Optimal. Leaf size=44 \[ \frac {\tanh ^{-1}\left (\sqrt {5}-2 \sqrt {2} x\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (2 \sqrt {2} x+\sqrt {5}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1161, 618, 206} \[ \frac {\tanh ^{-1}\left (\sqrt {5}-2 \sqrt {2} x\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (2 \sqrt {2} x+\sqrt {5}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+2 x^2}{1-6 x^2+4 x^4} \, dx &=\frac {1}{4} \int \frac {1}{\frac {1}{2}-\sqrt {\frac {5}{2}} x+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\sqrt {\frac {5}{2}} x+x^2} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}-x^2} \, dx,x,-\sqrt {\frac {5}{2}}+2 x\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}-x^2} \, dx,x,\sqrt {\frac {5}{2}}+2 x\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {5}-2 \sqrt {2} x\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\sqrt {5}+2 \sqrt {2} x\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.95 \[ \frac {\log \left (-2 x^2+\sqrt {2} x+1\right )-\log \left (2 x^2+\sqrt {2} x-1\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 47, normalized size = 1.07 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {4 \, x^{4} - 2 \, x^{2} - 2 \, \sqrt {2} {\left (2 \, x^{3} - x\right )} + 1}{4 \, x^{4} - 6 \, x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 77, normalized size = 1.75 \[ -\frac {1}{4} \, \sqrt {2} \log \left ({\left | x + \frac {1}{4} \, \sqrt {10} + \frac {1}{4} \, \sqrt {2} \right |}\right ) + \frac {1}{4} \, \sqrt {2} \log \left ({\left | x + \frac {1}{4} \, \sqrt {10} - \frac {1}{4} \, \sqrt {2} \right |}\right ) - \frac {1}{4} \, \sqrt {2} \log \left ({\left | x - \frac {1}{4} \, \sqrt {10} + \frac {1}{4} \, \sqrt {2} \right |}\right ) + \frac {1}{4} \, \sqrt {2} \log \left ({\left | x - \frac {1}{4} \, \sqrt {10} - \frac {1}{4} \, \sqrt {2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 82, normalized size = 1.86 \[ -\frac {2 \left (-5+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {8 x}{2 \sqrt {10}-2 \sqrt {2}}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}-\frac {2 \left (5+\sqrt {5}\right ) \sqrt {5}\, \arctanh \left (\frac {8 x}{2 \sqrt {10}+2 \sqrt {2}}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )} \]
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 \[ \int \frac {2 \, x^{2} + 1}{4 \, x^{4} - 6 \, x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 20, normalized size = 0.45 \[ -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{2\,x^2-1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 46, normalized size = 1.05 \[ \frac {\sqrt {2} \log {\left (x^{2} - \frac {\sqrt {2} x}{2} - \frac {1}{2} \right )}}{4} - \frac {\sqrt {2} \log {\left (x^{2} + \frac {\sqrt {2} x}{2} - \frac {1}{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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