Optimal. Leaf size=32 \[ \frac {e^{2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}}}{e^5-x} \]
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Rubi [F]
time = 106.30, 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 {\exp \left (2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}\right ) \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}\right ) \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{\left (405+5 e^2\right ) x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx\\ &=\int \frac {\exp \left (2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (5 e^{2+\frac {e^4}{5 \left (e+(3+x)^2\right )}}+4 e^9 (3+x)-4 e^4 x (3+x)+10 e^{1+\frac {e^4}{5 \left (e+(3+x)^2\right )}} (3+x)^2+5 e^{\frac {e^4}{5 \left (e+(3+x)^2\right )}} (3+x)^4\right )}{5 \left (e^5 (9+e)-\left (9+e-6 e^5\right ) x-\left (6-e^5\right ) x^2-x^3\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (5 e^{2+\frac {e^4}{5 \left (e+(3+x)^2\right )}}+4 e^9 (3+x)-4 e^4 x (3+x)+10 e^{1+\frac {e^4}{5 \left (e+(3+x)^2\right )}} (3+x)^2+5 e^{\frac {e^4}{5 \left (e+(3+x)^2\right )}} (3+x)^4\right )}{\left (e^5 (9+e)-\left (9+e-6 e^5\right ) x-\left (6-e^5\right ) x^2-x^3\right )^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {5 \exp \left (2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}+\frac {e^4}{5 \left (9+e+6 x+x^2\right )}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right )}{\left (e^5-x\right )^2}+\frac {4 \exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) (3+x)}{\left (e^5-x\right )^2 \left (9+e+6 x+x^2\right )^2}-\frac {4 \exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) x (3+x)}{\left (e^5-x\right )^2 \left (9+e+6 x+x^2\right )^2}\right ) \, dx\\ &=\frac {4}{5} \int \frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) (3+x)}{\left (e^5-x\right )^2 \left (9+e+6 x+x^2\right )^2} \, dx-\frac {4}{5} \int \frac {\exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) x (3+x)}{\left (e^5-x\right )^2 \left (9+e+6 x+x^2\right )^2} \, dx+\int \frac {\exp \left (2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}+\frac {e^4}{5 \left (9+e+6 x+x^2\right )}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right )}{\left (e^5-x\right )^2} \, dx\\ &=\frac {4}{5} \int \left (\frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (3+e^5\right )}{\left (9+e+6 e^5+e^{10}\right )^2 \left (e^5-x\right )^2}+\frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (27-e+18 e^5+3 e^{10}\right )}{\left (9+e+6 e^5+e^{10}\right )^3 \left (e^5-x\right )}+\frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (27-9 e+18 e^5-2 e^6+3 e^{10}+\left (9-e+6 e^5+e^{10}\right ) x\right )}{\left (9+e+6 e^5+e^{10}\right )^2 \left (9+e+6 x+x^2\right )^2}+\frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (135-9 e+108 e^5-2 e^6+27 e^{10}+2 e^{15}+\left (27-e+18 e^5+3 e^{10}\right ) x\right )}{\left (9+e+6 e^5+e^{10}\right )^3 \left (9+e+6 x+x^2\right )}\right ) \, dx-\frac {4}{5} \int \left (\frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (3+e^5\right )}{\left (9+e+6 e^5+e^{10}\right )^2 \left (e^5-x\right )^2}+\frac {\exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (-27-3 e-2 e^6+9 e^{10}+2 e^{15}\right )}{\left (9+e+6 e^5+e^{10}\right )^3 \left (e^5-x\right )}+\frac {\exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (-\left ((9+e) \left (9-e+6 e^5+e^{10}\right )\right )-\left (27+3 e+18 e^5+2 e^6+3 e^{10}\right ) x\right )}{\left (9+e+6 e^5+e^{10}\right )^2 \left (9+e+6 x+x^2\right )^2}+\frac {\exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (-162-18 