Optimal. Leaf size=45 \[ \frac {1}{8} \text {RootSum}\left [2-4 \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt {1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
[Out]
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Rubi [C] Result contains complex when optimal does not.
time = 5.17, 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 = {1600,
6857, 1743, 1223, 1212, 226, 1210, 1231, 1721, 1262, 749, 858, 221, 739, 212, 210}
\begin {gather*} \text {Too large to display} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
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*} \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 [A]
time = 0.22, 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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 9.64, size = 161, normalized size = 3.58
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 | \(\text {Expression too large to display}\) | \(1526\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.64, size = 1459, normalized size = 32.42 \begin {gather*} \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {\sqrt {2} + 2}} \arctan \left (\frac {{\left (4 \, x^{10} + 4 \, x^{6} - 2 \, \sqrt {2} {\left (x^{14} + x^{10} + x^{6} + x^{2}\right )} - {\left (x^{16} + 4 \, x^{12} + 4 \, x^{8} + 4 \, x^{4} - 2 \, \sqrt {2} {\left (x^{12} + 2 \, x^{8} + x^{4}\right )} + 1\right )} \sqrt {\sqrt {2} + 2}\right )} \sqrt {{\left (3 \, \sqrt {2} - 4\right )} \sqrt {\sqrt {2} + 2}} \sqrt {\sqrt {2} \sqrt {\sqrt {2} + 2}} - 2 \, {\left ({\left (2 \, x^{13} + 4 \, x^{9} + 4 \, x^{5} - \sqrt {2} {\left (x^{13} + 3 \, x^{9} + 3 \, x^{5} + x\right )} + 2 \, x\right )} \sqrt {x^{4} + 1} \sqrt {\sqrt {2} + 2} - 2 \, {\left (x^{11} + 2 \, x^{7} + x^{3} - \sqrt {2} {\left (x^{11} + x^{7} + x^{3}\right )}\right )} \sqrt {x^{4} + 1}\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {2} + 2}}}{2 \, {\left (x^{16} + 1\right )}}\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {-\sqrt {2} + 2}} \arctan \left (\frac {2 \, {\left (2 \, x^{11} + 4 \, x^{7} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{11} + x^{7} + x^{3}\right )} + {\left (2 \, x^{13} + 4 \, x^{9} + 4 \, x^{5} + \sqrt {2} {\left (x^{13} + 3 \, x^{9} + 3 \, x^{5} + x\right )} + 2 \, x\right )} \sqrt {-\sqrt {2} + 2}\right )} \sqrt {x^{4} + 1} \sqrt {\sqrt {2} \sqrt {-\sqrt {2} + 2}} + {\left (4 \, x^{10} + 4 \, x^{6} + 2 \, \sqrt {2} {\left (x^{14} + x^{10} + x^{6} + x^{2}\right )} + {\left (x^{16} + 4 \, x^{12} + 4 \, x^{8} + 4 \, x^{4} + 2 \, \sqrt {2} {\left (x^{12} + 2 \, x^{8} + x^{4}\right )} + 1\right )} \sqrt {-\sqrt {2} + 2}\right )} \sqrt {{\left (3 \, \sqrt {2} + 4\right )} \sqrt {-\sqrt {2} + 2}} \sqrt {\sqrt {2} \sqrt {-\sqrt {2} + 2}}}{2 \, {\left (x^{16} + 1\right )}}\right ) - \frac {1}{32} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {\sqrt {2} + 2}} \log \left (\frac {2 \, {\left (4 \, x^{11} + 6 \, x^{7} + 4 \, x^{3} - \sqrt {2} {\left (3 \, x^{11} + 4 \, x^{7} + 3 \, x^{3}\right )}\right )} \sqrt {x^{4} + 1} \sqrt {\sqrt {2} + 2} - 2 \, {\left (2 \, x^{13} + 4 \, x^{9} + 4 \, x^{5} - \sqrt {2} {\left (x^{13} + 