Optimal. Leaf size=100 \[ \frac {\sqrt {x^8+x^5+2 x^4+x+1}}{4 x}-\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {x^8+x^5+2 x^4+x+1}}{x^4+x+1}\right )+\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {x^8+x^5+2 x^4+x+1}}{x^4+x+1}\right ) \]
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Rubi [F] time = 1.45, 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+3 x^4\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{x^2 \left (4+x+4 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+3 x^4\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{x^2 \left (4+x+4 x^4\right )} \, dx &=\int \left (-\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{4 x^2}+\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{16 x}+\frac {\left (-1+64 x^2-4 x^3\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{16 \left (4+x+4 x^4\right )}\right ) \, dx\\ &=\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x} \, dx+\frac {1}{16} \int \frac {\left (-1+64 x^2-4 x^3\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4} \, dx-\frac {1}{4} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x^2} \, dx\\ &=\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x} \, dx+\frac {1}{16} \int \left (\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{-4-x-4 x^4}+\frac {64 x^2 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4}-\frac {4 x^3 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4}\right ) \, dx-\frac {1}{4} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x^2} \, dx\\ &=\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x} \, dx+\frac {1}{16} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{-4-x-4 x^4} \, dx-\frac {1}{4} \int \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{x^2} \, dx-\frac {1}{4} \int \frac {x^3 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4} \, dx+4 \int \frac {x^2 \sqrt {1+x+2 x^4+x^5+x^8}}{4+x+4 x^4} \, dx\\ \end {align*}
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Mathematica [F] time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+3 x^4\right ) \sqrt {1+x+2 x^4+x^5+x^8}}{x^2 \left (4+x+4 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.36, size = 100, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x+2 x^4+x^5+x^8}}{4 x}-\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right )+\frac {1}{8} \tanh ^{-1}\left (\frac {\sqrt {1+x+2 x^4+x^5+x^8}}{1+x+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.36, size = 94, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {3} x \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{4} - x + 2\right )}}{6 \, \sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1}}\right ) + x \log \left (\frac {2 \, x^{4} + x - 2 \, \sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1} + 2}{x}\right ) - 4 \, \sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1}}{16 \, x} \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*} \int \frac {\sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1} {\left (3 \, x^{4} - 1\right )}}{{\left (4 \, x^{4} + x + 4\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.59, size = 124, normalized size = 1.24
method | result | size |
trager | \(\frac {\sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x}+\frac {\ln \left (-\frac {2 x^{4}+2 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}+x +2}{x}\right )}{16}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+3\right )-6 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x^{4}+x +4}\right )}{16}\) | \(124\) |
risch | \(\frac {\sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x}-\frac {\ln \left (-\frac {-2 x^{4}+2 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}-x -2}{x}\right )}{16}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+3\right )+6 \sqrt {x^{8}+x^{5}+2 x^{4}+x +1}}{4 x^{4}+x +4}\right )}{16}\) | \(127\) |
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*} \int \frac {\sqrt {x^{8} + x^{5} + 2 \, x^{4} + x + 1} {\left (3 \, x^{4} - 1\right )}}{{\left (4 \, x^{4} + x + 4\right )} x^{2}}\,{d x} \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 (3\,x^4-1\right )\,\sqrt {x^8+x^5+2\,x^4+x+1}}{x^2\,\left (4\,x^4+x+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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