Optimal. Leaf size=133 \[ -2 \tan ^{-1}\left (\frac {\sqrt {x^4+x^2+1}}{x^2-x+1}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {x^4+x^2+1}}{\sqrt {2+\sqrt {5}} \left (x^2-x+1\right )}\right )+\sqrt {\frac {1}{5} \left (2 \sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {x^4+x^2+1}}{\sqrt {\sqrt {5}-2} \left (x^2-x+1\right )}\right ) \]
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Rubi [F] time = 1.00, 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^2\right ) \sqrt {1+x^2+x^4}}{\left (1+x^2\right ) \left (1+x+x^2+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+x^2+x^4}}{\left (1+x^2\right ) \left (1+x+x^2+x^3+x^4\right )} \, dx &=\int \left (-\frac {2 \sqrt {1+x^2+x^4}}{1+x^2}+\frac {\left (1+2 x+2 x^2\right ) \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {1+x^2+x^4}}{1+x^2} \, dx\right )+\int \frac {\left (1+2 x+2 x^2\right ) \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ &=-\left (2 \int \frac {x^2}{\sqrt {1+x^2+x^4}} \, dx\right )-2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \left (\frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}+\frac {2 x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}+\frac {2 x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx\right )+2 \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx-\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-\int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ &=-\frac {2 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=-\frac {2 x \sqrt {1+x^2+x^4}}{1+x^2}-\tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}+2 \int \frac {x \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx+\int \frac {\sqrt {1+x^2+x^4}}{1+x+x^2+x^3+x^4} \, dx\\ \end {align*}
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Mathematica [C] time = 139.37, size = 4461, normalized size = 33.54 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.06, size = 133, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {1+x^2+x^4}}{1-x+x^2}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-2+\sqrt {5}} \sqrt {1+x^2+x^4}}{1-x+x^2}\right )+\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {5}} \sqrt {1+x^2+x^4}}{1-x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 306, normalized size = 2.30 \begin {gather*} \frac {1}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{4} + \sqrt {5} x^{2} + x^{2} + 2\right )} \sqrt {2 \, \sqrt {5} + 2} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} + \sqrt {5} x - x + 2\right )} \sqrt {2 \, \sqrt {5} + 2}}{8 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} + {\left (x^{4} + 3 \, x^{2} + \sqrt {5} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {2 \, \sqrt {x^{4} + x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} - {\left (x^{4} + 3 \, x^{2} + \sqrt {5} {\left (x^{4} + x^{2} + 1\right )} + 1\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \arctan \left (\frac {x}{\sqrt {x^{4} + 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 {\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{4} + x^{3} + x^{2} + x + 1\right )} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.56, size = 186, normalized size = 1.40
method | result | size |
elliptic | \(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\left (\sqrt {x^{4}+x^{2}+1}-x^{2}\right )^{2}+\left (-15 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} \right ) \left (\sqrt {x^{4}+x^{2}+1}-x^{2}\right )-15 \textit {\_R}^{3}+5 \textit {\_R}^{2}-4 \textit {\_R} +2\right )\right )}{2}+\frac {\left (-\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}-\frac {2 \sqrt {5}\, \arctanh \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\right ) \sqrt {2}}{2}\) | \(186\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \ln \left (\frac {25 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{4} x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{2}-5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right )+2 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right )-5 \sqrt {x^{4}+x^{2}+1}}{5 x \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}-x^{2}-1}\right )}{5}-\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {-25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{5} x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{3} x^{2}-15 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{3} x -5 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{3}-3 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) x^{2}+\sqrt {x^{4}+x^{2}+1}-3 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{5 x \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+x^{2}+x +1}\right )-\RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )^{2}+5\right ) \RootOf \left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right ) x -\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\) | \(573\) |
default | \(\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {8 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticE \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}-\frac {2 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {8 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -2\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (6 \underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+11 x^{2}-6 \underline {\hspace {1.25 ex}}\alpha +4\right )}{22 \sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha }\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha }}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{10}\) | \(712\) |
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^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}}{{\left (x^{4} + x^{3} + x^{2} + x + 1\right )} {\left (x^{2} + 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 (x^2-1\right )\,\sqrt {x^4+x^2+1}}{\left (x^2+1\right )\,\left (x^4+x^3+x^2+x+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 {\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \left (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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