Optimal. Leaf size=58 \[ -2 \text {ArcTan}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {2 x^2}{1+2 x^2+x^4+\left (-1+x^2\right ) \sqrt {1+x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 46, normalized size of antiderivative = 0.79, number of steps
used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1701, 1712,
209, 1261, 738, 212} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {1-x^2}{2 \sqrt {x^4+x^2+1}}\right )-2 \text {ArcTan}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 738
Rule 1261
Rule 1701
Rule 1712
Rubi steps
\begin {align*} \int \frac {-2-x+2 x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx &=-\int \frac {x}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \frac {-2+2 x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx,x,x^2\right )\right )-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=-2 \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1-x^2}{\sqrt {1+x^2+x^4}}\right )\\ &=-2 \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {1-x^2}{2 \sqrt {1+x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 58, normalized size = 1.00 \begin {gather*} -2 \text {ArcTan}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {1}{2} \tanh ^{-1}\left (\frac {2 x^2}{1+2 x^2+x^4+\left (-1+x^2\right ) \sqrt {1+x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.95, size = 219, normalized size = 3.78
method | result | size |
elliptic | \(\frac {\arctanh \left (\frac {-x^{2}+1}{2 \sqrt {\left (x^{2}+1\right )^{2}-x^{2}}}\right )}{2}+2 \arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\) | \(46\) |
trager | \(-\frac {\ln \left (-\frac {6 \RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right ) x -8 x^{2}+4 \sqrt {x^{4}+x^{2}+1}-3 x +8}{\left (2 \RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right ) x -x +4\right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right )}{2}+\frac {\ln \left (-\frac {6 \RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right ) x -8 x^{2}+4 \sqrt {x^{4}+x^{2}+1}-3 x +8}{\left (2 \RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right ) x -x +4\right )^{2}}\right )}{2}+\frac {\RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right ) \ln \left (-\frac {-6 \RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right ) x -8 x^{2}+4 \sqrt {x^{4}+x^{2}+1}+3 x +8}{\left (2 \RootOf \left (4 \textit {\_Z}^{2}-4 \textit {\_Z} +17\right ) x -x -4\right )^{2}}\right )}{2}\) | \(210\) |
default | \(\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {\arctanh \left (\frac {x^{2}}{2 \sqrt {x^{4}+x^{2}+1}}-\frac {1}{2 \sqrt {x^{4}+x^{2}+1}}\right )}{2}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i x^{2} \sqrt {3}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i x^{2} \sqrt {3}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(219\) |
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 [A]
time = 0.45, size = 63, normalized size = 1.09 \begin {gather*} 2 \, \arctan \left (\frac {\sqrt {x^{4} + x^{2} + 1}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {5 \, x^{4} + 2 \, x^{2} - 4 \, \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )} + 5}{x^{4} + 2 \, x^{2} + 1}\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 {2 x^{2} - x - 2}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, 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 = -0.02 \begin {gather*} \int -\frac {-2\,x^2+x+2}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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