Optimal. Leaf size=27 \[ \left (16+e^{\left (-3 x+\left (5-e^{-2+x}\right ) x\right )^4}-\log (2)\right )^2 \]
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Rubi [A] time = 8.49, antiderivative size = 54, normalized size of antiderivative = 2.00, number of steps used = 8, number of rules used = 4, integrand size = 333, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6688, 12, 6706, 6} \begin {gather*} e^{\frac {2 \left (2 e^2-e^x\right )^4 x^4}{e^8}}+2 e^{\frac {\left (2 e^2-e^x\right )^4 x^4}{e^8}} (16-\log (2)) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 6688
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \exp \left (32 x^4-64 e^{-2+x} x^4+48 e^{-4+2 x} x^4-16 e^{-6+3 x} x^4+2 e^{-8+4 x} x^4\right ) \left (128 x^3+e^{-2+x} \left (-256 x^3-64 x^4\right )+e^{-6+3 x} \left (-64 x^3-48 x^4\right )+e^{-8+4 x} \left (8 x^3+8 x^4\right )+e^{-4+2 x} \left (192 x^3+96 x^4\right )\right ) \, dx+\int \exp \left (16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4\right ) \left (2048 x^3-128 x^3 \log (2)+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right ) \, dx\\ &=\int 8 e^{-8+\frac {2 \left (-2 e^2+e^x\right )^4 x^4}{e^8}} \left (2 e^2-e^x\right )^3 x^3 \left (2 e^2-e^x (1+x)\right ) \, dx+\int \exp \left (16 x^4-32 e^{-2+x} x^4+24 e^{-4+2 x} x^4-8 e^{-6+3 x} x^4+e^{-8+4 x} x^4\right ) \left (x^3 (2048-128 \log (2))+e^{-4+2 x} \left (3072 x^3+1536 x^4+\left (-192 x^3-96 x^4\right ) \log (2)\right )+e^{-8+4 x} \left (128 x^3+128 x^4+\left (-8 x^3-8 x^4\right ) \log (2)\right )+e^{-6+3 x} \left (-1024 x^3-768 x^4+\left (64 x^3+48 x^4\right ) \log (2)\right )+e^{-2+x} \left (-4096 x^3-1024 x^4+\left (256 x^3+64 x^4\right ) \log (2)\right )\right ) \, dx\\ &=8 \int e^{-8+\frac {2 \left (-2 e^2+e^x\right )^4 x^4}{e^8}} \left (2 e^2-e^x\right )^3 x^3 \left (2 e^2-e^x (1+x)\right ) \, dx+\int 8 e^{-8+\frac {\left (-2 e^2+e^x\right )^4 x^4}{e^8}} \left (2 e^2-e^x\right )^3 x^3 \left (2 e^2-e^x (1+x)\right ) (16-\log (2)) \, dx\\ &=e^{\frac {2 \left (2 e^2-e^x\right )^4 x^4}{e^8}}+(8 (16-\log (2))) \int e^{-8+\frac {\left (-2 e^2+e^x\right )^4 x^4}{e^8}} \left (2 e^2-e^x\right )^3 x^3 \left (2 e^2-e^x (1+x)\right ) \, dx\\ &=e^{\frac {2 \left (2 e^2-e^x\right )^4 x^4}{e^8}}+2 e^{\frac {\left (2 e^2-e^x\right )^4 x^4}{e^8}} (16-\log (2))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.48, size = 47, normalized size = 1.74 \begin {gather*} e^{\frac {\left (-2 e^2+e^x\right )^4 x^4}{e^8}} \left (32+e^{\frac {\left (-2 e^2+e^x\right )^4 x^4}{e^8}}-\log (4)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 104, normalized size = 3.85 \begin {gather*} -2 \, {\left (\log \relax (2) - 16\right )} e^{\left (x^{4} e^{\left (4 \, x - 8\right )} - 8 \, x^{4} e^{\left (3 \, x - 6\right )} + 24 \, x^{4} e^{\left (2 \, x - 4\right )} - 32 \, x^{4} e^{\left (x - 2\right )} + 16 \, x^{4}\right )} + e^{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} - 16 \, x^{4} e^{\left (3 \, x - 6\right )} + 48 \, x^{4} e^{\left (2 \, x - 4\right )} - 64 \, x^{4} e^{\left (x - 2\right )} + 32 \, x^{4}\right )} \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*} \int 8 \, {\left (16 \, x^{3} + {\left (x^{4} + x^{3}\right )} e^{\left (4 \, x - 8\right )} - 2 \, {\left (3 \, x^{4} + 4 \, x^{3}\right )} e^{\left (3 \, x - 6\right )} + 12 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{\left (2 \, x - 4\right )} - 8 \, {\left (x^{4} + 4 \, x^{3}\right )} e^{\left (x - 2\right )}\right )} e^{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} - 