Optimal. Leaf size=101 \[ -\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}-x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}+x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}} \]
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Rubi [F]
time = 0.43, 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 {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+2 x^6} \left (-1+4 x^6\right )}{2+x^4+8 x^6+8 x^{12}} \, dx &=\int \left (\frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}}+\frac {4 x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}}\right ) \, dx\\ &=4 \int \frac {x^6 \sqrt {1+2 x^6}}{2+x^4+8 x^6+8 x^{12}} \, dx+\int \frac {\sqrt {1+2 x^6}}{-2-x^4-8 x^6-8 x^{12}} \, dx\\ \end {align*}
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Mathematica [A]
time = 3.68, size = 92, normalized size = 0.91 \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}-x^2+2 \sqrt {2} x^6}\right )+\tanh ^{-1}\left (\frac {2^{3/4} x \sqrt {1+2 x^6}}{\sqrt {2}+x^2+2 \sqrt {2} x^6}\right )}{4 \sqrt [4]{2}} \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 = 2.47, size = 187, normalized size = 1.85
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{2}+4 \sqrt {2 x^{6}+1}\, x -2 \RootOf \left (\textit {\_Z}^{4}+2\right )}{4 x^{6}+x^{2} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}+2}\right )}{8}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{6}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}-4 \sqrt {2 x^{6}+1}\, x +2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{-4 x^{6}+x^{2} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-2}\right )}{8}\) | \(187\) |
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. 1041 vs.
\(2 (77) = 154\).
time = 0.57, size = 1041, normalized size = 10.31 \begin {gather*} \frac {1}{32} \cdot 8^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {128 \, x^{24} + 256 \, x^{18} + 32 \, x^{16} + 192 \, x^{12} + 32 \, x^{10} + 2 \, x^{8} + 64 \, x^{6} + 8 \, x^{4} + 8 \, \sqrt {2} {\left (16 \, x^{20} + 24 \, x^{14} + 2 \, x^{12} + 12 \, x^{8} + x^{6} + 2 \, x^{2}\right )} + \sqrt {2 \, x^{6} + 1} {\left (8^{\frac {3}{4}} \sqrt {2} {\left (24 \, x^{15} + 24 \, x^{9} - x^{7} + 6 \, x^{3}\right )} + 4 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (16 \, x^{19} + 24 \, x^{13} - 6 \, x^{11} + 12 \, x^{7} - 3 \, x^{5} + 2 \, x\right )}\right )} - {\left (8^{\frac {3}{4}} \sqrt {2} {\left (16 \, x^{20} + 24 \, x^{14} - 6 \, x^{12} + 12 \, x^{8} - 3 \, x^{6} + 2 \, x^{2}\right )} + 8^{\frac {1}{4}} \sqrt {2} {\left (64 \, x^{24} + 128 \, x^{18} - 64 \, x^{16} + 96 \, x^{12} - 64 \, x^{10} - x^{8} + 32 \, x^{6} - 16 \, x^{4} + 4\right )} + 8 \, {\left (8 \, x^{15} + 8 \, x^{9} + x^{7} + 2 \, x^{3} + 4 \, \sqrt {2} {\left (2 \, x^{11} + x^{5}\right )}\right )} \sqrt {2 \, x^{6} + 1}\right )} \sqrt {\frac {16 \, x^{8} + 8 \, x^{2} + \sqrt {2} {\left (8 \, x^{12} + 8 \, x^{6} + x^{4} + 2\right )} + \sqrt {2 \, x^{6} + 1} {\left (2 \cdot 8^{\frac {1}{4}} \sqrt {2} x^{3} + 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{7} + x\right )}\right )}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}} + 8}{2 \, {\left (64 \, x^{24} + 128 \, x^{18} - 112 \, x^{16} + 96 \, x^{12} - 112 \, x^{10} + x^{8} + 32 \, x^{6} - 28 \, x^{4} + 4\right )}}\right ) - \frac {1}{32} \cdot 8^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {128 \, x^{24} + 256 \, x^{18} + 32 \, x^{16} + 192 \, x^{12} + 32 \, x^{10} + 2 \, x^{8} + 64 \, x^{6} + 8 \, x^{4} + 8 \, \sqrt {2} {\left (16 \, x^{20} + 24 \, x^{14} + 2 \, x^{12} + 12 \, x^{8} + x^{6} + 2 \, x^{2}\right )} - \sqrt {2 \, x^{6} + 1} {\left (8^{\frac {3}{4}} \sqrt {2} {\left (24 \, x^{15} + 24 \, x^{9} - x^{7} + 6 \, x^{3}\right )} + 4 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (16 \, x^{19} + 24 \, x^{13} - 6 \, x^{11} + 12 \, x^{7} - 3 \, x^{5} + 2 \, x\right )}\right )} + {\left (8^{\frac {3}{4}} \sqrt {2} {\left (16 \, x^{20} + 24 \, x^{14} - 6 \, x^{12} + 12 \, x^{8} - 3 \, x^{6} + 2 \, x^{2}\right )} + 8^{\frac {1}{4}} \sqrt {2} {\left (64 \, x^{24} + 128 \, x^{18} - 64 \, x^{16} + 96 \, x^{12} - 64 \, x^{10} - x^{8} + 32 \, x^{6} - 16 \, x^{4} + 4\right )} - 8 \, {\left (8 \, x^{15} + 8 \, x^{9} + x^{7} + 2 \, x^{3} + 4 \, \sqrt {2} {\left (2 \, x^{11} + x^{5}\right )}\right )} \sqrt {2 \, x^{6} + 1}\right )} \sqrt {\frac {16 \, x^{8} + 8 \, x^{2} + \sqrt {2} {\left (8 \, x^{12} + 8 \, x^{6} + x^{4} + 2\right )} - \sqrt {2 \, x^{6} + 1} {\left (2 \cdot 8^{\frac {1}{4}} \sqrt {2} x^{3} + 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{7} + x\right )}\right )}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}} + 8}{2 \, {\left (64 \, x^{24} + 128 \, x^{18} - 112 \, x^{16} + 96 \, x^{12} - 112 \, x^{10} + x^{8} + 32 \, x^{6} - 28 \, x^{4} + 4\right )}}\right ) - \frac {1}{128} \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {64 \, {\left (16 \, x^{8} + 8 \, x^{2} + \sqrt {2} {\left (8 \, x^{12} + 8 \, x^{6} + x^{4} + 2\right )} + \sqrt {2 \, x^{6} + 1} {\left (2 \cdot 8^{\frac {1}{4}} \sqrt {2} x^{3} + 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{7} + x\right )}\right )}\right )}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\right ) + \frac {1}{128} \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {64 \, {\left (16 \, x^{8} + 8 \, x^{2} + \sqrt {2} {\left (8 \, x^{12} + 8 \, x^{6} + x^{4} + 2\right )} - \sqrt {2 \, x^{6} + 1} {\left (2 \cdot 8^{\frac {1}{4}} \sqrt {2} x^{3} + 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{7} + x\right )}\right )}\right )}}{8 \, x^{12} + 8 \, x^{6} + x^{4} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{3} - 1\right ) \left (2 x^{3} + 1\right ) \sqrt {2 x^{6} + 1}}{8 x^{12} + 8 x^{6} + x^{4} + 2}\, dx \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 {\sqrt {2\,x^6+1}\,\left (4\,x^6-1\right )}{8\,x^{12}+8\,x^6+x^4+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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