 16, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {1586, 6725, 1729, 1209, 1198, 220, 1196, 1217, 1707, 1248, 735, 844, 215, 725, 206, 204}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 204
Rule 206
Rule 215
Rule 220
Rule 725
Rule 735
Rule 844
Rule 1196
Rule 1198
Rule 1209
Rule 1217
Rule 1248
Rule 1586
Rule 1707
Rule 1729
Rule 6725
Rubi steps
\begin {align*} \int \frac {-1+x^{16}}{\sqrt {1+x^4} \left (1+x^{16}\right )} \, dx &=\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 )\\ &=-2 \left (-\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i-\sqrt [8]{-1}\right )}+\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}+i x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1+(-1)^{5/8}\right )}+\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left (-i+\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [8]{-1}\right )}+\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}-i x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1-(-1)^{5/8}\right )}-\frac {1}{16} \left ((-1)^{5/8} \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [8]{-1} \left (i+\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}+\sqrt [4]{-1}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (-i+\sqrt [4]{-1}\right )}+\frac {\left (1+\sqrt [8]{-1}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}+\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}-\frac {1}{16} \left ((-1)^{3/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [4]{-1} \left (1-\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )-2 \left (\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}-\sqrt [4]{-1}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (-i+\sqrt [4]{-1}\right )}-\frac {\left (i-(-1)^{5/8}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}-\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (i-\sqrt [4]{-1}\right )}+\frac {1}{16} \left ((-1)^{3/8} \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{16} \left (\sqrt [4]{-1} \left (1+\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (1-\sqrt [8]{-1}\right )}-\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}+x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1-\sqrt [8]{-1}\right )}-\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left ((-1)^{5/8} \left (1-\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\frac {\sqrt [8]{-1} \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (1+\sqrt [8]{-1}\right )}-\frac {\sqrt [8]{-1} \int \frac {1+x^2}{\left (\sqrt [8]{-1}-x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (1+\sqrt [8]{-1}\right )}+\frac {1}{16} \left ((-1)^{5/8} \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx-\frac {1}{16} \left ((-1)^{5/8} \left (1+\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (-\frac {\left (1+(-1)^{5/8}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}+(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}-\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}+(-1)^{3/4}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}+\frac {1}{16} \left ((-1)^{7/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [4]{-1} \left (1-(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )+2 \left (\frac {\left (1-(-1)^{5/8}\right ) \int \frac {1+x^2}{\left (\sqrt [8]{-1}-(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}-\frac {\left (\sqrt [8]{-1} \left (\sqrt [8]{-1}-(-1)^{3/4}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{8 \left (i+\sqrt [4]{-1}\right )}-\frac {1}{16} \left ((-1)^{7/8} \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx+\frac {1}{16} \left (\sqrt [4]{-1} \left (1+(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx\right )\\ &=-2 \left (-\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {(-1)^{3/8}-(-1)^{5/8}}}+\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\sqrt [8]{-1} \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i-\sqrt [8]{-1}\right ) \sqrt {1+x^4}}-\frac {\sqrt [8]{-1} \left (i-\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}+\frac {\left (1-(-1)^{5/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} (-1)^{3/8} \left (i-\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1+(-1)^{5/8}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {\sqrt [8]{-1}-(-1)^{7/8}}}-\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (1-\sqrt [8]{-1}\right ) \sqrt {1+x^4}}+\frac {(-1)^{5/8} \left (1-\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}-\frac {\left (1+\sqrt [8]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} (-1)^{7/8} \left (1-\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1-\sqrt [8]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {(-1)^{5/16} \left (i+\sqrt [8]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{3/16} \sqrt {-1+\sqrt [4]{-1}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {-1+\sqrt [4]{-1}} \left (1-(-1)^{5/8}\right )}-\frac {(-1)^{5/8} \left ((1+i)+i \sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (i+\sqrt [8]{-1}\right ) \sqrt {1+x^4}}+\frac {\sqrt [8]{-1} \left (i+\sqrt [8]{-1}\right ) \left ((1+i)+i \sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}+\frac {\left (1+(-1)^{5/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (-\frac {1}{4} (-1)^{3/8} \left (i+\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1-(-1)^{5/8}\right ) \sqrt {1+x^4}}\right )+2 \left (-\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}+\frac {(-1)^{15/16} \tan ^{-1}\left (\frac {\sqrt [16]{-1} \sqrt {-1+(-1)^{3/4}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {-1+(-1)^{3/4}}}+\frac {(-1)^{5/8} \left ((-1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\sqrt [8]{-1} \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \left (1+\sqrt [8]{-1}\right ) \sqrt {1+x^4}}-\frac {(-1)^{5/8} \left (1+\sqrt [8]{-1}\right ) \left ((-1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}-\frac {\left (1-\sqrt [8]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (-\frac {1}{4} (-1)^{7/8} \left (1+\sqrt [8]{-1}\right )^2;2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1+\sqrt [8]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (-\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {\sqrt [16]{-1} \left (1+\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{3/16} \sqrt {-1+\sqrt [4]{-1}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {-1+\sqrt [4]{-1}} \left (i+\sqrt [4]{-1}\right )}+\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (1+(-1)^{5/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1-(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1+(-1)^{5/8}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2+(-1)^{3/8}-(-1)^{5/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (i+\sqrt [4]{-1}\right ) \sqrt {1+x^4}}\right )+2 \left (\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {\tan ^{-1}\left (\frac {\sqrt {(-1)^{3/8}-(-1)^{5/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {(-1)^{3/8}-(-1)^{5/8}}}-\frac {(-1)^{7/8} \left ((1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}-\frac {\left (1-(-1)^{5/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1+(-1)^{5/8}\right ) \left ((1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}+\frac {\sqrt [4]{-1} \left (1-(-1)^{5/8}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-(-1)^{3/8}+(-1)^{5/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (i+\sqrt [4]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (-\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}-\frac {(-1)^{3/16} \left (1-\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {\sqrt [16]{-1} \sqrt {-1+(-1)^{3/4}} x}{\sqrt {1+x^4}}\right )}{16 \left (i-\sqrt [4]{-1}\right ) \sqrt {-1+(-1)^{3/4}}}+\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}+\frac {\left (1-\sqrt [8]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {2} \left ((-1-i)+\sqrt {2}\right ) \sqrt {1+x^4}}-\frac {\sqrt [4]{-1} \left (1+\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}+\frac {\left (i+\sqrt [4]{-1}-2 (-1)^{3/8}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2+\sqrt [8]{-1}-(-1)^{7/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (i-\sqrt [4]{-1}\right ) \sqrt {1+x^4}}\right )-2 \left (\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) x \sqrt {1+x^4}}{16 \left (1+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt [8]{-1}-(-1)^{7/8}} x}{\sqrt {1+x^4}}\right )}{16 \sqrt {\sqrt [8]{-1}-(-1)^{7/8}}}-\frac {(-1)^{3/8} \left ((-1-i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{16 \sqrt {1+x^4}}+\frac {\left (1+\sqrt [8]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{8 \sqrt {2} \left ((-1-i)+\sqrt {2}\right ) \sqrt {1+x^4}}-\frac {\sqrt [4]{-1} \left (1-\sqrt [8]{-1}\right ) \left ((-1-i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \sqrt {1+x^4}}-\frac {(-1)^{3/4} \left (1+\sqrt [8]{-1}\right )^2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\sqrt [8]{-1}+(-1)^{7/8}\right );2 \tan ^{-1}(x)|\frac {1}{2}\right )}{32 \left (1+(-1)^{3/4}\right ) \sqrt {1+x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 21.34, size = 186, normalized size = 4.13 \begin {gather*} \frac {1}{4} \sqrt [4]{-1} \left (-4 F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-\sqrt [8]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (\sqrt [8]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-(-1)^{3/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left ((-1)^{3/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-(-1)^{5/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left ((-1)^{5/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-(-1)^{7/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left ((-1)^{7/8};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 45, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 1459, normalized size = 32.42
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} + 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 [B] time = 11.92, size = 67, normalized size = 1.49
method | result | size |
elliptic | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (8 \textit {\_Z}^{8}-8 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}-\textit {\_R}^{2}\right ) \ln \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right ) \sqrt {2}}{16}\) | \(67\) |
default | \(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{16}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{14}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {x^{4}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+1}}-\frac {2 \left (-1\right )^{\frac {3}{4}} \underline {\hspace {1.25 ex}}\alpha ^{15} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , i \underline {\hspace {1.25 ex}}\alpha ^{14}, i\right )}{\sqrt {x^{4}+1}}\right )\right )}{16}\) | \(161\) |
trager | \(\text {Expression too large to display}\) | \(2481\) |
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} + 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 = -0.02 \begin {gather*} \int \frac {x^{16}-1}{\sqrt {x^4+1}\,\left (x^{16}+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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