Optimal. Leaf size=47 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+2 x^2-1}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+2 x^2-1}}\right ) \]
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Rubi [C] time = 2.61, antiderivative size = 1670, normalized size of antiderivative = 35.53, number of steps used = 52, number of rules used = 13, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {6725, 1208, 1187, 1098, 1184, 1214, 1456, 540, 421, 419, 538, 537, 6728}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 419
Rule 421
Rule 537
Rule 538
Rule 540
Rule 1098
Rule 1184
Rule 1187
Rule 1208
Rule 1214
Rule 1456
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right ) \sqrt {-1+2 x^2+x^4}}{\left (-1+x^4\right ) \left (-1+x^2+x^4\right )} \, dx &=\int \left (\frac {\sqrt {-1+2 x^2+x^4}}{-1+x^2}+\frac {\sqrt {-1+2 x^2+x^4}}{1+x^2}+\frac {\left (-1-2 x^2\right ) \sqrt {-1+2 x^2+x^4}}{-1+x^2+x^4}\right ) \, dx\\ &=\int \frac {\sqrt {-1+2 x^2+x^4}}{-1+x^2} \, dx+\int \frac {\sqrt {-1+2 x^2+x^4}}{1+x^2} \, dx+\int \frac {\left (-1-2 x^2\right ) \sqrt {-1+2 x^2+x^4}}{-1+x^2+x^4} \, dx\\ &=2 \int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx-2 \int \frac {1}{\left (1+x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx-\int \frac {-3-x^2}{\sqrt {-1+2 x^2+x^4}} \, dx-\int \frac {-1-x^2}{\sqrt {-1+2 x^2+x^4}} \, dx+\int \left (-\frac {2 \sqrt {-1+2 x^2+x^4}}{1-\sqrt {5}+2 x^2}-\frac {2 \sqrt {-1+2 x^2+x^4}}{1+\sqrt {5}+2 x^2}\right ) \, dx\\ &=2 \left (\frac {1}{2} \int \frac {2-2 \sqrt {2}+2 x^2}{\sqrt {-1+2 x^2+x^4}} \, dx\right )-2 \int \frac {\sqrt {-1+2 x^2+x^4}}{1-\sqrt {5}+2 x^2} \, dx-2 \int \frac {\sqrt {-1+2 x^2+x^4}}{1+\sqrt {5}+2 x^2} \, dx+\frac {\int \frac {2-2 \sqrt {2}+2 x^2}{\left (1+x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx}{\sqrt {2}}+\left (2+\sqrt {2}\right ) \int \frac {1}{\sqrt {-1+2 x^2+x^4}} \, dx-\frac {2 \int \frac {2-2 \sqrt {2}+2 x^2}{\left (-1+x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx}{-4+2 \sqrt {2}}+\frac {4 \int \frac {1}{\sqrt {-1+2 x^2+x^4}} \, dx}{-4+2 \sqrt {2}}\\ &=2 \left (\frac {x \left (1+\sqrt {2}+x^2\right )}{\sqrt {-1+2 x^2+x^4}}-\frac {2^{3/4} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} E\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}\right )-\frac {\sqrt [4]{2} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\left (2-\sqrt {2}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {1}{2} \int \frac {-3-\sqrt {5}-2 x^2}{\sqrt {-1+2 x^2+x^4}} \, dx+\frac {1}{2} \int \frac {-3+\sqrt {5}-2 x^2}{\sqrt {-1+2 x^2+x^4}} \, dx-\left (-1-\sqrt {5}\right ) \int \frac {1}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx-\left (-1+\sqrt {5}\right ) \int \frac {1}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx+\frac {\left (\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {\sqrt {2-2 \sqrt {2}+2 x^2}}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (1+x^2\right )} \, dx}{\sqrt {2} \sqrt {-1+2 x^2+x^4}}-\frac {\left (2 \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {\sqrt {2-2 \sqrt {2}+2 x^2}}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (-1+x^2\right )} \, dx}{\left (-4+2 \sqrt {2}\right ) \sqrt {-1+2 x^2+x^4}}\\ &=2 \left (\frac {x \left (1+\sqrt {2}+x^2\right )}{\sqrt {-1+2 x^2+x^4}}-\frac {2^{3/4} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} E\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}\right )-\frac {\sqrt [4]{2} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\left (2-\sqrt {2}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-2 \left (\frac {1}{2} \int \frac {2-2 \sqrt {2}+2 x^2}{\sqrt {-1+2 x^2+x^4}} \, dx\right )+\frac {1}{2} \left (-1-2 \sqrt {2}-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+2 x^2+x^4}} \, dx-\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+2 x^2+x^4}} \, dx}{1-2 \sqrt {2}-\sqrt {5}}+\frac {\left (1+\sqrt {5}\right ) \int \frac {2-2 \sqrt {2}+2 x^2}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx}{1-2 \sqrt {2}-\sqrt {5}}+\frac {1}{2} \left (-1-2 \sqrt {2}+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+2 x^2+x^4}} \, dx-\frac {\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+2 x^2+x^4}} \, dx}{1-2 \sqrt {2}+\sqrt {5}}+\frac {\left (1-\sqrt {5}\right ) \int \frac {2-2 \sqrt {2}+2 x^2}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+2 x^2+x^4}} \, dx}{1-2 \sqrt {2}+\sqrt {5}}-\frac {\left (2 \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (1+x^2\right ) \sqrt {2-2 \sqrt {2}+2 x^2}} \, dx}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\sqrt {2} \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}} \, dx}{\sqrt {-1+2 x^2+x^4}}-\frac {\left (4 \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}} \, dx}{\left (-4+2 \sqrt {2}\right ) \sqrt {-1+2 x^2+x^4}}-\frac {\left (4 \left (2-\sqrt {2}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {2-2 \sqrt {2}+2 x^2}} \, dx}{\left (-4+2 \sqrt {2}\right ) \sqrt {-1+2 x^2+x^4}}\\ &=-\frac {\sqrt [4]{2} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\left (2-\sqrt {2}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {\sqrt {2-2 \sqrt {2}+2 x^2}}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {\sqrt {2-2 \sqrt {2}+2 x^2}}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}-\frac {\left (2 \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (1+x^2\right ) \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\sqrt {2} \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\sqrt {-1+2 x^2+x^4}}-\frac {\left (4 \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\left (-4+2 \sqrt {2}\right ) \sqrt {-1+2 x^2+x^4}}-\frac {\left (4 \left (2-\sqrt {2}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\left (-4+2 \sqrt {2}\right ) \sqrt {-1+2 x^2+x^4}}\\ &=\frac {\sqrt {2} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\sqrt {2 \left (3-2 \sqrt {2}\right )} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}-\frac {\sqrt [4]{2} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\left (2-\sqrt {2}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {2 \sqrt {3-2 \sqrt {2}} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (1-\sqrt {2};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}-\frac {2 \sqrt {2 \left (\frac {3}{2}-\sqrt {2}\right )} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (-1+\sqrt {2};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}} \, dx}{\left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {2-2 \sqrt {2}+2 x^2}} \, dx}{\left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}\\ &=\frac {\sqrt {2} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\sqrt {2 \left (3-2 \sqrt {2}\right )} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}-\frac {\sqrt [4]{2} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\left (2-\sqrt {2}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {2 \sqrt {3-2 \sqrt {2}} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (1-\sqrt {2};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}-\frac {2 \sqrt {2 \left (\frac {3}{2}-\sqrt {2}\right )} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (-1+\sqrt {2};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (1-\sqrt {5}+2 x^2\right ) \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{2-2 \sqrt {2}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{2-2 \sqrt {2}}}} \, dx}{\left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}\\ &=\frac {\sqrt {2} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\sqrt {2 \left (3-2 \sqrt {2}\right )} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\sqrt {3-2 \sqrt {2}} \left (1+\sqrt {5}\right ) \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}+\frac {\sqrt {3-2 \sqrt {2}} \left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} F\left (\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {-1+2 