Optimal. Leaf size=30 \[ \left (2+e^{\frac {1}{5 \left (1+3 e^{4-e^x}\right )^2 x^2}}\right )^2+x \]
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Rubi [F] time = 54.22, 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 {5 x^3+135 e^{12-3 e^x} x^3+135 e^{8-2 e^x} x^3+45 e^{4-e^x} x^3+\exp \left (\frac {2}{5 x^2+45 e^{8-2 e^x} x^2+30 e^{4-e^x} x^2}\right ) \left (-4+e^{4-e^x} \left (-12+12 e^x x\right )\right )+\exp \left (\frac {1}{5 x^2+45 e^{8-2 e^x} x^2+30 e^{4-e^x} x^2}\right ) \left (-8+e^{4-e^x} \left (-24+24 e^x x\right )\right )}{5 x^3+135 e^{12-3 e^x} x^3+135 e^{8-2 e^x} x^3+45 e^{4-e^x} x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{3 e^x} \left (5 x^3+135 e^{12-3 e^x} x^3+135 e^{8-2 e^x} x^3+45 e^{4-e^x} x^3+\exp \left (\frac {2}{5 x^2+45 e^{8-2 e^x} x^2+30 e^{4-e^x} x^2}\right ) \left (-4+e^{4-e^x} \left (-12+12 e^x x\right )\right )+\exp \left (\frac {1}{5 x^2+45 e^{8-2 e^x} x^2+30 e^{4-e^x} x^2}\right ) \left (-8+e^{4-e^x} \left (-24+24 e^x x\right )\right )\right )}{5 \left (3 e^4+e^{e^x}\right )^3 x^3} \, dx\\ &=\frac {1}{5} \int \frac {e^{3 e^x} \left (5 x^3+135 e^{12-3 e^x} x^3+135 e^{8-2 e^x} x^3+45 e^{4-e^x} x^3+\exp \left (\frac {2}{5 x^2+45 e^{8-2 e^x} x^2+30 e^{4-e^x} x^2}\right ) \left (-4+e^{4-e^x} \left (-12+12 e^x x\right )\right )+\exp \left (\frac {1}{5 x^2+45 e^{8-2 e^x} x^2+30 e^{4-e^x} x^2}\right ) \left (-8+e^{4-e^x} \left (-24+24 e^x x\right )\right )\right )}{\left (3 e^4+e^{e^x}\right )^3 x^3} \, dx\\ &=\frac {1}{5} \int \left (\frac {5 e^{3 e^x}}{\left (3 e^4+e^{e^x}\right )^3}+\frac {45 e^{4+2 e^x}}{\left (3 e^4+e^{e^x}\right )^3}+\frac {135 e^{3 e^x-3 \left (-4+e^x\right )}}{\left (3 e^4+e^{e^x}\right )^3}+\frac {135 e^{3 e^x-2 \left (-4+e^x\right )}}{\left (3 e^4+e^{e^x}\right )^3}-\frac {8 e^{2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}} \left (3 e^4+e^{e^x}-3 e^{4+x} x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^3}-\frac {4 \exp \left (2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}\right ) \left (3 e^4+e^{e^x}-3 e^{4+x} x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^3}\right ) \, dx\\ &=-\left (\frac {4}{5} \int \frac {\exp \left (2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}\right ) \left (3 e^4+e^{e^x}-3 e^{4+x} x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^3} \, dx\right )-\frac {8}{5} \int \frac {e^{2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}} \left (3 e^4+e^{e^x}-3 e^{4+x} x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^3} \, dx+9 \int \frac {e^{4+2 e^x}}{\left (3 e^4+e^{e^x}\right )^3} \, dx+27 \int \frac {e^{3 e^x-3 \left (-4+e^x\right )}}{\left (3 e^4+e^{e^x}\right )^3} \, dx+27 \int \frac {e^{3 e^x-2 \left (-4+e^x\right )}}{\left (3 e^4+e^{e^x}\right )^3} \, dx+\int \frac {e^{3 e^x}}{\left (3 e^4+e^{e^x}\right )^3} \, dx\\ &=-\left (\frac {4}{5} \int \left (\frac {\exp \left (2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}\right )}{\left (3 e^4+e^{e^x}\right )^2 x^3}-\frac {3 \exp \left (4+2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}+x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^2}\right ) \, dx\right )-\frac {8}{5} \int \left (\frac {e^{2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}}}{\left (3 e^4+e^{e^x}\right )^2 x^3}-\frac {3 \exp \left (4+2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}+x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^2}\right ) \, dx+9 \operatorname {Subst}\left (\int \frac {e^{4+2 x}}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )+27 \operatorname {Subst}\left (\int \frac {e^{12}}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )+27 \operatorname {Subst}\left (\int \frac {e^{8+x}}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {e^{3 x}}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )\\ &=-\left (\frac {4}{5} \int \frac {\exp \left (2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}\right )}{\left (3 e^4+e^{e^x}\right )^2 x^3} \, dx\right )-\frac {8}{5} \int \frac {e^{2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}}}{\left (3 e^4+e^{e^x}\right )^2 x^3} \, dx+\frac {12}{5} \int \frac {\exp \left (4+2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}+x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^2} \, dx+\frac {24}{5} \int \frac {\exp \left (4+2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}+x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^2} \, dx+9 \operatorname {Subst}\left (\int \frac {e^{4+2 x}}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )+\left (27 e^8\right ) \operatorname {Subst}\left (\int \frac {e^x}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )+\left (27 e^{12}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \left (\frac {1}{x}-\frac {27 e^{12}}{\left (3 e^4+e^x\right )^3 x}+\frac {27 e^8}{\left (3 e^4+e^x\right )^2 x}-\frac {9 e^4}{\left (3 e^4+e^x\right ) x}\right ) \, dx,x,e^x\right )\\ &=-\frac {27 e^{8-x}}{2 \left (3 e^4+e^{e^x}\right )^2}+x-\frac {4}{5} \int \frac {\exp \left (2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}\right )}{\left (3 e^4+e^{e^x}\right )^2 x^3} \, dx-\frac {8}{5} \int \frac {e^{2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}}}{\left (3 e^4+e^{e^x}\right )^2 x^3} \, dx+\frac {12}{5} \int \frac {\exp \left (4+2 e^x+\frac {2 e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}+x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^2} \, dx+\frac {24}{5} \int \frac {\exp \left (4+2 e^x+\frac {e^{2 e^x}}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}+x\right )}{\left (3 e^4+e^{e^x}\right )^3 x^2} \, dx+9 \operatorname {Subst}\left (\int \frac {e^{4+2 x}}{\left (3 e^4+e^x\right )^3 x} \, dx,x,e^x\right )-\left (9 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3 e^4+e^x\right ) x} \, dx,x,e^x\right )-\frac {1}{2} \left (27 e^8\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3 e^4+e^x\right )^2 x^2} \, dx,x,e^x\right )+\left (27 e^8\right ) \operatorname {Subst}\left (\int \frac {1}{\left (3 e^4+e^x\right )^2 x} \, dx,x,e^x\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 2.01, size = 124, normalized size = 4.13 \begin {gather*} \frac {1}{5} \left (5 e^{\frac {2}{5 x^2}+\frac {18 e^8}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}-\frac {12 e^4}{5 \left (3 e^4+e^{e^x}\right ) x^2}}+20 e^{\frac {1}{5 x^2}+\frac {9 e^8}{5 \left (3 e^4+e^{e^x}\right )^2 x^2}-\frac {6 e^4}{5 \left (3 e^4+e^{e^x}\right ) x^2}}+5 x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 70, normalized size = 2.33 \begin {gather*} x + e^{\left (\frac {2}{5 \, {\left (6 \, x^{2} e^{\left (-e^{x} + 4\right )} + 9 \, x^{2} e^{\left (-2 \, e^{x} + 8\right )} + x^{2}\right )}}\right )} + 4 \, e^{\left (\frac {1}{5 \, {\left (6 \, x^{2} e^{\left (-e^{x} + 4\right )} + 9 \, x^{2} e^{\left (-2 \, e^{x} + 8\right )} + x^{2}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 61, normalized size = 2.03
method | result | size |
risch | \({\mathrm e}^{\frac {2}{5 x^{2} \left (9 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+8}+6 \,{\mathrm e}^{-{\mathrm e}^{x}+4}+1\right )}}+x +4 \,{\mathrm e}^{\frac {1}{5 x^{2} \left (9 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+8}+6 \,{\mathrm e}^{-{\mathrm e}^{x}+4}+1\right )}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 78, normalized size = 2.60 \begin {gather*} x + e^{\left (\frac {2 \, e^{\left (2 \, e^{x}\right )}}{5 \, {\left (9 \, x^{2} e^{8} + x^{2} e^{\left (2 \, e^{x}\right )} + 6 \, x^{2} e^{\left (e^{x} + 4\right )}\right )}}\right )} + 4 \, e^{\left (\frac {e^{\left (2 \, e^{x}\right )}}{5 \, {\left (9 \, x^{2} e^{8} + x^{2} e^{\left (2 \, e^{x}\right )} + 6 \, x^{2} e^{\left (e^{x} + 4\right )}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.73, size = 72, normalized size = 2.40 \begin {gather*} x+{\mathrm {e}}^{\frac {2}{5\,x^2+30\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^{-{\mathrm {e}}^x}+45\,x^2\,{\mathrm {e}}^8\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}}}+4\,{\mathrm {e}}^{\frac {1}{5\,x^2+30\,x^2\,{\mathrm {e}}^4\,{\mathrm {e}}^{-{\mathrm {e}}^x}+45\,x^2\,{\mathrm {e}}^8\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.87, size = 70, normalized size = 2.33 \begin {gather*} x + e^{\frac {2}{30 x^{2} e^{4 - e^{x}} + 45 x^{2} e^{8 - 2 e^{x}} + 5 x^{2}}} + 4 e^{\frac {1}{30 x^{2} e^{4 - e^{x}} + 45 x^{2} e^{8 - 2 e^{x}} + 5 x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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