Optimal. Leaf size=23 \[ -\frac {1}{5} e^{\frac {x}{2 (-2+x)}} x+3 x \log (4) \]
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Rubi [F]
time = 0.22, 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 {e^{\frac {2 x}{-8+4 x}} \left (-4+5 x-x^2\right )+\left (60-60 x+15 x^2\right ) \log (4)}{20-20 x+5 x^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 {e^{\frac {2 x}{-8+4 x}} \left (-4+5 x-x^2\right )+\left (60-60 x+15 x^2\right ) \log (4)}{5 (-2+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {e^{\frac {2 x}{-8+4 x}} \left (-4+5 x-x^2\right )+\left (60-60 x+15 x^2\right ) \log (4)}{(-2+x)^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {e^{\frac {x}{2 (-2+x)}} \left (4-5 x+x^2\right )}{(-2+x)^2}+15 \log (4)\right ) \, dx\\ &=3 x \log (4)-\frac {1}{5} \int \frac {e^{\frac {x}{2 (-2+x)}} \left (4-5 x+x^2\right )}{(-2+x)^2} \, dx\\ &=3 x \log (4)-\frac {1}{5} \int \left (e^{\frac {x}{2 (-2+x)}}+\frac {e^{\frac {x}{2 (-2+x)}}}{2-x}-\frac {2 e^{\frac {x}{2 (-2+x)}}}{(-2+x)^2}\right ) \, dx\\ &=3 x \log (4)-\frac {1}{5} \int e^{\frac {x}{2 (-2+x)}} \, dx-\frac {1}{5} \int \frac {e^{\frac {x}{2 (-2+x)}}}{2-x} \, dx+\frac {2}{5} \int \frac {e^{\frac {x}{2 (-2+x)}}}{(-2+x)^2} \, dx\\ &=3 x \log (4)-\frac {1}{5} \int e^{\frac {x}{2 (-2+x)}} \, dx-\frac {1}{5} \int \frac {e^{\frac {1}{2}+\frac {1}{-2+x}}}{2-x} \, dx+\frac {2}{5} \int \frac {e^{\frac {1}{2}+\frac {1}{-2+x}}}{(-2+x)^2} \, dx\\ &=-\frac {2}{5} e^{\frac {1}{2}+\frac {1}{-2+x}}-\frac {1}{5} \sqrt {e} \text {Ei}\left (\frac {1}{-2+x}\right )+3 x \log (4)-\frac {1}{5} \int e^{\frac {x}{2 (-2+x)}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 25, normalized size = 1.09 \begin {gather*} -\frac {1}{5} e^{\frac {1}{2} \left (1+\frac {2}{-2+x}\right )} x+x \log (64) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 40, normalized size = 1.74
method | result | size |
risch | \(6 x \ln \left (2\right )-\frac {x \,{\mathrm e}^{\frac {x}{2 x -4}}}{5}\) | \(19\) |
derivativedivides | \(-\frac {{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}} \left (x -2\right )}{5}+6 \left (x -2\right ) \ln \left (2\right )-\frac {2 \,{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}}}{5}\) | \(40\) |
default | \(-\frac {{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}} \left (x -2\right )}{5}+6 \left (x -2\right ) \ln \left (2\right )-\frac {2 \,{\mathrm e}^{\frac {1}{2}+\frac {1}{x -2}}}{5}\) | \(40\) |
norman | \(\frac {\frac {2 x \,{\mathrm e}^{\frac {2 x}{4 x -8}}}{5}-\frac {x^{2} {\mathrm e}^{\frac {2 x}{4 x -8}}}{5}+6 x^{2} \ln \left (2\right )-24 \ln \left (2\right )}{x -2}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 18, normalized size = 0.78 \begin {gather*} -\frac {1}{5} \, x e^{\left (\frac {x}{2 \, {\left (x - 2\right )}}\right )} + 6 \, x \log \left (2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 19, normalized size = 0.83 \begin {gather*} - \frac {x e^{\frac {2 x}{4 x - 8}}}{5} + 6 x \log {\left (2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 34, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (\frac {x e^{\left (\frac {x}{2 \, {\left (x - 2\right )}}\right )}}{x - 2} - 30 \, \log \left (2\right )\right )}}{5 \, {\left (\frac {x}{x - 2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 18, normalized size = 0.78 \begin {gather*} x\,\left (\ln \left (64\right )-\frac {{\mathrm {e}}^{\frac {2\,x}{4\,x-8}}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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