Optimal. Leaf size=171 \[ -\frac {x}{8 \sqrt {-1+x^4}}+\frac {1}{32} \text {ArcTan}\left (\frac {-\frac {1}{2}-x^2+\frac {x^4}{2}}{x \sqrt {-1+x^4}}\right )+\frac {\text {ArcTan}\left (\frac {\frac {1}{2^{3/4}}+\frac {x^2}{\sqrt [4]{2}}-\frac {x^4}{2^{3/4}}}{x \sqrt {-1+x^4}}\right )}{8\ 2^{3/4}}-\frac {1}{32} \tanh ^{-1}\left (\frac {-\frac {1}{2}+x^2+\frac {x^4}{2}}{x \sqrt {-1+x^4}}\right )+\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {-1+x^4}}{-1+\sqrt {2} x^2+x^4}\right )}{8\ 2^{3/4}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.79, antiderivative size = 833, normalized size of antiderivative = 4.87, number of steps
used = 57, number of rules used = 19, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.950, Rules used = {6857, 1166,
425, 434, 438, 435, 259, 228, 418, 1225, 1713, 212, 209, 1443, 1229, 1471, 554, 552, 551}
\begin {gather*} -\frac {x \left (1-x^2\right )}{16 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left (-\sqrt [4]{-1};\text {ArcSin}(x)|-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left (\sqrt [4]{-1};\text {ArcSin}(x)|-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left (-(-1)^{3/4};\text {ArcSin}(x)|-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}+\frac {\sqrt {x^2+1} \Pi \left ((-1)^{3/4};\text {ArcSin}(x)|-1\right ) \sqrt {1-x^2}}{8 \sqrt {x^4-1}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \text {ArcTan}\left (\frac {(1+i) x}{\sqrt {x^4-1}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {x^4-1}}\right )+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{4 \left ((2-2 i)+2 \sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((1+i)+\sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((1-i)+\sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((-1+i)+\sqrt {2}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \left ((-1-i)+\sqrt {2}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {x^4-1}}+\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {x^4-1}}-\frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{4 \sqrt {2} \sqrt {x^4-1}}-\frac {x \left (x^2+1\right )}{16 \sqrt {x^4-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 228
Rule 259
Rule 418
Rule 425
Rule 434
Rule 435
Rule 438
Rule 551
Rule 552
Rule 554
Rule 1166
Rule 1225
Rule 1229
Rule 1443
Rule 1471
Rule 1713
Rule 6857
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {-1+x^4} \left (-1+x^{16}\right )} \, dx &=\int \left (\frac {1}{8 \left (-1+x^2\right ) \sqrt {-1+x^4}}-\frac {1}{8 \left (1+x^2\right ) \sqrt {-1+x^4}}-\frac {1}{4 \sqrt {-1+x^4} \left (1+x^4\right )}+\frac {1}{2 \sqrt {-1+x^4} \left (1+x^8\right )}\right ) \, dx\\ &=\frac {1}{8} \int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{8} \int \frac {1}{\left (1+x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{4} \int \frac {1}{\sqrt {-1+x^4} \left (1+x^4\right )} \, dx+\frac {1}{2} \int \frac {1}{\sqrt {-1+x^4} \left (1+x^8\right )} \, dx\\ &=\frac {1}{4} i \int \frac {1}{\left (i-x^4\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{4} i \int \frac {1}{\sqrt {-1+x^4} \left (i+x^4\right )} \, dx-\frac {1}{8} \int \frac {1}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{8} \int \frac {1}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )^{3/2}} \, dx}{8 \sqrt {-1+x^4}}+\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\left (-1+x^2\right )^{3/2} \sqrt {1+x^2}} \, dx}{8 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-2 \left (\frac {1}{16} \int \frac {1}{\sqrt {-1+x^4}} \, dx\right )-\frac {1}{16} \int \frac {1-i x^2}{\left (1+i x^2\right ) \sqrt {-1+x^4}} \, dx-\frac {1}{16} \int \frac {1+i x^2}{\left (1-i x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {1}{8} \int \frac {1}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx+\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {-1+x^2}}{\sqrt {1+x^2}} \, dx}{16 \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}} \, dx}{16 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \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 )}{8 \sqrt {2} \sqrt {-1+x^4}}-\frac {1}{16} \text {Subst}\left (\int \frac {1}{1-2 i x^2} \, dx,x,\frac {x}{\sqrt {-1+x^4}}\right )-\frac {1}{16} \text {Subst}\left (\int \frac {1}{1+2 i x^2} \, dx,x,\frac {x}{\sqrt {-1+x^4}}\right )+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1-\sqrt [4]{-1}\right )}-\frac {\sqrt [4]{-1} \int \frac {1-x^2}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx}{8 \left (1-\sqrt [4]{-1}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+\sqrt [4]{-1}\right )}+\frac {\sqrt [4]{-1} \int \frac {1-x^2}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx}{8 \left (1+\sqrt [4]{-1}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1-(-1)^{3/4}\right )}+\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}+\frac {(-1)^{3/4} \int \frac {1-x^2}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}+\frac {\int \frac {1-x^2}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right )}-\frac {\left (\sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx}{16 \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}} \, dx}{16 \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}{8 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {1-x^2} \sqrt {1+x^2} E\left (\left .