Optimal. Leaf size=95 \[ \tan ^{-1}\left (\frac {x \sqrt {-x^8-2 x^3-2 x^2-1}}{x^8+2 x^3+2 x^2+1}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-x^8-2 x^3-2 x^2-1}}{x^8+2 x^3+2 x^2+1}\right ) \]
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Rubi [F] time = 4.34, 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^2-2 x^3-x^8} \left (-1+x^3+3 x^8\right )}{\left (1+2 x^3+x^8\right ) \left (1+x^2+2 x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8} \left (-1+x^3+3 x^8\right )}{\left (1+2 x^3+x^8\right ) \left (1+x^2+2 x^3+x^8\right )} \, dx &=\int \left (-\frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{2 (1+x)}+\frac {\left (1+4 x-3 x^2+4 x^3-5 x^4+6 x^5+x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{2 \left (1-x+x^2+x^3-x^4+x^5-x^6+x^7\right )}+\frac {\left (-1-3 x-4 x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1+x} \, dx\right )+\frac {1}{2} \int \frac {\left (1+4 x-3 x^2+4 x^3-5 x^4+6 x^5+x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+\int \frac {\left (-1-3 x-4 x^6\right ) \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8} \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1+x} \, dx\right )+\frac {1}{2} \int \left (\frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {4 x \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}-\frac {3 x^2 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {4 x^3 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}-\frac {5 x^4 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {6 x^5 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}+\frac {x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7}\right ) \, dx+\int \left (\frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{-1-x^2-2 x^3-x^8}-\frac {3 x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8}-\frac {4 x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1+x} \, dx\right )+\frac {1}{2} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+\frac {1}{2} \int \frac {x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx-\frac {3}{2} \int \frac {x^2 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+2 \int \frac {x \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+2 \int \frac {x^3 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx-\frac {5}{2} \int \frac {x^4 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx+3 \int \frac {x^5 \sqrt {-1-2 x^2-2 x^3-x^8}}{1-x+x^2+x^3-x^4+x^5-x^6+x^7} \, dx-3 \int \frac {x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8} \, dx-4 \int \frac {x^6 \sqrt {-1-2 x^2-2 x^3-x^8}}{1+x^2+2 x^3+x^8} \, dx+\int \frac {\sqrt {-1-2 x^2-2 x^3-x^8}}{-1-x^2-2 x^3-x^8} \, dx\\ \end {align*}
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Mathematica [F] time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1-2 x^2-2 x^3-x^8} \left (-1+x^3+3 x^8\right )}{\left (1+2 x^3+x^8\right ) \left (1+x^2+2 x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.58, size = 95, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-1-2 x^2-2 x^3-x^8}}{1+2 x^2+2 x^3+x^8}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [C] time = 1.47, size = 241, normalized size = 2.54 \begin {gather*} -\frac {1}{4} \, \sqrt {-2} \log \left (-\frac {2 \, {\left (\sqrt {-2} {\left (x^{8} + 2 \, x^{3} + 4 \, x^{2} + 1\right )} + 4 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x\right )}}{x^{8} + 2 \, x^{3} + 1}\right ) + \frac {1}{4} \, \sqrt {-2} \log \left (\frac {2 \, {\left (\sqrt {-2} {\left (x^{8} + 2 \, x^{3} + 4 \, x^{2} + 1\right )} - 4 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x\right )}}{x^{8} + 2 \, x^{3} + 1}\right ) - \frac {1}{4} i \, \log \left (\frac {i \, x^{8} + 2 i \, x^{3} + 3 i \, x^{2} - 2 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x + i}{x^{8} + 2 \, x^{3} + x^{2} + 1}\right ) + \frac {1}{4} i \, \log \left (\frac {-i \, x^{8} - 2 i \, x^{3} - 3 i \, x^{2} - 2 \, \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1} x - i}{x^{8} + 2 \, x^{3} + x^{2} + 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 {{\left (3 \, x^{8} + x^{3} - 1\right )} \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1}}{{\left (x^{8} + 2 \, x^{3} + x^{2} + 1\right )} {\left (x^{8} + 2 \, x^{3} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.26, size = 195, normalized size = 2.05
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{8}-2 x^{3}-2 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{8}+2 x^{3}+x^{2}+1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {-x^{8}-2 x^{3}-2 x^{2}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (1+x \right ) \left (x^{7}-x^{6}+x^{5}-x^{4}+x^{3}+x^{2}-x +1\right )}\right )}{2}\) | \(195\) |
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 {{\left (3 \, x^{8} + x^{3} - 1\right )} \sqrt {-x^{8} - 2 \, x^{3} - 2 \, x^{2} - 1}}{{\left (x^{8} + 2 \, x^{3} + x^{2} + 1\right )} {\left (x^{8} + 2 \, x^{3} + 1\right )}}\,{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^8+x^3-1\right )\,\sqrt {-x^8-2\,x^3-2\,x^2-1}}{\left (x^8+2\,x^3+1\right )\,\left (x^8+2\,x^3+x^2+1\right )} \,d x \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 (3 x^{8} + x^{3} - 1\right ) \sqrt {- x^{8} - 2 x^{3} - 2 x^{2} - 1}}{\left (x + 1\right ) \left (x^{8} + 2 x^{3} + x^{2} + 1\right ) \left (x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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