Optimal. Leaf size=28 \[ -4+\frac {1-x}{-x+e^{-4+x} \log (4)}+\log \left (e^{2+x}\right ) \]
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Rubi [F]
time = 2.69, 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^{8-2 x} \left (1+x^2\right )+e^{4-x} (-2-x) \log (4)+\log ^2(4)}{e^{8-2 x} x^2-2 e^{4-x} x \log (4)+\log ^2(4)} \, 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^{2 x} \left (e^{8-2 x} \left (1+x^2\right )+e^{4-x} (-2-x) \log (4)+\log ^2(4)\right )}{\left (e^4 x-e^x \log (4)\right )^2} \, dx\\ &=\int \left (\frac {1+x^2}{x^2}+\frac {e^{-4+x} \left (2-2 x+x^2\right ) \log (4)}{x^3}+\frac {e^{2 x} (-1+x)^2 \log ^2(4)}{x^2 \left (e^4 x-e^x \log (4)\right )^2}+\frac {e^{-4+2 x} \left (2-2 x+x^2\right ) \log ^2(4)}{x^3 \left (e^4 x-e^x \log (4)\right )}\right ) \, dx\\ &=\log (4) \int \frac {e^{-4+x} \left (2-2 x+x^2\right )}{x^3} \, dx+\log ^2(4) \int \frac {e^{2 x} (-1+x)^2}{x^2 \left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{-4+2 x} \left (2-2 x+x^2\right )}{x^3 \left (e^4 x-e^x \log (4)\right )} \, dx+\int \frac {1+x^2}{x^2} \, dx\\ &=\log (4) \int \left (\frac {2 e^{-4+x}}{x^3}-\frac {2 e^{-4+x}}{x^2}+\frac {e^{-4+x}}{x}\right ) \, dx+\log ^2(4) \int \left (\frac {e^{2 x}}{\left (e^4 x-e^x \log (4)\right )^2}+\frac {e^{2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )^2}-\frac {2 e^{2 x}}{x \left (e^4 x-e^x \log (4)\right )^2}\right ) \, dx+\log ^2(4) \int \left (\frac {2 e^{-4+2 x}}{x^3 \left (e^4 x-e^x \log (4)\right )}-\frac {2 e^{-4+2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )}+\frac {e^{-4+2 x}}{x \left (e^4 x-e^x \log (4)\right )}\right ) \, dx+\int \left (1+\frac {1}{x^2}\right ) \, dx\\ &=-\frac {1}{x}+x+\log (4) \int \frac {e^{-4+x}}{x} \, dx+(2 \log (4)) \int \frac {e^{-4+x}}{x^3} \, dx-(2 \log (4)) \int \frac {e^{-4+x}}{x^2} \, dx+\log ^2(4) \int \frac {e^{2 x}}{\left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{-4+2 x}}{x \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{2 x}}{x \left (e^4 x-e^x \log (4)\right )^2} \, dx+\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^3 \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )} \, dx\\ &=-\frac {1}{x}+x-\frac {e^{-4+x} \log (4)}{x^2}+\frac {2 e^{-4+x} \log (4)}{x}+\frac {\text {Ei}(x) \log (4)}{e^4}+\log (4) \int \frac {e^{-4+x}}{x^2} \, dx-(2 \log (4)) \int \frac {e^{-4+x}}{x} \, dx+\log ^2(4) \int \frac {e^{2 x}}{\left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{-4+2 x}}{x \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{2 x}}{x \left (e^4 x-e^x \log (4)\right )^2} \, dx+\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^3 \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )} \, dx\\ &=-\frac {1}{x}+x-\frac {e^{-4+x} \log (4)}{x^2}+\frac {e^{-4+x} \log (4)}{x}-\frac {\text {Ei}(x) \log (4)}{e^4}+\log (4) \int \frac {e^{-4+x}}{x} \, dx+\log ^2(4) \int \frac {e^{2 x}}{\left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{-4+2 x}}{x \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{2 x}}{x \left (e^4 x-e^x \log (4)\right )^2} \, dx+\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^3 \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )} \, dx\\ &=-\frac {1}{x}+x-\frac {e^{-4+x} \log (4)}{x^2}+\frac {e^{-4+x} \log (4)}{x}+\log ^2(4) \int \frac {e^{2 x}}{\left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )^2} \, dx+\log ^2(4) \int \frac {e^{-4+2 x}}{x \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{2 x}}{x \left (e^4 x-e^x \log (4)\right )^2} \, dx+\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^3 \left (e^4 x-e^x \log (4)\right )} \, dx-\left (2 \log ^2(4)\right ) \int \frac {e^{-4+2 x}}{x^2 \left (e^4 x-e^x \log (4)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 24, normalized size = 0.86 \begin {gather*} x+\frac {e^4 (-1+x)}{e^4 x-e^x \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.45, size = 34, normalized size = 1.21
method | result | size |
risch | \(x -\frac {1}{x}-\frac {2 \left (x -1\right ) \ln \left (2\right )}{x \left (-x \,{\mathrm e}^{-x +4}+2 \ln \left (2\right )\right )}\) | \(34\) |
norman | \(\frac {2 x \ln \left (2\right )-x^{2} {\mathrm e}^{-x +4}-2 \ln \left (2\right )+{\mathrm e}^{-x +4}}{-x \,{\mathrm e}^{-x +4}+2 \ln \left (2\right )}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 36, normalized size = 1.29 \begin {gather*} \frac {x^{2} e^{4} - 2 \, x e^{x} \log \left (2\right ) + x e^{4} - e^{4}}{x e^{4} - 2 \, e^{x} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 36, normalized size = 1.29 \begin {gather*} \frac {{\left (x^{2} - 1\right )} e^{\left (-x + 4\right )} - 2 \, {\left (x - 1\right )} \log \left (2\right )}{x e^{\left (-x + 4\right )} - 2 \, \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 = 31, normalized size = 1.11 \begin {gather*} x + \frac {2 x \log {\left (2 \right )} - 2 \log {\left (2 \right )}}{x^{2} e^{4 - x} - 2 x \log {\left (2 \right )}} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 44, normalized size = 1.57 \begin {gather*} \frac {x^{2} e^{\left (-x + 4\right )} - 2 \, x \log \left (2\right ) - e^{\left (-x + 4\right )} + 2 \, \log \left (2\right )}{x e^{\left (-x + 4\right )} - 2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 28, normalized size = 1.00 \begin {gather*} \frac {x+x^2-x\,{\mathrm {e}}^{x-4}\,\ln \left (4\right )-1}{x-2\,{\mathrm {e}}^{x-4}\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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