\sin ^{-1}(x)\right |-1\right )}{16 \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 )}{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 )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\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 )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\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 )}{8 \sqrt {2} \left (1-(-1)^{3/4}\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 )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {1}{8} \int \frac {1}{\sqrt {-1+x^4}} \, dx-\frac {\left (\sqrt [4]{-1} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\left (\sqrt [4]{-1} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\left ((-1)^{3/4} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx}{8 \left (1+(-1)^{3/4}\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-(-1)^{3/4} x^2\right )} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\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}} \, dx}{16 \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\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} \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 )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\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 )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\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 )}{8 \sqrt {2} \left (1-(-1)^{3/4}\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 )}{8 \sqrt {2} \left (1+(-1)^{3/4}\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}{8 \left (1-\sqrt [4]{-1}\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}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \left (-1+(-1)^{3/4}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1+\sqrt [4]{-1}\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}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\left ((-1)^{3/4} \left (-1+\sqrt [4]{-1}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \left (1+(-1)^{3/4}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt [4]{-1}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\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} \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 )}{8 \sqrt {2} \left (1-\sqrt [4]{-1}\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 )}{8 \sqrt {2} \left (1+\sqrt [4]{-1}\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 )}{8 \sqrt {2} \left (1-(-1)^{3/4}\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 )}{8 \sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1-\sqrt [4]{-1}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+\sqrt [4]{-1}\right )}-\frac {\int \frac {1}{\sqrt {-1+x^4}} \, dx}{8 \left (1+(-1)^{3/4}\right )}-\frac {\sqrt [4]{-1} \int \frac {1}{\sqrt {-1+x^4}} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right )}-\frac {\left (\sqrt [4]{-1} \left (-1+(-1)^{3/4}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}-\frac {\left ((-1)^{3/4} \left (-1+\sqrt [4]{-1}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\left (\sqrt [4]{-1} \left (1+(-1)^{3/4}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx}{8 \left (1-\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt [4]{-1}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx}{4 \sqrt {2} \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}\\ &=-\frac {x \left (1-x^2\right )}{16 \sqrt {-1+x^4}}-\frac {x \left (1+x^2\right )}{16 \sqrt {-1+x^4}}-\left (\frac {1}{32}-\frac {i}{32}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )-\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} \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 )}{8 \sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-1+x^4}}-\frac {\sqrt [4]{-1} \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 \left ((1+i)+\sqrt {2}\right ) \sqrt {-1+x^4}}+\frac {\sqrt [4]{-1} \left (1-(-1)^{3/4}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right )}{8 \left (1+\sqrt [4]{-1}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}(x)\right |-1\right )}{8 \sqrt {-1+x^4}}+\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1}\right ) \sqrt {1-x^2} \sqrt {1+x^2} \Pi \left (-(-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right )}{8 \left (1+(-1)^{3/4}\right ) \sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \Pi \left ((-1)^{3/4};\left .\sin ^{-1}(x)\right |-1\right )}{8 \sqrt {-1+x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.57, size = 143, normalized size = 0.84 \begin {gather*} \frac {1}{32} \left (-\frac {4 x}{\sqrt {-1+x^4}}-(1-i) \text {ArcTan}\left (\frac {(1+i) x}{\sqrt {-1+x^4}}\right )+(1+i) \text {ArcTan}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^4}}{x}\right )-2 \sqrt [4]{2} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt {-1+x^4}}{1+\sqrt {2} x^2-x^4}\right )+2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {-1+x^4}}{-1+\sqrt {2} x^2+x^4}\right )\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
3.
