Optimal. Leaf size=72 \[ -\frac {\sqrt {x^8+2 x^4+x^2+1}}{2 \left (x^4-x+1\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{x^4+\sqrt {x^8+2 x^4+x^2+1}+x+1}\right )}{\sqrt {2}} \]
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Rubi [F] time = 2.17, 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+2 x^4+x^8}}{\left (1-x+x^4\right )^2 \left (1+x+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+2 x^4+x^8}}{\left (1-x+x^4\right )^2 \left (1+x+x^4\right )} \, dx &=\int \left (\frac {\left (-1+4 x^3\right ) \sqrt {1+x^2+2 x^4+x^8}}{2 \left (1-x+x^4\right )^2}+\frac {\left (-1-4 x^2+x^3\right ) \sqrt {1+x^2+2 x^4+x^8}}{4 \left (1-x+x^4\right )}+\frac {\left (-1+4 x^2-x^3\right ) \sqrt {1+x^2+2 x^4+x^8}}{4 \left (1+x+x^4\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (-1-4 x^2+x^3\right ) \sqrt {1+x^2+2 x^4+x^8}}{1-x+x^4} \, dx+\frac {1}{4} \int \frac {\left (-1+4 x^2-x^3\right ) \sqrt {1+x^2+2 x^4+x^8}}{1+x+x^4} \, dx+\frac {1}{2} \int \frac {\left (-1+4 x^3\right ) \sqrt {1+x^2+2 x^4+x^8}}{\left (1-x+x^4\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {\sqrt {1+x^2+2 x^4+x^8}}{-1+x-x^4}-\frac {4 x^2 \sqrt {1+x^2+2 x^4+x^8}}{1-x+x^4}+\frac {x^3 \sqrt {1+x^2+2 x^4+x^8}}{1-x+x^4}\right ) \, dx+\frac {1}{4} \int \left (\frac {\sqrt {1+x^2+2 x^4+x^8}}{-1-x-x^4}+\frac {4 x^2 \sqrt {1+x^2+2 x^4+x^8}}{1+x+x^4}-\frac {x^3 \sqrt {1+x^2+2 x^4+x^8}}{1+x+x^4}\right ) \, dx+\frac {1}{2} \int \left (-\frac {\sqrt {1+x^2+2 x^4+x^8}}{\left (1-x+x^4\right )^2}+\frac {4 x^3 \sqrt {1+x^2+2 x^4+x^8}}{\left (1-x+x^4\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sqrt {1+x^2+2 x^4+x^8}}{-1-x-x^4} \, dx+\frac {1}{4} \int \frac {\sqrt {1+x^2+2 x^4+x^8}}{-1+x-x^4} \, dx+\frac {1}{4} \int \frac {x^3 \sqrt {1+x^2+2 x^4+x^8}}{1-x+x^4} \, dx-\frac {1}{4} \int \frac {x^3 \sqrt {1+x^2+2 x^4+x^8}}{1+x+x^4} \, dx-\frac {1}{2} \int \frac {\sqrt {1+x^2+2 x^4+x^8}}{\left (1-x+x^4\right )^2} \, dx+2 \int \frac {x^3 \sqrt {1+x^2+2 x^4+x^8}}{\left (1-x+x^4\right )^2} \, dx-\int \frac {x^2 \sqrt {1+x^2+2 x^4+x^8}}{1-x+x^4} \, dx+\int \frac {x^2 \sqrt {1+x^2+2 x^4+x^8}}{1+x+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+2 x^4+x^8}}{\left (1-x+x^4\right )^2 \left (1+x+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.52, size = 72, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+x^2+2 x^4+x^8}}{2 \left (1-x+x^4\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{1+x+x^4+\sqrt {1+x^2+2 x^4+x^8}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 120, normalized size = 1.67 \begin {gather*} \frac {\sqrt {2} {\left (x^{4} - x + 1\right )} \log \left (\frac {3 \, x^{8} - 2 \, x^{5} + 6 \, x^{4} + 2 \, \sqrt {2} \sqrt {x^{8} + 2 \, x^{4} + x^{2} + 1} {\left (x^{4} - x + 1\right )} + 3 \, x^{2} - 2 \, x + 3}{x^{8} + 2 \, x^{5} + 2 \, x^{4} + x^{2} + 2 \, x + 1}\right ) - 4 \, \sqrt {x^{8} + 2 \, x^{4} + x^{2} + 1}}{8 \, {\left (x^{4} - x + 1\right )}} \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} + 2 \, x^{4} + x^{2} + 1} {\left (3 \, x^{4} - 1\right )}}{{\left (x^{4} + x + 1\right )} {\left (x^{4} - x + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.75, size = 90, normalized size = 1.25
method | result | size |
trager | \(-\frac {\sqrt {x^{8}+2 x^{4}+x^{2}+1}}{2 \left (x^{4}-x +1\right )}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\RootOf \left (\textit {\_Z}^{2}-2\right )-2 \sqrt {x^{8}+2 x^{4}+x^{2}+1}}{x^{4}+x +1}\right )}{4}\) | \(90\) |
risch | \(-\frac {\sqrt {x^{8}+2 x^{4}+x^{2}+1}}{2 \left (x^{4}-x +1\right )}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}-2\right ) x +\RootOf \left (\textit {\_Z}^{2}-2\right )-2 \sqrt {x^{8}+2 x^{4}+x^{2}+1}}{x^{4}+x +1}\right )}{4}\) | \(90\) |
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} + 2 \, x^{4} + x^{2} + 1} {\left (3 \, x^{4} - 1\right )}}{{\left (x^{4} + x + 1\right )} {\left (x^{4} - x + 1\right )}^{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+2\,x^4+x^2+1}}{{\left (x^4-x+1\right )}^2\,\left (x^4+x+1\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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