Optimal. Leaf size=26 \[ x+x^2-\frac {5 x+\log (\log (4))}{\left (\frac {1}{x}-x\right ) x} \]
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Rubi [A]
time = 0.04, antiderivative size = 23, normalized size of antiderivative = 0.88, number of steps
used = 5, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6, 28, 1828,
1600} \begin {gather*} x^2-\frac {5 x+\log (\log (4))}{1-x^2}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 28
Rule 1600
Rule 1828
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-7 x^2-4 x^3+x^4+2 x^5+x (2-2 \log (\log (4)))}{1-2 x^2+x^4} \, dx\\ &=\int \frac {-4-7 x^2-4 x^3+x^4+2 x^5+x (2-2 \log (\log (4)))}{\left (-1+x^2\right )^2} \, dx\\ &=-\frac {5 x+\log (\log (4))}{1-x^2}+\frac {1}{2} \int \frac {-2-4 x+2 x^2+4 x^3}{-1+x^2} \, dx\\ &=-\frac {5 x+\log (\log (4))}{1-x^2}+\frac {1}{2} \int (2+4 x) \, dx\\ &=x+x^2-\frac {5 x+\log (\log (4))}{1-x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.77 \begin {gather*} x+x^2+\frac {5 x+\log (\log (4))}{-1+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.02, size = 38, normalized size = 1.46
method | result | size |
risch | \(x^{2}+x +\frac {\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )+5 x}{x^{2}-1}\) | \(23\) |
gosper | \(\frac {x^{4}+x^{3}+\ln \left (2 \ln \left (2\right )\right )+4 x -1}{x^{2}-1}\) | \(25\) |
norman | \(\frac {x^{3}+x^{4}+4 x -1+\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}{x^{2}-1}\) | \(25\) |
default | \(x^{2}+x -\frac {-\frac {5}{2}+\frac {\ln \left (2 \ln \left (2\right )\right )}{2}}{x +1}-\frac {-\frac {5}{2}-\frac {\ln \left (2 \ln \left (2\right )\right )}{2}}{x -1}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 22, normalized size = 0.85 \begin {gather*} x^{2} + x + \frac {5 \, x + \log \left (2 \, \log \left (2\right )\right )}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 28, normalized size = 1.08 \begin {gather*} \frac {x^{4} + x^{3} - x^{2} + 4 \, x + \log \left (2 \, \log \left (2\right )\right )}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 20, normalized size = 0.77 \begin {gather*} x^{2} + x + \frac {5 x + \log {\left (\log {\left (2 \right )} \right )} + \log {\left (2 \right )}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 22, normalized size = 0.85 \begin {gather*} x^{2} + x + \frac {5 \, x + \log \left (2 \, \log \left (2\right )\right )}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 20, normalized size = 0.77 \begin {gather*} x+\frac {5\,x+\ln \left (\ln \left (4\right )\right )}{x^2-1}+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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