Optimal. Leaf size=38 \[ \frac {\tanh ^{-1}\left (\frac {2 x+1}{\sqrt {5}}\right )}{\sqrt {5}}-\frac {\tanh ^{-1}\left (\frac {1-2 x}{\sqrt {5}}\right )}{\sqrt {5}} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1161, 618, 206} \[ \frac {\tanh ^{-1}\left (\frac {2 x+1}{\sqrt {5}}\right )}{\sqrt {5}}-\frac {\tanh ^{-1}\left (\frac {1-2 x}{\sqrt {5}}\right )}{\sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1-x^2}{1-3 x^2+x^4} \, dx &=-\left (\frac {1}{2} \int \frac {1}{-1-x+x^2} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+x+x^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{5-x^2} \, dx,x,-1+2 x\right )+\operatorname {Subst}\left (\int \frac {1}{5-x^2} \, dx,x,1+2 x\right )\\ &=\frac {\tanh ^{-1}\left (\frac {-1+2 x}{\sqrt {5}}\right )}{\sqrt {5}}+\frac {\tanh ^{-1}\left (\frac {1+2 x}{\sqrt {5}}\right )}{\sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.05 \[ \frac {\log \left (x^2+\sqrt {5} x+1\right )-\log \left (-x^2+\sqrt {5} x-1\right )}{2 \sqrt {5}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 39, normalized size = 1.03 \[ \frac {1}{10} \, \sqrt {5} \log \left (\frac {x^{4} + 7 \, x^{2} + 2 \, \sqrt {5} {\left (x^{3} + x\right )} + 1}{x^{4} - 3 \, x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 39, normalized size = 1.03 \[ -\frac {1}{10} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {5} + \frac {2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt {5} + \frac {2}{x} \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.89 \[ \frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x +1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x -1\right ) \sqrt {5}}{5}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.46, size = 55, normalized size = 1.45 \[ -\frac {1}{10} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} + 1}{2 \, x + \sqrt {5} + 1}\right ) - \frac {1}{10} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} - 1}{2 \, x + \sqrt {5} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 18, normalized size = 0.47 \[ \frac {\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,x}{x^2+1}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 39, normalized size = 1.03 \[ - \frac {\sqrt {5} \log {\left (x^{2} - \sqrt {5} x + 1 \right )}}{10} + \frac {\sqrt {5} \log {\left (x^{2} + \sqrt {5} x + 1 \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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