Optimal. Leaf size=53 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {x^4-2}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {x^4-2}}\right )}{2 \sqrt [4]{2}} \]
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Rubi [C] time = 1.43, antiderivative size = 647, normalized size of antiderivative = 12.21, number of steps used = 40, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6728, 406, 223, 409, 1215, 1457, 540, 253, 538, 537} \begin {gather*} \frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {x^2+\sqrt {2}}{\sqrt {2}-x^2}} \sqrt {\sqrt {2} x^2-2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {\sqrt {2} x^2-2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {x^4-2}}-\frac {\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \Pi \left (-\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{4 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {x^4-2}}-\frac {\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \Pi \left (\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{4 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {x^4-2}}-\frac {\left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \Pi \left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}}-\frac {\left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {\sqrt {2} x^2+2} \Pi \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {x^4-2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 223
Rule 253
Rule 406
Rule 409
Rule 537
Rule 538
Rule 540
Rule 1215
Rule 1457
Rule 6728
Rubi steps
\begin {align*} \int \frac {\sqrt {-2+x^4} \left (2+x^4\right )}{4-6 x^4+x^8} \, dx &=\int \left (\frac {\left (1+\sqrt {5}\right ) \sqrt {-2+x^4}}{-6-2 \sqrt {5}+2 x^4}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-2+x^4}}{-6+2 \sqrt {5}+2 x^4}\right ) \, dx\\ &=\left (1-\sqrt {5}\right ) \int \frac {\sqrt {-2+x^4}}{-6+2 \sqrt {5}+2 x^4} \, dx+\left (1+\sqrt {5}\right ) \int \frac {\sqrt {-2+x^4}}{-6-2 \sqrt {5}+2 x^4} \, dx\\ &=\frac {1}{2} \left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-2+x^4}} \, dx+\left (2 \left (3-\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {-2+x^4} \left (-6+2 \sqrt {5}+2 x^4\right )} \, dx+\frac {1}{2} \left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-2+x^4}} \, dx+\left (2 \left (3+\sqrt {5}\right )\right ) \int \frac {1}{\sqrt {-2+x^4} \left (-6-2 \sqrt {5}+2 x^4\right )} \, dx\\ &=\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx\\ &=\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2 \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right )}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2 \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right )}-\frac {\int \frac {\sqrt {2}-x^2}{\left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right )}-\frac {\int \frac {\sqrt {2}-x^2}{\left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right ) \sqrt {-2+x^4}} \, dx}{2 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right )}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2-\sqrt {2 \left (3+\sqrt {5}\right )}}+\frac {\sqrt {3+\sqrt {5}} \int \frac {\sqrt {2}-x^2}{\left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right )}-\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2+\sqrt {2 \left (3+\sqrt {5}\right )}}-\frac {\sqrt {3+\sqrt {5}} \int \frac {\sqrt {2}-x^2}{\left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right ) \sqrt {-2+x^4}} \, dx}{2 \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}\\ &=\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}-\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}+\frac {\left (\sqrt {3+\sqrt {5}} \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{2 \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\sqrt {3+\sqrt {5}} \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {\sqrt {\sqrt {2}-x^2}}{\sqrt {-\sqrt {2}-x^2} \left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{2 \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}\\ &=\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2} \left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \sqrt {-2+x^4}}-\frac {\left (\sqrt {3+\sqrt {5}} \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}+\frac {\left (\left (-\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2} \left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}+\frac {\left (\sqrt {3+\sqrt {5}} \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, dx}{2 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}+\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, dx}{\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\left (1-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2} \left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}+\frac {\left (\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}} \, dx}{\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\left (1+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {2}-x^2} \sqrt {\sqrt {2}-x^2} \left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}\\ &=\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{2} \left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {3+\sqrt {5}} \int \frac {1}{\sqrt {-2+x^4}} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right )}+\frac {\sqrt {3+\sqrt {5}} \int \frac {1}{\sqrt {-2+x^4}} \, dx}{2 \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right )}+\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2-\sqrt {2 \left (3+\sqrt {5}\right )}}+\frac {\int \frac {1}{\sqrt {-2+x^4}} \, dx}{2+\sqrt {2 \left (3+\sqrt {5}\right )}}-\frac {\left (\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}} \left (1+\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \sqrt {-2+x^4}}+\frac {\left (\left (-\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}} \left (1-\frac {x^2}{\sqrt {3+\sqrt {5}}}\right )} \, dx}{2 \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {-2+x^4}}-\frac {\left (\left (1-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}} \left (1-\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2-\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}-\frac {\left (\left (1+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}\right ) \sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}}\right ) \int \frac {1}{\sqrt {\sqrt {2}-x^2} \sqrt {1+\frac {x^2}{\sqrt {2}}} \left (1+\frac {1}{2} \sqrt {3+\sqrt {5}} x^2\right )} \, dx}{\left (2+\sqrt {2 \left (3+\sqrt {5}\right )}\right ) \sqrt {-2+x^4}}\\ &=\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1-\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (1+\sqrt {\frac {2}{3+\sqrt {5}}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (\sqrt {2}-\sqrt {3+\sqrt {5}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}+\frac {\sqrt {3+\sqrt {5}} \sqrt {\frac {\sqrt {2}+x^2}{\sqrt {2}-x^2}} \sqrt {-2+\sqrt {2} x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} x}{\sqrt {-2+\sqrt {2} x^2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} \left (\sqrt {2}+\sqrt {3+\sqrt {5}}\right ) \sqrt {\frac {1}{2-\sqrt {2} x^2}} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (-\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}-\frac {\sqrt {\sqrt {2}-x^2} \sqrt {2+\sqrt {2} x^2} \Pi \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )}{2 \sqrt {2} \sqrt {-2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 154, normalized size = 2.91 \begin {gather*} \frac {\sqrt {2-x^4} \left (2 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (-\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (\sqrt {\frac {2}{3+\sqrt {5}}};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )-\Pi \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )};\left .\sin ^{-1}\left (\frac {x}{\sqrt [4]{2}}\right )\right |-1\right )\right )}{2 \sqrt [4]{2} \sqrt {x^4-2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 53, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt {-2+x^4}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 226, normalized size = 4.26 \begin {gather*} \frac {1}{4} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} - 2 \, x^{2}\right )} + 2^{\frac {1}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )}\right )} - 4 \, \sqrt {x^{4} - 2} {\left (2^{\frac {3}{4}} x^{3} + 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x\right )}\right )}}{2 \, {\left (x^{8} - 6 \, x^{4} + 4\right )}}\right ) - \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} + 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{8} - 2 \, x^{4} + 4\right )} - 4 \, {\left (x^{5} + \sqrt {2} x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} - 2 \, x^{2}\right )}}{x^{8} - 6 \, x^{4} + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.15, size = 79, normalized size = 1.49
method | result | size |
default | \(\frac {\left (\frac {2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )}{2}-\frac {2^{\frac {1}{4}} \ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )}{4}\right ) \sqrt {2}}{2}\) | \(79\) |
elliptic | \(\frac {\left (\frac {2^{\frac {1}{4}} \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}-2}}{2 x}\right )}{2}-\frac {2^{\frac {1}{4}} \ln \left (\frac {\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {x^{4}-2}\, \sqrt {2}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )}{4}\right ) \sqrt {2}}{2}\) | \(79\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 x^{4} \RootOf \left (\textit {\_Z}^{4}-8\right )+8 \sqrt {x^{4}-2}\, x -4 \RootOf \left (\textit {\_Z}^{4}-8\right )}{\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x^{4}+4}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+8 \sqrt {x^{4}-2}\, x +4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2 x^{4}-4}\right )}{8}\) | \(182\) |
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 {{\left (x^{4} + 2\right )} \sqrt {x^{4} - 2}}{x^{8} - 6 \, x^{4} + 4}\,{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 {\sqrt {x^4-2}\,\left (x^4+2\right )}{x^8-6\,x^4+4} \,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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