Optimal. Leaf size=139 \[ -\frac {x}{2 \sqrt [4]{-1-x^4+x^8}}+\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-1-x^4+x^8}}{\sqrt {2} x^2-\sqrt {-1-x^4+x^8}}\right )}{4\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {-1-x^4+x^8}}\right )}{4\ 2^{3/4}} \]
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Rubi [F]
time = 0.91, 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^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^8\right ) \left (1+x^8\right )}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx &=\int \frac {-1+x^{16}}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx\\ &=\int \left (\frac {1}{\sqrt [4]{-1-x^4+x^8}}-\frac {2-3 x^8}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt [4]{-1-x^4+x^8}} \, dx-\int \frac {2-3 x^8}{\sqrt [4]{-1-x^4+x^8} \left (1-3 x^8+x^{16}\right )} \, dx\\ &=\frac {\left (\sqrt [4]{1+\frac {2 x^4}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^4}{-1+\sqrt {5}}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {2 x^4}{-1-\sqrt {5}}} \sqrt [4]{1+\frac {2 x^4}{-1+\sqrt {5}}}} \, dx}{\sqrt [4]{-1-x^4+x^8}}-\int \left (\frac {-3-\sqrt {5}}{\sqrt [4]{-1-x^4+x^8} \left (-3-\sqrt {5}+2 x^8\right )}+\frac {-3+\sqrt {5}}{\sqrt [4]{-1-x^4+x^8} \left (-3+\sqrt {5}+2 x^8\right )}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {5}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^4}{1+\sqrt {5}},\frac {2 x^4}{1-\sqrt {5}}\right )}{\sqrt [4]{-1-x^4+x^8}}-\left (-3-\sqrt {5}\right ) \int \frac {1}{\sqrt [4]{-1-x^4+x^8} \left (-3-\sqrt {5}+2 x^8\right )} \, dx-\left (-3+\sqrt {5}\right ) \int \frac {1}{\sqrt [4]{-1-x^4+x^8} \left (-3+\sqrt {5}+2 x^8\right )} \, dx\\ &=\frac {x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {5}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^4}{1+\sqrt {5}},\frac {2 x^4}{1-\sqrt {5}}\right )}{\sqrt [4]{-1-x^4+x^8}}-\left (-3-\sqrt {5}\right ) \int \left (\frac {\sqrt {3+\sqrt {5}}}{2 \left (-3-\sqrt {5}\right ) \left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}}+\frac {\sqrt {3+\sqrt {5}}}{2 \left (-3-\sqrt {5}\right ) \left (\sqrt {3+\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}}\right ) \, dx-\left (-3+\sqrt {5}\right ) \int \left (\frac {\sqrt {3-\sqrt {5}}}{2 \left (-3+\sqrt {5}\right ) \left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}}+\frac {\sqrt {3-\sqrt {5}}}{2 \left (-3+\sqrt {5}\right ) \left (\sqrt {3-\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}}\right ) \, dx\\ &=\frac {x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {5}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^4}{1+\sqrt {5}},\frac {2 x^4}{1-\sqrt {5}}\right )}{\sqrt [4]{-1-x^4+x^8}}-\frac {1}{2} \sqrt {3-\sqrt {5}} \int \frac {1}{\left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {1}{2} \sqrt {3-\sqrt {5}} \int \frac {1}{\left (\sqrt {3-\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {1}{2} \sqrt {3+\sqrt {5}} \int \frac {1}{\left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {1}{2} \sqrt {3+\sqrt {5}} \int \frac {1}{\left (\sqrt {3+\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx\\ &=\frac {x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {5}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^4}{1+\sqrt {5}},\frac {2 x^4}{1-\sqrt {5}}\right )}{\sqrt [4]{-1-x^4+x^8}}-\frac {1}{2} \sqrt {3-\sqrt {5}} \int \frac {1}{\left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {1}{2} \sqrt {3+\sqrt {5}} \int \frac {1}{\left (\sqrt {3+\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {\left (\sqrt {3+\sqrt {5}} \sqrt [4]{-\frac {1}{\sqrt {3+\sqrt {5}}}-\frac {x^4}{\sqrt {2}}} \sqrt [4]{\sqrt {3+\sqrt {5}}-\sqrt {2} x^4}\right ) \int \frac {1}{\sqrt [4]{-\frac {1}{\sqrt {3+\sqrt {5}}}-\frac {x^4}{\sqrt {2}}} \left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^4\right )^{5/4}} \, dx}{2 \sqrt [4]{-1-x^4+x^8}}-\frac {\left (\sqrt {3-\sqrt {5}} \sqrt [4]{-\frac {1}{\sqrt {3-\sqrt {5}}}+\frac {x^4}{\sqrt {2}}} \sqrt [4]{\sqrt {3-\sqrt {5}}+\sqrt {2} x^4}\right ) \int \frac {1}{\sqrt [4]{-\frac {1}{\sqrt {3-\sqrt {5}}}+\frac {x^4}{\sqrt {2}}} \left (\sqrt {3-\sqrt {5}}+\sqrt {2} x^4\right )^{5/4}} \, dx}{2 \sqrt [4]{-1-x^4+x^8}}\\ &=\frac {x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {5}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^4}{1+\sqrt {5}},\frac {2 x^4}{1-\sqrt {5}}\right )}{\sqrt [4]{-1-x^4+x^8}}-\frac {1}{2} \sqrt {3-\sqrt {5}} \int \frac {1}{\left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {1}{2} \sqrt {3+\sqrt {5}} \int \frac {1}{\left (\sqrt {3+\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {\left (\sqrt {3-\sqrt {5}} \sqrt [4]{\sqrt {3-\sqrt {5}}+\sqrt {2} x^4} \sqrt [4]{1-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x^4}\right ) \int \frac {1}{\left (\sqrt {3-\sqrt {5}}+\sqrt {2} x^4\right )^{5/4} \sqrt [4]{1-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x^4}} \, dx}{2 \sqrt [4]{-1-x^4+x^8}}-\frac {\left (\sqrt {3+\sqrt {5}} \sqrt [4]{\sqrt {3+\sqrt {5}}-\sqrt {2} x^4} \sqrt [4]{1+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4}\right ) \int \frac {1}{\left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^4\right )^{5/4} \sqrt [4]{1+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4}} \, dx}{2 \sqrt [4]{-1-x^4+x^8}}\\ &=\frac {x \sqrt [4]{1-\frac {2 x^4}{1-\sqrt {5}}} \sqrt [4]{1-\frac {2 x^4}{1+\sqrt {5}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 x^4}{1+\sqrt {5}},\frac {2 x^4}{1-\sqrt {5}}\right )}{\sqrt [4]{-1-x^4+x^8}}-\frac {x \sqrt [4]{\sqrt {3+\sqrt {5}}-\sqrt {2} x^4} \sqrt [4]{1+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4} F_1\left (\frac {1}{4};\frac {5}{4},\frac {1}{4};\frac {5}{4};\sqrt {\frac {2}{3+\sqrt {5}}} x^4,-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4\right )}{2 \sqrt [8]{3+\sqrt {5}} \sqrt [4]{-1-x^4+x^8}}-\frac {x \sqrt [4]{\sqrt {3-\sqrt {5}}+\sqrt {2} x^4} \sqrt [4]{1-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x^4} F_1\left (\frac {1}{4};\frac {5}{4},\frac {1}{4};\frac {5}{4};-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4,\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )} x^4\right )}{2 \sqrt [8]{3-\sqrt {5}} \sqrt [4]{-1-x^4+x^8}}-\frac {1}{2} \sqrt {3-\sqrt {5}} \int \frac {1}{\left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx-\frac {1}{2} \sqrt {3+\sqrt {5}} \int \frac {1}{\left (\sqrt {3+\sqrt {5}}+\sqrt {2} x^4\right ) \sqrt [4]{-1-x^4+x^8}} \, dx\\ \end {align*}
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Mathematica [A]
time = 1.47, size = 136, normalized size = 0.98 \begin {gather*} \frac {1}{8} \left (-\frac {4 x}{\sqrt [4]{-1-x^4+x^8}}+\sqrt [4]{2} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-1-x^4+x^8}}{\sqrt {2} x^2-\sqrt {-1-x^4+x^8}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-1-x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {-1-x^4+x^8}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.91, size = 285, normalized size = 2.05
method | result | size |
trager | \(-\frac {x}{2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{8}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{8}-x^{4}-1}\, x^{2}+2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{4}-4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+8\right )}{x^{8}+x^{4}-1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {x^{8} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{8}-x^{4}-1}\, x^{2}-2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}-4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{x^{8}+x^{4}-1}\right )}{16}\) | \(285\) |
risch | \(-\frac {x}{2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}}}-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{8}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{8}-x^{4}-1}\, x^{2}+2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{4}-4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+8\right )}{x^{8}+x^{4}-1}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {x^{8} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{8}-x^{4}-1}\, x^{2}-2 \left (x^{8}-x^{4}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}-4 \left (x^{8}-x^{4}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{x^{8}+x^{4}-1}\right )}{16}\) | \(285\) |
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. 658 vs.
\(2 (111) = 222\).
time = 14.53, size = 658, normalized size = 4.73 \begin {gather*} -\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{8} + x^{4} - 1\right )}\right )} \sqrt {\frac {x^{8} + x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{8} + x^{4} - 1}}}{2 \, {\left (x^{8} - 3 \, x^{4} - 1\right )}}\right ) + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \arctan \left (\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{8} + x^{4} - 1\right )}\right )} \sqrt {\frac {x^{8} + x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1}{x^{8} + x^{4} - 1}}}{2 \, {\left (x^{8} - 3 \, x^{4} - 1\right )}}\right ) + 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \log \left (\frac {2 \, {\left (x^{8} + x^{4} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{8} + x^{4} - 1}\right ) - 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )} \log \left (\frac {2 \, {\left (x^{8} + x^{4} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {2} \sqrt {x^{8} - x^{4} - 1} x^{2} - 2 \cdot 2^{\frac {3}{4}} {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{8} + x^{4} - 1}\right ) + 16 \, {\left (x^{8} - x^{4} - 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{8} - x^{4} - 1\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^8-1\right )\,\left (x^8+1\right )}{{\left (x^8-x^4-1\right )}^{1/4}\,\left (x^{16}-3\,x^8+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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