Optimal. Leaf size=48 \[ \frac {\tanh ^{-1}\left (\frac {2 \sqrt {2} x+1}{\sqrt {5}}\right )}{\sqrt {10}}-\frac {\tanh ^{-1}\left (\frac {1-2 \sqrt {2} x}{\sqrt {5}}\right )}{\sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 48, 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} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 \sqrt {2} x+1}{\sqrt {5}}\right )}{\sqrt {10}}-\frac {\tanh ^{-1}\left (\frac {1-2 \sqrt {2} x}{\sqrt {5}}\right )}{\sqrt {10}} \end {gather*}
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 &=-\left (\frac {1}{4} \int \frac {1}{-\frac {1}{2}-\frac {x}{\sqrt {2}}+x^2} \, dx\right )-\frac {1}{4} \int \frac {1}{-\frac {1}{2}+\frac {x}{\sqrt {2}}+x^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {5}{2}-x^2} \, dx,x,-\frac {1}{\sqrt {2}}+2 x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {5}{2}-x^2} \, dx,x,\frac {1}{\sqrt {2}}+2 x\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {1-2 \sqrt {2} x}{\sqrt {5}}\right )}{\sqrt {10}}+\frac {\tanh ^{-1}\left (\frac {1+2 \sqrt {2} x}{\sqrt {5}}\right )}{\sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.88 \begin {gather*} \frac {\log \left (2 x^2+\sqrt {10} x+1\right )-\log \left (-2 x^2+\sqrt {10} x-1\right )}{2 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x^2}{1-6 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.59, size = 45, normalized size = 0.94 \begin {gather*} \frac {1}{20} \, \sqrt {10} \log \left (\frac {4 \, x^{4} + 14 \, x^{2} + 2 \, \sqrt {10} {\left (2 \, x^{3} + x\right )} + 1}{4 \, x^{4} - 6 \, x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 77, normalized size = 1.60 \begin {gather*} \frac {1}{20} \, \sqrt {10} \log \left ({\left | x + \frac {1}{4} \, \sqrt {10} + \frac {1}{4} \, \sqrt {2} \right |}\right ) + \frac {1}{20} \, \sqrt {10} \log \left ({\left | x + \frac {1}{4} \, \sqrt {10} - \frac {1}{4} \, \sqrt {2} \right |}\right ) - \frac {1}{20} \, \sqrt {10} \log \left ({\left | x - \frac {1}{4} \, \sqrt {10} + \frac {1}{4} \, \sqrt {2} \right |}\right ) - \frac {1}{20} \, \sqrt {10} \log \left ({\left | x - \frac {1}{4} \, \sqrt {10} - \frac {1}{4} \, \sqrt {2} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 82, normalized size = 1.71 \begin {gather*} \frac {2 \left (\sqrt {5}-1\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 (\sqrt {5}+1\right ) \sqrt {5}\, \arctanh \left (\frac {8 x}{2 \sqrt {10}+2 \sqrt {2}}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )} \end {gather*}
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} - 1}{4 \, x^{4} - 6 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 20, normalized size = 0.42 \begin {gather*} \frac {\sqrt {10}\,\mathrm {atanh}\left (\frac {\sqrt {10}\,x}{2\,x^2+1}\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 46, normalized size = 0.96 \begin {gather*} - \frac {\sqrt {10} \log {\left (x^{2} - \frac {\sqrt {10} x}{2} + \frac {1}{2} \right )}}{20} + \frac {\sqrt {10} \log {\left (x^{2} + \frac {\sqrt {10} x}{2} + \frac {1}{2} \right )}}{20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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