Optimal. Leaf size=93 \[ \frac {1}{10} \left (-5-\sqrt {5}\right ) \text {ArcTan}\left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {-1-x^2+x^4}}\right )+\frac {1}{10} \left (5-\sqrt {5}\right ) \text {ArcTan}\left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {-1-x^2+x^4}}\right ) \]
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Rubi [F]
time = 0.45, 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 ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx &=\int \left (\frac {\sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8}+\frac {x^4 \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8}\right ) \, dx\\ &=\int \frac {\sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx+\int \frac {x^4 \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 77, normalized size = 0.83 \begin {gather*} \frac {1}{10} \left (-\left (\left (-5+\sqrt {5}\right ) \text {ArcTan}\left (\frac {\left (-1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right )\right )-\left (5+\sqrt {5}\right ) \text {ArcTan}\left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs.
\(2(66)=132\).
time = 0.95, size = 209, normalized size = 2.25
method | result | size |
default | \(\frac {\left (-\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{2 \sqrt {10}-2 \sqrt {2}}+\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{2 \sqrt {10}+2 \sqrt {2}}+\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right ) \sqrt {2}}{2}\) | \(209\) |
elliptic | \(\frac {\left (-\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{2 \sqrt {10}-2 \sqrt {2}}+\frac {4 \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{2 \sqrt {10}+2 \sqrt {2}}+\frac {12 \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right ) \sqrt {2}}{2}\) | \(209\) |
trager | \(-20 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} \ln \left (-\frac {1400 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5} x^{2}-70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+460 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}-30 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+36 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}-4 x \sqrt {x^{4}-x^{2}-1}+9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )-3 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {1400 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5} x^{2}-70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+460 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}-30 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +70 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+36 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}-4 x \sqrt {x^{4}-x^{2}-1}+9 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )+\RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {1200 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5} x^{2}+60 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}-20 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}+2 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}-60 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x -60 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}-2 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}-x \sqrt {x^{4}-x^{2}-1}-2 \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \RootOf \left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}-x^{4}+x^{2}+1}\right )\) | \(674\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 313 vs.
\(2 (61) = 122\).
time = 0.51, size = 313, normalized size = 3.37 \begin {gather*} \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} + 3} \arctan \left (-\frac {2 \, \sqrt {10} \sqrt {x^{4} - x^{2} - 1} {\left (5 \, x^{3} + \sqrt {5} {\left (2 \, x^{5} - 5 \, x^{3} - 2 \, x\right )}\right )} \sqrt {\sqrt {5} + 3} + \sqrt {10} {\left (15 \, x^{8} - 65 \, x^{6} + 5 \, x^{4} + 65 \, x^{2} - \sqrt {5} {\left (7 \, x^{8} - 29 \, x^{6} + x^{4} + 29 \, x^{2} + 7\right )} + 15\right )} \sqrt {4 \, \sqrt {5} + 9} \sqrt {\sqrt {5} + 3}}{20 \, {\left (x^{8} - 5 \, x^{6} + 3 \, x^{4} + 5 \, x^{2} + 1\right )}}\right ) + \frac {1}{10} \, \sqrt {10} \sqrt {-\sqrt {5} + 3} \arctan \left (-\frac {40 \, \sqrt {10} \sqrt {x^{4} - x^{2} - 1} {\left (5 \, x^{3} - \sqrt {5} {\left (2 \, x^{5} - 5 \, x^{3} - 2 \, x\right )}\right )} \sqrt {-\sqrt {5} + 3} + \sqrt {10} {\left (15 \, x^{8} - 65 \, x^{6} + 5 \, x^{4} + 65 \, x^{2} + \sqrt {5} {\left (7 \, x^{8} - 29 \, x^{6} + x^{4} + 29 \, x^{2} + 7\right )} + 15\right )} \sqrt {-\sqrt {5} + 3} \sqrt {-1600 \, \sqrt {5} + 3600}}{400 \, {\left (x^{8} - 5 \, x^{6} + 3 \, x^{4} + 5 \, x^{2} + 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^4+1\right )\,\sqrt {x^4-x^2-1}}{x^8+x^6-3\,x^4-x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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