Optimal. Leaf size=65 \[ \frac {1}{2} \log \left (-2 x-\sqrt {5}+1\right )+\frac {1}{2} \log \left (-2 x+\sqrt {5}+1\right )-\frac {1}{2} \log \left (2 x-\sqrt {5}+1\right )-\frac {1}{2} \log \left (2 x+\sqrt {5}+1\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1161, 616, 31} \begin {gather*} \frac {1}{2} \log \left (-2 x-\sqrt {5}+1\right )+\frac {1}{2} \log \left (-2 x+\sqrt {5}+1\right )-\frac {1}{2} \log \left (2 x-\sqrt {5}+1\right )-\frac {1}{2} \log \left (2 x+\sqrt {5}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+x^2}{1-3 x^2+x^4} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {5} x+x^2} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {5} x+x^2} \, dx\\ &=\frac {1}{2} \int \frac {1}{\frac {1}{2} \left (-1-\sqrt {5}\right )+x} \, dx-\frac {1}{2} \int \frac {1}{\frac {1}{2} \left (1-\sqrt {5}\right )+x} \, dx+\frac {1}{2} \int \frac {1}{\frac {1}{2} \left (-1+\sqrt {5}\right )+x} \, dx-\frac {1}{2} \int \frac {1}{\frac {1}{2} \left (1+\sqrt {5}\right )+x} \, dx\\ &=\frac {1}{2} \log \left (1-\sqrt {5}-2 x\right )+\frac {1}{2} \log \left (1+\sqrt {5}-2 x\right )-\frac {1}{2} \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{2} \log \left (1+\sqrt {5}+2 x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.45 \begin {gather*} \frac {1}{2} \log \left (-x^2+x+1\right )-\frac {1}{2} \log \left (-x^2-x+1\right ) \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+x^2}{1-3 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.20, size = 21, normalized size = 0.32 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac {1}{2} \, \log \left (x^{2} - x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 43, normalized size = 0.66 \begin {gather*} -\frac {1}{4} \, \log \left ({\left | x + \frac {1}{x - \frac {1}{x}} - \frac {1}{x} + 2 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x + \frac {1}{x - \frac {1}{x}} - \frac {1}{x} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.34 \begin {gather*} \frac {\ln \left (x^{2}-x -1\right )}{2}-\frac {\ln \left (x^{2}+x -1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 21, normalized size = 0.32 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} + x - 1\right ) + \frac {1}{2} \, \log \left (x^{2} - x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 12, normalized size = 0.18 \begin {gather*} -\mathrm {atanh}\left (\frac {x}{x^2-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.29 \begin {gather*} \frac {\log {\left (x^{2} - x - 1 \right )}}{2} - \frac {\log {\left (x^{2} + x - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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