Integrand size = 28, antiderivative size = 26 \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {\sqrt {-1+x^4}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^4}}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.63 (sec) , antiderivative size = 291, normalized size of antiderivative = 11.19, number of steps used = 32, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {6860, 283, 312, 228, 1199, 1223, 1202, 1229, 1471, 554, 259, 552, 551} \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {\left (1+\sqrt {5}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4-1}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {x^4-1}}-\frac {\sqrt {2} \sqrt {x^2-1} \sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right ),\frac {1}{2}\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-\sqrt {5}\right ),\arcsin (x),-1\right )}{\sqrt {x^4-1}}+\frac {\sqrt {1-x^2} \sqrt {x^2+1} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin (x),-1\right )}{\sqrt {x^4-1}}+\frac {\sqrt {x^4-1}}{x} \]
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Rule 228
Rule 259
Rule 283
Rule 312
Rule 551
Rule 552
Rule 554
Rule 1199
Rule 1202
Rule 1223
Rule 1229
Rule 1471
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {-1+x^4}}{x^2}+\frac {\left (1+2 x^2\right ) \sqrt {-1+x^4}}{-1+x^2+x^4}\right ) \, dx \\ & = -\int \frac {\sqrt {-1+x^4}}{x^2} \, dx+\int \frac {\left (1+2 x^2\right ) \sqrt {-1+x^4}}{-1+x^2+x^4} \, dx \\ & = \frac {\sqrt {-1+x^4}}{x}-2 \int \frac {x^2}{\sqrt {-1+x^4}} \, dx+\int \left (\frac {2 \sqrt {-1+x^4}}{1-\sqrt {5}+2 x^2}+\frac {2 \sqrt {-1+x^4}}{1+\sqrt {5}+2 x^2}\right ) \, dx \\ & = \frac {\sqrt {-1+x^4}}{x}-2 \int \frac {1}{\sqrt {-1+x^4}} \, dx+2 \int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx+2 \int \frac {\sqrt {-1+x^4}}{1-\sqrt {5}+2 x^2} \, dx+2 \int \frac {\sqrt {-1+x^4}}{1+\sqrt {5}+2 x^2} \, dx \\ & = -\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {1}{2} \int \frac {1-\sqrt {5}-2 x^2}{\sqrt {-1+x^4}} \, dx-\frac {1}{2} \int \frac {1+\sqrt {5}-2 x^2}{\sqrt {-1+x^4}} \, dx+\left (1-\sqrt {5}\right ) \int \frac {1}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx+\left (1+\sqrt {5}\right ) \int \frac {1}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx \\ & = -\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {-1+x^4}}-\frac {1}{2} \left (-1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}+\frac {\left (2 \left (1-\sqrt {5}\right )\right ) \int \frac {1-x^2}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}-\frac {1}{2} \left (-1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx+\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}+\frac {\left (2 \left (1+\sqrt {5}\right )\right ) \int \frac {1-x^2}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}-2 \int \frac {1-x^2}{\sqrt {-1+x^4}} \, dx \\ & = -\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (2 \left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (2 \left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1+x^4}} \\ & = -\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}-\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}-\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {-1-x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx}{\left (3+\sqrt {5}\right ) \sqrt {-1+x^4}} \\ & = -\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (3-\sqrt {5}\right ) \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \left (3+\sqrt {5}\right ) \sqrt {-1+x^4}}-\frac {\left (1-\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3-\sqrt {5}}-\frac {\left (1+\sqrt {5}\right ) \int \frac {1}{\sqrt {-1+x^4}} \, dx}{3+\sqrt {5}}+\frac {\left (\left (1-\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt {1-x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx}{\sqrt {-1+x^4}} \\ & = -\frac {2 x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {-1+x^4}}{x}+\frac {2 \sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}-2 \left (-\frac {x \left (1+x^2\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} E\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{\sqrt {-1+x^4}}\right )-\frac {\sqrt {2} \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{\sqrt {-1+x^4}}+\frac {\left (1-\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\left (1+\sqrt {5}\right ) \sqrt {-1+x^2} \sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right ),\frac {1}{2}\right )}{2 \sqrt {2} \sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-\sqrt {5}\right ),\arcsin (x),-1\right )}{\sqrt {-1+x^4}}+\frac {\sqrt {1-x^2} \sqrt {1+x^2} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin (x),-1\right )}{\sqrt {-1+x^4}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {\sqrt {-1+x^4}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^4}}\right ) \]
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Time = 3.68 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {\sqrt {x^{4}-1}}{x}-\arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right )\) | \(27\) |
default | \(\frac {-\arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right ) x +\sqrt {x^{4}-1}}{x}\) | \(28\) |
pseudoelliptic | \(\frac {-\arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right ) x +\sqrt {x^{4}-1}}{x}\) | \(28\) |
elliptic | \(\frac {\left (\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{x}-\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}-1}}{x}\right )\right ) \sqrt {2}}{2}\) | \(38\) |
trager | \(\frac {\sqrt {x^{4}-1}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{4}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{2}-1}\right )}{2}\) | \(72\) |
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Time = 0.33 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\frac {x \arctan \left (\frac {2 \, \sqrt {x^{4} - 1} x}{x^{4} - x^{2} - 1}\right ) + 2 \, \sqrt {x^{4} - 1}}{2 \, x} \]
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\[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}{x^{2} \left (x^{4} + x^{2} - 1\right )}\, dx \]
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\[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {x^{4} - 1}}{{\left (x^{4} + x^{2} - 1\right )} x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )} \sqrt {x^{4} - 1}}{{\left (x^{4} + x^{2} - 1\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^2 \left (-1+x^2+x^4\right )} \, dx=\int \frac {\sqrt {x^4-1}\,\left (x^4+1\right )}{x^2\,\left (x^4+x^2-1\right )} \,d x \]
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