Optimal. Leaf size=26 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^4+x^2+1}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1698, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^4+x^2+1}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1698
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^2+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-3 x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^2+x^4}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 1.32, size = 522, normalized size = 20.08 \[ \frac {(-1)^{2/3} \left (-\sqrt [3]{-1} \left (\sqrt [3]{-1}-1\right )^2 \left (1+\sqrt [3]{-1}\right ) \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )-2 i \sqrt {3} \sqrt {\frac {(-1)^{2/3}-x}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \sqrt {\frac {x+(-1)^{2/3}}{-\sqrt [3]{-1} x+x-1}} \sqrt {\frac {-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \left (\sqrt [3]{-1}-x\right )^2 \left (\left (1+\sqrt [3]{-1}\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right ),-3\right )-2 \sqrt [3]{-1} \Pi \left (-1;\left .\sin ^{-1}\left (\sqrt {\frac {-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right )\right |-3\right )\right )+2 i \sqrt {3} \sqrt {\frac {(-1)^{2/3}-x}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \sqrt {\frac {x+(-1)^{2/3}}{-\sqrt [3]{-1} x+x-1}} \sqrt {\frac {-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}} \left (\sqrt [3]{-1}-x\right )^2 \left (\left (\sqrt [3]{-1}-1\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right ),-3\right )-2 \sqrt [3]{-1} \Pi \left (3;\left .\sin ^{-1}\left (\sqrt {\frac {-\sqrt [3]{-1} x+x+1}{\left (1+\sqrt [3]{-1}\right ) \left (\sqrt [3]{-1}-x\right )}}\right )\right |-3\right )\right )\right )}{\left (1-(-1)^{2/3}\right ) \sqrt {x^4+x^2+1}} \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.21, size = 26, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^4+x^2+1}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 45, normalized size = 1.73 \[ \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{4} + 2 \, \sqrt {3} \sqrt {x^{4} + x^{2} + 1} x + 4 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) \]
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 \[ \int -\frac {x^{2} + 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 31, normalized size = 1.19
method | result | size |
elliptic | \(\frac {\arctanh \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}\, \sqrt {6}}{6 x}\right ) \sqrt {6}\, \sqrt {2}}{6}\) | \(31\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x +\sqrt {x^{4}+x^{2}+1}}{\left (1+x \right ) \left (-1+x \right )}\right )}{3}\) | \(41\) |
default | \(-\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{\sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {1}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \sqrt {x^{4}+x^{2}+1}}\) | \(184\) |
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 \[ -\int \frac {x^{2} + 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int -\frac {x^2+1}{\left (x^2-1\right )\,\sqrt {x^4+x^2+1}} \,d x \]
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 \[ - \int \frac {x^{2}}{x^{2} \sqrt {x^{4} + x^{2} + 1} - \sqrt {x^{4} + x^{2} + 1}}\, dx - \int \frac {1}{x^{2} \sqrt {x^{4} + x^{2} + 1} - \sqrt {x^{4} + x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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