3 \, x^{9} + 3 \, x^{5} + x\right )} + 2 \, x\right )} \sqrt {x^{4} + 1} + {\left (x^{16} + 8 \, x^{12} + 12 \, x^{8} + 8 \, x^{4} - \sqrt {2} {\left (x^{16} + 6 \, x^{12} + 8 \, x^{8} + 6 \, x^{4} + 1\right )} - 2 \, {\left (3 \, x^{14} + 7 \, x^{10} + 7 \, x^{6} + 3 \, x^{2} - \sqrt {2} {\left (2 \, x^{14} + 5 \, x^{10} + 5 \, x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {\sqrt {2} + 2} + 1\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {2} + 2}}}{x^{16} + 1}\right ) + \frac {1}{32} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {\sqrt {2} + 2}} \log \left (\frac {2 \, {\left (4 \, x^{11} + 6 \, x^{7} + 4 \, x^{3} - \sqrt {2} {\left (3 \, x^{11} + 4 \, x^{7} + 3 \, x^{3}\right )}\right )} \sqrt {x^{4} + 1} \sqrt {\sqrt {2} + 2} - 2 \, {\left (2 \, x^{13} + 4 \, x^{9} + 4 \, x^{5} - \sqrt {2} {\left (x^{13} + 3 \, x^{9} + 3 \, x^{5} + x\right )} + 2 \, x\right )} \sqrt {x^{4} + 1} - {\left (x^{16} + 8 \, x^{12} + 12 \, x^{8} + 8 \, x^{4} - \sqrt {2} {\left (x^{16} + 6 \, x^{12} + 8 \, x^{8} + 6 \, x^{4} + 1\right )} - 2 \, {\left (3 \, x^{14} + 7 \, x^{10} + 7 \, x^{6} + 3 \, x^{2} - \sqrt {2} {\left (2 \, x^{14} + 5 \, x^{10} + 5 \, x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {\sqrt {2} + 2} + 1\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {2} + 2}}}{x^{16} + 1}\right ) - \frac {1}{32} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {-\sqrt {2} + 2}} \log \left (-\frac {2 \, {\left (2 \, x^{13} + 4 \, x^{9} + 4 \, x^{5} + \sqrt {2} {\left (x^{13} + 3 \, x^{9} + 3 \, x^{5} + x\right )} + {\left (4 \, x^{11} + 6 \, x^{7} + 4 \, x^{3} + \sqrt {2} {\left (3 \, x^{11} + 4 \, x^{7} + 3 \, x^{3}\right )}\right )} \sqrt {-\sqrt {2} + 2} + 2 \, x\right )} \sqrt {x^{4} + 1} + {\left (x^{16} + 8 \, x^{12} + 12 \, x^{8} + 8 \, x^{4} + \sqrt {2} {\left (x^{16} + 6 \, x^{12} + 8 \, x^{8} + 6 \, x^{4} + 1\right )} + 2 \, {\left (3 \, x^{14} + 7 \, x^{10} + 7 \, x^{6} + 3 \, x^{2} + \sqrt {2} {\left (2 \, x^{14} + 5 \, x^{10} + 5 \, x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {-\sqrt {2} + 2} + 1\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {2} + 2}}}{x^{16} + 1}\right ) + \frac {1}{32} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {-\sqrt {2} + 2}} \log \left (-\frac {2 \, {\left (2 \, x^{13} + 4 \, x^{9} + 4 \, x^{5} + \sqrt {2} {\left (x^{13} + 3 \, x^{9} + 3 \, x^{5} + x\right )} + {\left (4 \, x^{11} + 6 \, x^{7} + 4 \, x^{3} + \sqrt {2} {\left (3 \, x^{11} + 4 \, x^{7} + 3 \, x^{3}\right )}\right )} \sqrt {-\sqrt {2} + 2} + 2 \, x\right )} \sqrt {x^{4} + 1} - {\left (x^{16} + 8 \, x^{12} + 12 \, x^{8} + 8 \, x^{4} + \sqrt {2} {\left (x^{16} + 6 \, x^{12} + 8 \, x^{8} + 6 \, x^{4} + 1\right )} + 2 \, {\left (3 \, x^{14} + 7 \, x^{10} + 7 \, x^{6} + 3 \, x^{2} + \sqrt {2} {\left (2 \, x^{14} + 5 \, x^{10} + 5 \, x^{6} + 2 \, x^{2}\right )}\right )} \sqrt {-\sqrt {2} + 2} + 1\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {2} + 2}}}{x^{16} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \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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