x^2+x^4}}-\frac {\sqrt [4]{2} \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\left (2-\sqrt {2}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}-\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2^{3/4} \left (1-2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {\left (1+2 \sqrt {2}+\sqrt {5}\right ) \sqrt {\frac {1-\left (1-\sqrt {2}\right ) x^2}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+\left (1+\sqrt {2}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-1+\left (1+\sqrt {2}\right ) x^2}}\right )|\frac {1}{4} \left (2+\sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {\frac {1}{1-\left (1+\sqrt {2}\right ) x^2}} \sqrt {-1+2 x^2+x^4}}-\frac {2 \sqrt {3-2 \sqrt {2}} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (1-\sqrt {2};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}-\frac {2 \sqrt {2 \left (\frac {3}{2}-\sqrt {2}\right )} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (-1+\sqrt {2};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\sqrt {3-2 \sqrt {2}} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (\frac {2 \left (1-\sqrt {2}\right )}{1-\sqrt {5}};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}+\frac {\sqrt {3-2 \sqrt {2}} \sqrt {1+\sqrt {2}+x^2} \sqrt {1+\frac {x^2}{1-\sqrt {2}}} \Pi \left (\frac {2 \left (1-\sqrt {2}\right )}{1+\sqrt {5}};\sin ^{-1}\left (\sqrt {1+\sqrt {2}} x\right )|-3+2 \sqrt {2}\right )}{\sqrt {-1+2 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 5.04, size = 1280, normalized size = 27.23 \begin {gather*} \frac {(x-1) (x+1) \left (x^2+1\right ) \left (x^4+x^2-1\right ) \left (-\frac {4 \left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) \sqrt {-\frac {x+i \sqrt {1+\sqrt {2}}}{i \sqrt {1+\sqrt {2}} x-\sqrt {-1+\sqrt {2}} x+\sqrt {2}-(1+i)}} \sqrt {\frac {i \sqrt {1+\sqrt {2}}-x}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}} \sqrt {\frac {\left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right ) x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}} \left (\sqrt {-1+\sqrt {2}} \left (\Pi \left (\frac {\left (-1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right )}{\left (1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right )};\left .\sin ^{-1}\left (\sqrt {\frac {-i \sqrt {1+\sqrt {2}} x+\sqrt {-1+\sqrt {2}} x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}}\right )\right |-i\right )-\Pi \left (\frac {\left (1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right )}{\left (-1+\sqrt {-1+\sqrt {2}}\right ) \left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right )};\left .\sin ^{-1}\left (\sqrt {\frac {-i \sqrt {1+\sqrt {2}} x+\sqrt {-1+\sqrt {2}} x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}}\right )\right |-i\right )\right )-F\left (\left .\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right ) x+\sqrt {2}-(1+i)}{-\left (\left (\sqrt {-1+\sqrt {2}}+i \sqrt {1+\sqrt {2}}\right ) x\right )+\sqrt {2}-(1-i)}}\right )\right |-i\right )\right ) \left (\sqrt {-1+\sqrt {2}}-x\right )^2}{\left (-2+\sqrt {2}\right ) \left (\sqrt {-1+\sqrt {2}}-i \sqrt {1+\sqrt {2}}\right )}-i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} F\left (i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )+2 i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} \Pi \left (1+\sqrt {2};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )-\frac {i \sqrt {5 \left (1+\sqrt {2}\right )} \sqrt {-x^4-2 x^2+1} \Pi \left (-\frac {2 \left (1+\sqrt {2}\right )}{-1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{-3+\sqrt {5}}+\frac {3 i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} \Pi \left (-\frac {2 \left (1+\sqrt {2}\right )}{-1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{-3+\sqrt {5}}-\frac {i \sqrt {5 \left (1+\sqrt {2}\right )} \sqrt {-x^4-2 x^2+1} \Pi \left (\frac {2 \left (1+\sqrt {2}\right )}{1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{3+\sqrt {5}}-\frac {3 i \sqrt {1+\sqrt {2}} \sqrt {-x^4-2 x^2+1} \Pi \left (\frac {2 \left (1+\sqrt {2}\right )}{1+\sqrt {5}};i \sinh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )|-3-2 \sqrt {2}\right )}{3+\sqrt {5}}\right )}{\sqrt {x^4+2 x^2-1} \left (x^8+x^6-2 x^4-x^2+1\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.39, size = 47, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {-1+2 x^2+x^4}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+2 x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 109, normalized size = 2.32 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {x^{8} + 16 \, x^{6} + 30 \, x^{4} - 4 \, \sqrt {2} {\left (x^{5} + 4 \, x^{3} - x\right )} \sqrt {x^{4} + 2 \, x^{2} - 1} - 16 \, x^{2} + 1}{x^{8} - 2 \, x^{4} + 1}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} + 3 \, x^{2} + 2 \, \sqrt {x^{4} + 2 \, x^{2} - 1} x - 1}{x^{4} + x^{2} - 1}\right ) \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*} \int \frac {\sqrt {x^{4} + 2 \, x^{2} - 1} {\left (x^{4} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 76, normalized size = 1.62
method | result | size |
elliptic | \(\frac {\left (-\ln \left (1+\frac {\sqrt {x^{4}+2 x^{2}-1}\, \sqrt {2}}{2 x}\right )+\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+2 x^{2}-1}}{x}\right )+\ln \left (-1+\frac {\sqrt {x^{4}+2 x^{2}-1}\, \sqrt {2}}{2 x}\right )\right ) \sqrt {2}}{2}\) | \(76\) |
trager | \(\frac {\ln \left (-\frac {x^{4}+2 \sqrt {x^{4}+2 x^{2}-1}\, x +3 x^{2}-1}{x^{4}+x^{2}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{4}+2 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\) | \(114\) |
default | \(-\frac {2 \sqrt {1-\left (1-\sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (1+\sqrt {2}\right ) x^{2}}\, \left (\EllipticF \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )-\EllipticE \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}\, \left (2+2 \sqrt {2}\right )}-\frac {2 \sqrt {1-\left (1-\sqrt {2}\right ) x^{2}}\, \sqrt {1-\left (1+\sqrt {2}\right ) x^{2}}\, \EllipticPi \left (x \sqrt {1-\sqrt {2}}, \frac {1}{1-\sqrt {2}}, \frac {\sqrt {1+\sqrt {2}}}{\sqrt {1-\sqrt {2}}}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}+\frac {\sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticF \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}+\frac {2 \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticF \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}\, \left (2+2 \sqrt {2}\right )}-\frac {2 \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticE \left (x \sqrt {1-\sqrt {2}}, i+i \sqrt {2}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}\, \left (2+2 \sqrt {2}\right )}-\frac {2 \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticPi \left (x \sqrt {1-\sqrt {2}}, -\frac {1}{1-\sqrt {2}}, \frac {\sqrt {1+\sqrt {2}}}{\sqrt {1-\sqrt {2}}}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}+1\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+2 x^{2}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {\sqrt {2}\, x^{2}-x^{2}+1}\, \sqrt {-\sqrt {2}\, x^{2}-x^{2}+1}\, \EllipticPi \left (x \sqrt {1-\sqrt {2}}, -\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {2}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {2}-1, \frac {\sqrt {1+\sqrt {2}}}{\sqrt {1-\sqrt {2}}}\right )}{\sqrt {1-\sqrt {2}}\, \sqrt {x^{4}+2 x^{2}-1}}\right )\right )}{4}\) | \(712\) |
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 {\sqrt {x^{4} + 2 \, x^{2} - 1} {\left (x^{4} + 1\right )}}{{\left (x^{4} + x^{2} - 1\right )} {\left (x^{4} - 1\right )}}\,{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 {\left (x^4+1\right )\,\sqrt {x^4+2\,x^2-1}}{\left (x^4-1\right )\,\left (x^4+x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{4} + 1\right ) \sqrt {x^{4} + 2 x^{2} - 1}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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