Optimal. Leaf size=29 \[ \frac {x}{e \left (e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)\right )} \]
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Rubi [F] time = 35.54, 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^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, 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^{-5+\frac {64 x^2}{e^4}} \left (-e^4 x-e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )-e^4 (4-x) \log (4-x)\right )}{(4-x) \left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\\ &=\int \left (\frac {e^{-5+\frac {64 x^2}{e^4}} \left (e^4+64 x^2-4 x^4\right )}{e^{\frac {\left (16+x^2\right )^2}{e^4}}-e^{\frac {64 x^2}{e^4}} \log (4-x)}-\frac {e^{-5+\frac {64 x^2}{e^4}} x \left (-e^4+256 x \log (4-x)-64 x^2 \log (4-x)-16 x^3 \log (4-x)+4 x^4 \log (4-x)\right )}{(-4+x) \left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}\right ) \, dx\\ &=\int \frac {e^{-5+\frac {64 x^2}{e^4}} \left (e^4+64 x^2-4 x^4\right )}{e^{\frac {\left (16+x^2\right )^2}{e^4}}-e^{\frac {64 x^2}{e^4}} \log (4-x)} \, dx-\int \frac {e^{-5+\frac {64 x^2}{e^4}} x \left (-e^4+256 x \log (4-x)-64 x^2 \log (4-x)-16 x^3 \log (4-x)+4 x^4 \log (4-x)\right )}{(-4+x) \left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\\ &=\int \frac {e^4+64 x^2-4 x^4}{e^5 \left (e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)\right )} \, dx-\int \frac {e^{-5+\frac {64 x^2}{e^4}} x \left (e^4-4 (-4+x)^2 x (4+x) \log (4-x)\right )}{(4-x) \left (e^{\frac {256+x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\\ &=\frac {\int \frac {e^4+64 x^2-4 x^4}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e^5}-\int \left (\frac {e^{-5+\frac {64 x^2}{e^4}} \left (-e^4+256 x \log (4-x)-64 x^2 \log (4-x)-16 x^3 \log (4-x)+4 x^4 \log (4-x)\right )}{\left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}+\frac {4 e^{-5+\frac {64 x^2}{e^4}} \left (-e^4+256 x \log (4-x)-64 x^2 \log (4-x)-16 x^3 \log (4-x)+4 x^4 \log (4-x)\right )}{(-4+x) \left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {e^{-5+\frac {64 x^2}{e^4}} \left (-e^4+256 x \log (4-x)-64 x^2 \log (4-x)-16 x^3 \log (4-x)+4 x^4 \log (4-x)\right )}{(-4+x) \left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\right )+\frac {\int \left (\frac {e^4}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)}+\frac {64 x^2}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)}-\frac {4 x^4}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)}\right ) \, dx}{e^5}-\int \frac {e^{-5+\frac {64 x^2}{e^4}} \left (-e^4+256 x \log (4-x)-64 x^2 \log (4-x)-16 x^3 \log (4-x)+4 x^4 \log (4-x)\right )}{\left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\\ &=-\left (4 \int \frac {e^{-5+\frac {64 x^2}{e^4}} \left (e^4-4 (-4+x)^2 x (4+x) \log (4-x)\right )}{(4-x) \left (e^{\frac {256+x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\right )-\frac {4 \int \frac {x^4}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e^5}+\frac {64 \int \frac {x^2}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e^5}+\frac {\int \frac {1}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e}-\int \frac {e^{-5+\frac {64 x^2}{e^4}} \left (-e^4+4 (-4+x)^2 x (4+x) \log (4-x)\right )}{\left (e^{\frac {256+x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\\ &=-\left (4 \int \left (-\frac {e^{-1+\frac {64 x^2}{e^4}}}{(-4+x) \left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}+\frac {256 e^{-5+\frac {64 x^2}{e^4}} x \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}-\frac {64 e^{-5+\frac {64 x^2}{e^4}} x^2 \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}-\frac {16 e^{-5+\frac {64 x^2}{e^4}} x^3 \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}+\frac {4 e^{-5+\frac {64 x^2}{e^4}} x^4 \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}\right ) \, dx\right )-\frac {4 \int \frac {x^4}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e^5}+\frac {64 \int \frac {x^2}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e^5}+\frac {\int \frac {1}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e}-\int \left (-\frac {e^{-1+\frac {64 x^2}{e^4}}}{\left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}+\frac {256 e^{-5+\frac {64 x^2}{e^4}} x \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}-\frac {64 e^{-5+\frac {64 x^2}{e^4}} x^2 \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}-\frac {16 e^{-5+\frac {64 x^2}{e^4}} x^3 \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}+\frac {4 e^{-5+\frac {64 x^2}{e^4}} x^4 \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}\right ) \, dx\\ &=4 \int \frac {e^{-1+\frac {64 x^2}{e^4}}}{(-4+x) \left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx-4 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x^4 \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx+16 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x^3 \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx-16 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x^4 \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx+64 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x^2 \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx+64 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x^3 \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx-256 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x \log (4-x)}{\left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx+256 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x^2 \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx-1024 \int \frac {e^{-5+\frac {64 x^2}{e^4}} x \log (4-x)}{(-4+x) \left (-e^{\frac {256}{e^4}+\frac {x^4}{e^4}}+e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx-\frac {4 \int \frac {x^4}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e^5}+\frac {64 \int \frac {x^2}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e^5}+\frac {\int \frac {1}{e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)} \, dx}{e}+\int \frac {e^{-1+\frac {64 x^2}{e^4}}}{\left (e^{\frac {256}{e^4}+\frac {x^4}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.28, size = 29, normalized size = 1.00 \begin {gather*} \frac {x}{e \left (e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 37, normalized size = 1.28 \begin {gather*} -\frac {x e^{4}}{e^{5} \log \left (-x + 4\right ) - e^{\left ({\left (x^{4} - 32 \, x^{2} + 5 \, e^{4} + 256\right )} e^{\left (-4\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 30, normalized size = 1.03
method | result | size |
risch | \(\frac {{\mathrm e}^{-1} x}{{\mathrm e}^{\left (x -4\right )^{2} \left (4+x \right )^{2} {\mathrm e}^{-4}}-\ln \left (-x +4\right )}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.68, size = 46, normalized size = 1.59 \begin {gather*} -\frac {x e^{\left (32 \, x^{2} e^{\left (-4\right )}\right )}}{e^{\left (32 \, x^{2} e^{\left (-4\right )} + 1\right )} \log \left (-x + 4\right ) - e^{\left (x^{4} e^{\left (-4\right )} + 256 \, e^{\left (-4\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 158, normalized size = 5.45 \begin {gather*} \frac {x\,{\mathrm {e}}^{-1}\,{\left (4\,{\mathrm {e}}^4-x\,{\mathrm {e}}^4\right )}^2-{\mathrm {e}}^{-5}\,\ln \left (4-x\right )\,{\left (4\,{\mathrm {e}}^4-x\,{\mathrm {e}}^4\right )}^2\,\left (4\,x^5-16\,x^4-64\,x^3+256\,x^2\right )}{\left (\ln \left (4-x\right )-{\mathrm {e}}^{{\mathrm {e}}^{-4}\,x^4-32\,{\mathrm {e}}^{-4}\,x^2+256\,{\mathrm {e}}^{-4}}\right )\,\left (x-4\right )\,\left (4\,{\mathrm {e}}^8-x\,{\mathrm {e}}^8+512\,x^2\,{\mathrm {e}}^4\,\ln \left (4-x\right )-32\,x^4\,{\mathrm {e}}^4\,\ln \left (4-x\right )+4\,x^5\,{\mathrm {e}}^4\,\ln \left (4-x\right )-1024\,x\,{\mathrm {e}}^4\,\ln \left (4-x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 27, normalized size = 0.93 \begin {gather*} \frac {x}{e e^{\frac {x^{4} - 32 x^{2} + 256}{e^{4}}} - e \log {\left (4 - x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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