time = 0.68, size = 282, normalized size = 1.65
method | result | size |
risch | \(-\frac {x}{8 \sqrt {x^{4}-1}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{8 \sqrt {x^{4}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}\, \sqrt {x^{4}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}}-\frac {2 i \underline {\hspace {1.25 ex}}\alpha ^{7} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{6}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{32}+\frac {i \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) \sqrt {-2}}{2 \sqrt {x^{4}-1}}\right )-\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{2}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{64}\) | \(214\) |
elliptic | \(\frac {\left (-\frac {2^{\frac {3}{4}} \arctan \left (\frac {\sqrt {x^{4}-1}\, 2^{\frac {1}{4}}}{x}+1\right )}{16}-\frac {2^{\frac {3}{4}} \arctan \left (\frac {\sqrt {x^{4}-1}\, 2^{\frac {1}{4}}}{x}-1\right )}{16}-\frac {2^{\frac {3}{4}} \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {2^{\frac {3}{4}} \sqrt {x^{4}-1}}{2 x}+\frac {\sqrt {2}}{2}}{\frac {x^{4}-1}{2 x^{2}}+\frac {2^{\frac {3}{4}} \sqrt {x^{4}-1}}{2 x}+\frac {\sqrt {2}}{2}}\right )}{32}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}+1\right )}{32}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}-1\right )}{32}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {x^{4}-1}{2 x^{2}}-\frac {\sqrt {x^{4}-1}}{x}+1}{\frac {x^{4}-1}{2 x^{2}}+\frac {\sqrt {x^{4}-1}}{x}+1}\right )}{64}-\frac {\sqrt {2}\, x}{8 \sqrt {x^{4}-1}}\right ) \sqrt {2}}{2}\) | \(229\) |
default | \(\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-i \sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) \sqrt {-2}}{2 \sqrt {x^{4}-1}}\right )-\frac {4 i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{2}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{64}-\frac {x^{3}-x^{2}+x -1}{32 \sqrt {\left (1+x \right ) \left (x^{3}-x^{2}+x -1\right )}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{8 \sqrt {x^{4}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}\, \sqrt {x^{4}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}}-\frac {2 i \underline {\hspace {1.25 ex}}\alpha ^{7} \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{6}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{32}-\frac {x^{3}+x^{2}+x +1}{32 \sqrt {\left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )}}+\frac {\left (x^{2}-1\right ) x}{16 \sqrt {\left (x^{2}-1\right ) \left (x^{2}+1\right )}}\) | \(282\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 740 vs.
\(2 (132) = 264\).
time = 0.55, size = 740, normalized size = 4.33 \begin {gather*} \frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \arctan \left (\frac {2 \, x^{16} + 4 \, x^{8} + \sqrt {2} {\left (2^{\frac {3}{4}} {\left (x^{16} - 20 \, x^{12} + 34 \, x^{8} - 20 \, x^{4} + 1\right )} + 8 \, {\left (x^{11} + x^{3} + 4 \, \sqrt {2} {\left (x^{9} - x^{5}\right )}\right )} \sqrt {x^{4} - 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{14} - 9 \, x^{10} + 9 \, x^{6} - x^{2}\right )}\right )} \sqrt {\frac {8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} + 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}}{x^{8} + 1}} + 8 \, \sqrt {2} {\left (x^{14} - x^{10} + x^{6} - x^{2}\right )} + 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} {\left (x^{13} - 9 \, x^{9} + 9 \, x^{5} - x\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{11} - 8 \, x^{7} + 3 \, x^{3}\right )}\right )} + 2}{2 \, {\left (x^{16} - 32 \, x^{12} + 66 \, x^{8} - 32 \, x^{4} + 1\right )}}\right ) - 4 \cdot 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \arctan \left (\frac {2 \, x^{16} + 4 \, x^{8} - \sqrt {2} {\left (2^{\frac {3}{4}} {\left (x^{16} - 20 \, x^{12} + 34 \, x^{8} - 20 \, x^{4} + 1\right )} - 8 \, {\left (x^{11} + x^{3} + 4 \, \sqrt {2} {\left (x^{9} - x^{5}\right )}\right )} \sqrt {x^{4} - 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{14} - 9 \, x^{10} + 9 \, x^{6} - x^{2}\right )}\right )} \sqrt {\frac {8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} - 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}}{x^{8} + 1}} + 8 \, \sqrt {2} {\left (x^{14} - x^{10} + x^{6} - x^{2}\right )} - 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} {\left (x^{13} - 9 \, x^{9} + 9 \, x^{5} - x\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{11} - 8 \, x^{7} + 3 \, x^{3}\right )}\right )} + 2}{2 \, {\left (x^{16} - 32 \, x^{12} + 66 \, x^{8} - 32 \, x^{4} + 1\right )}}\right ) + 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {8 \, {\left (8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} + 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}\right )}}{x^{8} + 1}\right ) - 2^{\frac {1}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {8 \, {\left (8 \, x^{6} - 8 \, x^{2} + \sqrt {2} {\left (x^{8} + 1\right )} - 4 \, \sqrt {x^{4} - 1} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - x\right )}\right )}\right )}}{x^{8} + 1}\right ) + 4 \, {\left (x^{4} - 1\right )} \arctan \left (\frac {\sqrt {x^{4} - 1} x}{x^{2} + 1}\right ) + 2 \, {\left (x^{4} - 1\right )} \log \left (\frac {x^{4} + 2 \, x^{2} - 2 \, \sqrt {x^{4} - 1} x - 1}{x^{4} + 1}\right ) - 8 \, \sqrt {x^{4} - 1} x}{64 \, {\left (x^{4} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right ) \left (x^{8} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^8}{\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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