16 \, x^{4} e^{\left (3 \, x - 6\right )} + 48 \, x^{4} e^{\left (2 \, x - 4\right )} - 64 \, x^{4} e^{\left (x - 2\right )} + 32 \, x^{4}\right )} - 8 \, {\left (16 \, x^{3} \log \relax (2) - 256 \, x^{3} - {\left (16 \, x^{4} + 16 \, x^{3} - {\left (x^{4} + x^{3}\right )} \log \relax (2)\right )} e^{\left (4 \, x - 8\right )} + 2 \, {\left (48 \, x^{4} + 64 \, x^{3} - {\left (3 \, x^{4} + 4 \, x^{3}\right )} \log \relax (2)\right )} e^{\left (3 \, x - 6\right )} - 12 \, {\left (16 \, x^{4} + 32 \, x^{3} - {\left (x^{4} + 2 \, x^{3}\right )} \log \relax (2)\right )} e^{\left (2 \, x - 4\right )} + 8 \, {\left (16 \, x^{4} + 64 \, x^{3} - {\left (x^{4} + 4 \, x^{3}\right )} \log \relax (2)\right )} e^{\left (x - 2\right )}\right )} e^{\left (x^{4} e^{\left (4 \, x - 8\right )} - 8 \, x^{4} e^{\left (3 \, x - 6\right )} + 24 \, x^{4} e^{\left (2 \, x - 4\right )} - 32 \, x^{4} e^{\left (x - 2\right )} + 16 \, x^{4}\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 114, normalized size = 4.22
method | result | size |
risch | \({\mathrm e}^{2 x^{4} \left (-32 \,{\mathrm e}^{x -2}+{\mathrm e}^{4 x -8}-8 \,{\mathrm e}^{3 x -6}+24 \,{\mathrm e}^{2 x -4}+16\right )}-2 \,{\mathrm e}^{x^{4} \left (-32 \,{\mathrm e}^{x -2}+{\mathrm e}^{4 x -8}-8 \,{\mathrm e}^{3 x -6}+24 \,{\mathrm e}^{2 x -4}+16\right )} \ln \relax (2)+32 \,{\mathrm e}^{x^{4} \left (-32 \,{\mathrm e}^{x -2}+{\mathrm e}^{4 x -8}-8 \,{\mathrm e}^{3 x -6}+24 \,{\mathrm e}^{2 x -4}+16\right )}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.83, size = 104, normalized size = 3.85 \begin {gather*} -2 \, {\left (\log \relax (2) - 16\right )} e^{\left (x^{4} e^{\left (4 \, x - 8\right )} - 8 \, x^{4} e^{\left (3 \, x - 6\right )} + 24 \, x^{4} e^{\left (2 \, x - 4\right )} - 32 \, x^{4} e^{\left (x - 2\right )} + 16 \, x^{4}\right )} + e^{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} - 16 \, x^{4} e^{\left (3 \, x - 6\right )} + 48 \, x^{4} e^{\left (2 \, x - 4\right )} - 64 \, x^{4} e^{\left (x - 2\right )} + 32 \, x^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\mathrm {e}}^{48\,x^4\,{\mathrm {e}}^{2\,x-4}-64\,x^4\,{\mathrm {e}}^{x-2}-16\,x^4\,{\mathrm {e}}^{3\,x-6}+2\,x^4\,{\mathrm {e}}^{4\,x-8}+32\,x^4}\,\left ({\mathrm {e}}^{4\,x-8}\,\left (8\,x^4+8\,x^3\right )-{\mathrm {e}}^{x-2}\,\left (64\,x^4+256\,x^3\right )-{\mathrm {e}}^{3\,x-6}\,\left (48\,x^4+64\,x^3\right )+{\mathrm {e}}^{2\,x-4}\,\left (96\,x^4+192\,x^3\right )+128\,x^3\right )-{\mathrm {e}}^{24\,x^4\,{\mathrm {e}}^{2\,x-4}-32\,x^4\,{\mathrm {e}}^{x-2}-8\,x^4\,{\mathrm {e}}^{3\,x-6}+x^4\,{\mathrm {e}}^{4\,x-8}+16\,x^4}\,\left ({\mathrm {e}}^{x-2}\,\left (4096\,x^3-\ln \relax (2)\,\left (64\,x^4+256\,x^3\right )+1024\,x^4\right )-{\mathrm {e}}^{4\,x-8}\,\left (128\,x^3-\ln \relax (2)\,\left (8\,x^4+8\,x^3\right )+128\,x^4\right )+{\mathrm {e}}^{3\,x-6}\,\left (1024\,x^3-\ln \relax (2)\,\left (48\,x^4+64\,x^3\right )+768\,x^4\right )-{\mathrm {e}}^{2\,x-4}\,\left (3072\,x^3-\ln \relax (2)\,\left (96\,x^4+192\,x^3\right )+1536\,x^4\right )+128\,x^3\,\ln \relax (2)-2048\,x^3\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.76, size = 109, normalized size = 4.04 \begin {gather*} e^{- 64 x^{4} e^{x - 2} + 48 x^{4} e^{2 x - 4} - 16 x^{4} e^{3 x - 6} + 2 x^{4} e^{4 x - 8} + 32 x^{4}} + \left (32 - 2 \log {\relax (2 )}\right ) e^{- 32 x^{4} e^{x - 2} + 24 x^{4} e^{2 x - 4} - 8 x^{4} e^{3 x - 6} + x^{4} e^{4 x - 8} + 16 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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