Optimal. Leaf size=28 \[ \frac {2 x}{x-x^2}+\frac {4 \log ^{e^{2 x} x}(3)}{x} \]
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Rubi [A]
time = 0.77, antiderivative size = 51, normalized size of antiderivative = 1.82, number of steps
used = 5, number of rules used = 4, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1608, 27, 6820,
2326} \begin {gather*} \frac {2}{1-x}+\frac {4 e^{2 x} (2 x+1) \log ^{e^{2 x} x}(3)}{x \left (2 e^{2 x} x+e^{2 x}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1608
Rule 2326
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x^2+\log ^{e^{2 x} x}(3) \left (-4+8 x-4 x^2+e^{2 x} \left (4 x-12 x^3+8 x^4\right ) \log (\log (3))\right )}{x^2 \left (1-2 x+x^2\right )} \, dx\\ &=\int \frac {2 x^2+\log ^{e^{2 x} x}(3) \left (-4+8 x-4 x^2+e^{2 x} \left (4 x-12 x^3+8 x^4\right ) \log (\log (3))\right )}{(-1+x)^2 x^2} \, dx\\ &=\int \left (\frac {2}{(-1+x)^2}+\frac {4 \log ^{e^{2 x} x}(3) \left (-1+e^{2 x} x (1+2 x) \log (\log (3))\right )}{x^2}\right ) \, dx\\ &=\frac {2}{1-x}+4 \int \frac {\log ^{e^{2 x} x}(3) \left (-1+e^{2 x} x (1+2 x) \log (\log (3))\right )}{x^2} \, dx\\ &=\frac {2}{1-x}+\frac {4 e^{2 x} (1+2 x) \log ^{e^{2 x} x}(3)}{x \left (e^{2 x}+2 e^{2 x} x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 49, normalized size = 1.75 \begin {gather*} -\frac {2}{-1+x}+\frac {4 e^{2 x} (1+2 x) \log ^{e^{2 x} x}(3)}{x \left (e^{2 x}+2 e^{2 x} x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 23, normalized size = 0.82
method | result | size |
risch | \(-\frac {2}{x -1}+\frac {4 \ln \left (3\right )^{x \,{\mathrm e}^{2 x}}}{x}\) | \(23\) |
norman | \(\frac {-2 x +4 \,{\mathrm e}^{x \,{\mathrm e}^{2 x} \ln \left (\ln \left (3\right )\right )} x -4 \,{\mathrm e}^{x \,{\mathrm e}^{2 x} \ln \left (\ln \left (3\right )\right )}}{x \left (x -1\right )}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 29, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (2 \, {\left (x - 1\right )} \log \left (3\right )^{x e^{\left (2 \, x\right )}} - x\right )}}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 29, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (2 \, {\left (x - 1\right )} \log \left (3\right )^{x e^{\left (2 \, x\right )}} - x\right )}}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 20, normalized size = 0.71 \begin {gather*} - \frac {2}{x - 1} + \frac {4 e^{x e^{2 x} \log {\left (\log {\left (3 \right )} \right )}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 22, normalized size = 0.79 \begin {gather*} \frac {4\,{\ln \left (3\right )}^{x\,{\mathrm {e}}^{2\,x}}}{x}-\frac {2}{x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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