e-54 e^5-18 e^6+27 e^{10}-3 e^{11}+12 e^{15}+e^{20}-\left (27+3 e+2 e^6-9 e^{10}-2 e^{15}\right ) x\right )}{\left (9+e+6 e^5+e^{10}\right )^3 \left (9+e+6 x+x^2\right )}\right ) \, dx+\int \frac {e^{2 e^{-\frac {e^4}{5 \left (9+e+6 x+x^2\right )}}}}{\left (e^5-x\right )^2} \, dx\\ &=\frac {4 \int \frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (135-9 e+108 e^5-2 e^6+27 e^{10}+2 e^{15}+\left (27-e+18 e^5+3 e^{10}\right ) x\right )}{9+e+6 x+x^2} \, dx}{5 \left (9+e+6 e^5+e^{10}\right )^3}-\frac {4 \int \frac {\exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (-162-18 e-54 e^5-18 e^6+27 e^{10}-3 e^{11}+12 e^{15}+e^{20}-\left (27+3 e+2 e^6-9 e^{10}-2 e^{15}\right ) x\right )}{9+e+6 x+x^2} \, dx}{5 \left (9+e+6 e^5+e^{10}\right )^3}+\frac {4 \int \frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (27-9 e+18 e^5-2 e^6+3 e^{10}+\left (9-e+6 e^5+e^{10}\right ) x\right )}{\left (9+e+6 x+x^2\right )^2} \, dx}{5 \left (9+e+6 e^5+e^{10}\right )^2}-\frac {4 \int \frac {\exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right ) \left (-\left ((9+e) \left (9-e+6 e^5+e^{10}\right )\right )-\left (27+3 e+18 e^5+2 e^6+3 e^{10}\right ) x\right )}{\left (9+e+6 x+x^2\right )^2} \, dx}{5 \left (9+e+6 e^5+e^{10}\right )^2}+\frac {\left (4 \left (27-e+18 e^5+3 e^{10}\right )\right ) \int \frac {\exp \left (9+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right )}{e^5-x} \, dx}{5 \left (9+e+6 e^5+e^{10}\right )^3}+\frac {\left (4 \left (27+3 e+2 e^6-9 e^{10}-2 e^{15}\right )\right ) \int \frac {\exp \left (4+2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}-\frac {e^4}{5 \left (e+(3+x)^2\right )}\right )}{e^5-x} \, dx}{5 \left (9+e+6 e^5+e^{10}\right )^3}+\int \frac {e^{2 e^{-\frac {e^4}{5 \left (9+e+6 x+x^2\right )}}}}{\left (e^5-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 = 0.16, size = 33, normalized size = 1.03 \begin {gather*} -\frac {e^{2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}}}{-e^5+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.54, size = 30, normalized size = 0.94
method | result | size |
risch | \(\frac {{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \left (x^{2}+{\mathrm e}+6 x +9\right )}}}}{{\mathrm e}^{5}-x}\) | \(30\) |
norman | \(\frac {\left (x^{2} {\mathrm e}^{\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}+\left ({\mathrm e}+9\right ) {\mathrm e}^{\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}+6 x \,{\mathrm e}^{\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}{\left (x^{2}+{\mathrm e}+6 x +9\right ) \left ({\mathrm e}^{5}-x \right )}\) | \(192\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (28) = 56\).
time = 0.39, size = 94, normalized size = 2.94 \begin {gather*} -\frac {e^{\left (\frac {{\left (10 \, x^{2} + 60 \, x + 10 \, e - e^{\left (\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}} + 4\right )} + 90\right )} e^{\left (-\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}} + \frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}}{x - e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.56, size = 29, normalized size = 0.91 \begin {gather*} - \frac {e^{2 e^{- \frac {e^{4}}{5 x^{2} + 30 x + 5 e + 45}}}}{x - e^{5}} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.06, size = 34, normalized size = 1.06 \begin {gather*} -\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^4}{5\,\left (x^2+6\,x+\mathrm {e}+9\right )}}}}{x-{\mathrm {e}}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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