Optimal. Leaf size=58 \[ \frac {1}{2} \tanh ^{-1}\left (\frac {2 x^2}{x^4+2 x^2+\left (x^2-1\right ) \sqrt {x^4+x^2+1}+1}\right )-2 \tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]
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Rubi [A] time = 0.10, 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 = {1687, 1698, 203, 1247, 724, 206} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {1-x^2}{2 \sqrt {x^4+x^2+1}}\right )-2 \tan ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 724
Rule 1247
Rule 1687
Rule 1698
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} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+x+x^2}} \, dx,x,x^2\right )\right )-2 \operatorname {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 )+\operatorname {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 [C] time = 0.21, size = 170, normalized size = 2.93 \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {1-x^2}{2 \sqrt {x^4+x^2+1}}\right )+\frac {2 (-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{\sqrt {x^4+x^2+1}}-\frac {4 (-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \Pi \left (\sqrt [3]{-1};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{\sqrt {x^4+x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 58, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\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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fricas [A] time = 0.56, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} - x - 2}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.15, size = 46, normalized size = 0.79
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 | \(-\frac {\ln \left (-\frac {6 \RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right ) x +8 x^{2}+4 \sqrt {x^{4}+x^{2}+1}+3 x -8}{\left (2 \RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right ) x +x -4\right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right )}{2}-\frac {\ln \left (-\frac {6 \RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right ) x +8 x^{2}+4 \sqrt {x^{4}+x^{2}+1}+3 x -8}{\left (2 \RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right ) x +x -4\right )^{2}}\right )}{2}+\frac {\RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right ) \ln \left (\frac {6 \RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right ) x -8 x^{2}-4 \sqrt {x^{4}+x^{2}+1}+3 x +8}{\left (2 \RootOf \left (4 \textit {\_Z}^{2}+4 \textit {\_Z} +17\right ) x +x +4\right )^{2}}\right )}{2}\) | \(203\) |
default | \(\frac {4 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {\arctanh \left (\frac {x^{2}}{2 \sqrt {x^{4}+x^{2}+1}}-\frac {1}{2 \sqrt {x^{4}+x^{2}+1}}\right )}{2}-\frac {4 \sqrt {1+\frac {x^{2}}{2}-\frac {i \sqrt {3}\, x^{2}}{2}}\, \sqrt {1+\frac {x^{2}}{2}+\frac {i \sqrt {3}\, x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , -\frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(219\) |
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*} \int \frac {2 \, x^{2} - x - 2}{\sqrt {x^{4} + x^{2} + 1} {\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 = -0.02 \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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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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