Optimal. Leaf size=25 \[ e^{-2-\frac {-1+\frac {5}{x}}{x-\log (2)}}-x \]
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Rubi [F] time = 3.86, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-x^4+e^{\frac {5-x+2 x^2-2 x \log (2)}{-x^2+x \log (2)}} \left (10 x-x^2-5 \log (2)\right )+2 x^3 \log (2)-x^2 \log ^2(2)}{x^4-2 x^3 \log (2)+x^2 \log ^2(2)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^4+e^{\frac {5-x+2 x^2-2 x \log (2)}{-x^2+x \log (2)}} \left (10 x-x^2-5 \log (2)\right )+2 x^3 \log (2)-x^2 \log ^2(2)}{x^2 \left (x^2-2 x \log (2)+\log ^2(2)\right )} \, dx\\ &=\int \frac {-x^4+e^{\frac {5-x+2 x^2-2 x \log (2)}{-x^2+x \log (2)}} \left (10 x-x^2-5 \log (2)\right )+2 x^3 \log (2)-x^2 \log ^2(2)}{x^2 (x-\log (2))^2} \, dx\\ &=\int \left (-1+\frac {4^{\frac {1}{x-\log (2)}} e^{\frac {-5+x-2 x^2}{x (x-\log (2))}} \left (10 x-x^2-\log (32)\right )}{x^2 (x-\log (2))^2}\right ) \, dx\\ &=-x+\int \frac {4^{\frac {1}{x-\log (2)}} e^{\frac {-5+x-2 x^2}{x (x-\log (2))}} \left (10 x-x^2-\log (32)\right )}{x^2 (x-\log (2))^2} \, dx\\ &=-x+\int \left (-\frac {4^{\frac {1}{x-\log (2)}} e^{\frac {-5+x-2 x^2}{x (x-\log (2))}} \log (32)}{x^2 \log ^2(2)}+\frac {4^{\frac {1}{x-\log (2)}} e^{\frac {-5+x-2 x^2}{x (x-\log (2))}} \left (-\log ^2(2)+\log (32)\right )}{(x-\log (2))^2 \log ^2(2)}\right ) \, dx\\ &=-x-\frac {\log (32) \int \frac {4^{\frac {1}{x-\log (2)}} e^{\frac {-5+x-2 x^2}{x (x-\log (2))}}}{x^2} \, dx}{\log ^2(2)}+\left (-1+\frac {\log (32)}{\log ^2(2)}\right ) \int \frac {4^{\frac {1}{x-\log (2)}} e^{\frac {-5+x-2 x^2}{x (x-\log (2))}}}{(x-\log (2))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 3.10, size = 30, normalized size = 1.20 \begin {gather*} e^{\frac {-5+x-2 x^2+x \log (4)}{x (x-\log (2))}}-x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 33, normalized size = 1.32 \begin {gather*} -x + e^{\left (-\frac {2 \, x^{2} - 2 \, x \log \relax (2) - x + 5}{x^{2} - x \log \relax (2)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 64, normalized size = 2.56 \begin {gather*} -x + e^{\left (-\frac {2 \, x^{2}}{x^{2} - x \log \relax (2)} + \frac {2 \, x \log \relax (2)}{x^{2} - x \log \relax (2)} + \frac {x}{x^{2} - x \log \relax (2)} - \frac {5}{x^{2} - x \log \relax (2)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 32, normalized size = 1.28
method | result | size |
risch | \(-x +{\mathrm e}^{-\frac {2 x \ln \relax (2)-2 x^{2}+x -5}{x \left (\ln \relax (2)-x \right )}}\) | \(32\) |
norman | \(\frac {x^{3}-x \ln \relax (2)^{2}+x \ln \relax (2) {\mathrm e}^{\frac {-2 x \ln \relax (2)+2 x^{2}-x +5}{x \ln \relax (2)-x^{2}}}-x^{2} {\mathrm e}^{\frac {-2 x \ln \relax (2)+2 x^{2}-x +5}{x \ln \relax (2)-x^{2}}}}{\left (\ln \relax (2)-x \right ) x}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 89, normalized size = 3.56 \begin {gather*} -2 \, {\left (\frac {\log \relax (2)}{x - \log \relax (2)} - \log \left (x - \log \relax (2)\right )\right )} \log \relax (2) - 2 \, \log \relax (2) \log \left (x - \log \relax (2)\right ) - x + \frac {2 \, \log \relax (2)^{2}}{x - \log \relax (2)} + e^{\left (-\frac {5}{x \log \relax (2) - \log \relax (2)^{2}} + \frac {1}{x - \log \relax (2)} + \frac {5}{x \log \relax (2)} - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 73, normalized size = 2.92 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {x}{x\,\ln \relax (2)-x^2}}\,{\mathrm {e}}^{\frac {2\,x^2}{x\,\ln \relax (2)-x^2}}\,{\mathrm {e}}^{\frac {5}{x\,\ln \relax (2)-x^2}}}{2^{\frac {2\,x}{x\,\ln \relax (2)-x^2}}}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 26, normalized size = 1.04 \begin {gather*} - x + e^{\frac {2 x^{2} - 2 x \log {\relax (2 )} - x + 5}{- x^{2} + x \log {\relax (2 )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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