Optimal. Leaf size=128 \[ -\sqrt {\frac {1}{2} \left (3-i \sqrt {3}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {3}{2}-\frac {i \sqrt {3}}{2}} x}{\sqrt {-1+x^2+x^4}}\right )-\sqrt {\frac {1}{2} \left (3+i \sqrt {3}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {3}{2}+\frac {i \sqrt {3}}{2}} x}{\sqrt {-1+x^2+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 1.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{\left (-1+x^4\right ) \left (1+x^2-x^4-x^6+x^8\right )} \, dx &=\int \left (\frac {\left (-1+x^2+x^4\right )^{3/2}}{-1-x^2}+\frac {\left (-1+x^2+x^4\right )^{3/2}}{-1+x^2}+\frac {\left (1+2 x^2-2 x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}\right ) \, dx\\ &=\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{-1-x^2} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{-1+x^2} \, dx+\int \frac {\left (1+2 x^2-2 x^4\right ) \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=-\int x^2 \sqrt {-1+x^2+x^4} \, dx-\int \left (-2-x^2\right ) \sqrt {-1+x^2+x^4} \, dx-\int \frac {\sqrt {-1+x^2+x^4}}{-1-x^2} \, dx+\int \frac {\sqrt {-1+x^2+x^4}}{-1+x^2} \, dx+\int \left (\frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}+\frac {2 x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}-\frac {2 x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8}\right ) \, dx\\ &=-\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}-\frac {1}{15} \int \frac {19-2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+\frac {1}{15} \int \frac {-1+8 x^2}{\sqrt {-1+x^2+x^4}} \, dx+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx+\int \frac {x^2}{\sqrt {-1+x^2+x^4}} \, dx-\int \frac {-2-x^2}{\sqrt {-1+x^2+x^4}} \, dx+\int \frac {1}{\left (-1-x^2\right ) \sqrt {-1+x^2+x^4}} \, dx+\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1+x^2+x^4}} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=-\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+\frac {4}{15} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx+2 \left (\frac {1}{2} \int \frac {1-\sqrt {5}+2 x^2}{\sqrt {-1+x^2+x^4}} \, dx\right )+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-\frac {1}{2} \left (-3-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {1}{15} \left (20-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {\int \frac {1-\sqrt {5}+2 x^2}{\left (-1+x^2\right ) \sqrt {-1+x^2+x^4}} \, dx}{-3+\sqrt {5}}+\frac {2 \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx}{-3+\sqrt {5}}+\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx-\frac {\int \frac {1-\sqrt {5}+2 x^2}{\left (-1-x^2\right ) \sqrt {-1+x^2+x^4}} \, dx}{1+\sqrt {5}}-\frac {2 \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx}{1+\sqrt {5}}+\frac {1}{15} \left (-5+4 \sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^2+x^4}} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-\frac {\left (\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {\sqrt {1-\sqrt {5}+2 x^2}}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right )} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {\sqrt {1-\sqrt {5}+2 x^2}}{\left (-1-x^2\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-\frac {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (3-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\frac {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (-1-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}\right ) \int \frac {1}{\left (-1-x^2\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1-\sqrt {5}+2 x^2}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-\frac {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (3-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (-3+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\frac {\left (2 \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (\left (-1-\sqrt {5}\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}\right ) \int \frac {1}{\left (-1-x^2\right ) \sqrt {-\frac {1}{1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}}} \, dx}{\left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ &=\frac {x \left (1+\sqrt {5}+2 x^2\right )}{3 \sqrt {-1+x^2+x^4}}-\frac {1}{15} x \left (1+3 x^2\right ) \sqrt {-1+x^2+x^4}+\frac {1}{15} x \left (11+3 x^2\right ) \sqrt {-1+x^2+x^4}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{3 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+2 \left (\frac {x \left (1+\sqrt {5}+2 x^2\right )}{2 \sqrt {-1+x^2+x^4}}-\frac {\sqrt [4]{5} \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{2 \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}\right )-\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{\sqrt {2} \left (1+\sqrt {5}\right ) \sqrt {-1+x^2+x^4}}+\frac {\left (1-4 \sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\left (1-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (3-\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (4-\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{6\ 5^{3/4} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}-\frac {\sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1+\sqrt {5}\right ) \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {\frac {2-\left (1-\sqrt {5}\right ) x^2}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {5}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} x}{\sqrt {-2+\left (1+\sqrt {5}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {5}\right )\right )}{4 \sqrt [4]{5} \sqrt {\frac {1}{2-\left (1+\sqrt {5}\right ) x^2}} \sqrt {-1+x^2+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} \Pi \left (\frac {1}{2} \left (1-\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{2 \sqrt {2} \sqrt {-1+x^2+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{1-\sqrt {5}}} \Pi \left (\frac {1}{2} \left (-1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )}{2 \sqrt {2} \sqrt {-1+x^2+x^4}}+2 \int \frac {x^2 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx-2 \int \frac {x^4 \left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx+\int \frac {\left (-1+x^2+x^4\right )^{3/2}}{1+x^2-x^4-x^6+x^8} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.04, size = 124, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (-\sqrt {6-2 i \sqrt {3}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (-3-i \sqrt {3}\right )} x}{\sqrt {-1+x^2+x^4}}\right )-\sqrt {6+2 i \sqrt {3}} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} i \left (3 i+\sqrt {3}\right )} x}{\sqrt {-1+x^2+x^4}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2+x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.69, size = 871, normalized size = 6.80
method | result | size |
trager | \(\frac {\ln \left (-\frac {-x^{4}+2 x \sqrt {x^{4}+x^{2}-1}-2 x^{2}+1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \ln \left (-\frac {16 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{4} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{4}+8 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) x^{4}+24 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \sqrt {x^{4}+x^{2}-1}\, x +4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )-3 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right ) x^{2}+6 x \sqrt {x^{4}+x^{2}-1}+3 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+3\right )}{4 x^{2} \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}+x^{4}+x^{2}-1}\right )}{2}+\RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right ) \ln \left (-\frac {8 x^{2} \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{5}+2 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{4}+8 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3} x^{2}-6 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2} \sqrt {x^{4}+x^{2}-1}\, x -2 \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{3}-3 x \sqrt {x^{4}+x^{2}-1}}{4 x^{2} \RootOf \left (16 \textit {\_Z}^{4}+12 \textit {\_Z}^{2}+3\right )^{2}-x^{4}+2 x^{2}+1}\right )\) | \(569\) |
elliptic | \(\frac {\left (\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}\, \sqrt {3}}{4}-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}-\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}-\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}\, \sqrt {3}}{4}+\frac {\ln \left (\frac {x^{4}+x^{2}-1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}-1}\, \sqrt {2}\, \sqrt {6+4 \sqrt {3}}}{2 x}+\sqrt {3}\right ) \sqrt {6+4 \sqrt {3}}}{2}+\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right ) \sqrt {3}}{2 \sqrt {4 \sqrt {3}-6}}-\frac {\arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \left (6+4 \sqrt {3}\right )}{\sqrt {4 \sqrt {3}-6}}+\frac {2 \arctan \left (\frac {\frac {2 \sqrt {x^{4}+x^{2}-1}\, \sqrt {2}}{x}+\sqrt {6+4 \sqrt {3}}}{\sqrt {4 \sqrt {3}-6}}\right ) \sqrt {3}}{\sqrt {4 \sqrt {3}-6}}-\sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+x^{2}-1}}{x}\right )\right ) \sqrt {2}}{2}\) | \(599\) |
default | \(\text {Expression too large to display}\) | \(871\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4669 vs.
\(2 (84) = 168\).
time = 1.99, size = 4669, normalized size = 36.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^4+1\right )\,{\left (x^4+x^2-1\right )}^{3/2}}{\left (x^4-1\right )\,\left (x^8-x^6-x^4+x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________