Optimal. Leaf size=27 \[ \left (x-2 \left (\left (3+e^{\frac {36}{e^2 x^2}}\right )^2-x+x^2\right )\right )^2 \]
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Rubi [B] time = 0.28, antiderivative size = 81, normalized size of antiderivative = 3.00, number of steps used = 8, number of rules used = 5, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {12, 14, 2209, 629, 2288} \begin {gather*} \left (2 x^2-3 x+18\right )^2+24 e^{\frac {36}{e^2 x^2}} \left (2 x^2-3 x+18\right )+48 e^{\frac {108}{e^2 x^2}}+4 e^{\frac {144}{e^2 x^2}}+4 e^{\frac {72}{e^2 x^2}} \left (2 x^2-3 x+54\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 629
Rule 2209
Rule 2288
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{x^3} \, dx}{e^2}\\ &=\frac {\int \left (-\frac {10368 e^{\frac {108}{e^2 x^2}}}{x^3}-\frac {1152 e^{\frac {144}{e^2 x^2}}}{x^3}+2 e^2 (-3+4 x) \left (18-3 x+2 x^2\right )+\frac {4 e^{\frac {72}{e^2 x^2}} \left (-7776+432 x-288 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3}+\frac {24 e^{\frac {36}{e^2 x^2}} \left (-1296+216 x-144 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3}\right ) \, dx}{e^2}\\ &=2 \int (-3+4 x) \left (18-3 x+2 x^2\right ) \, dx+\frac {4 \int \frac {e^{\frac {72}{e^2 x^2}} \left (-7776+432 x-288 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3} \, dx}{e^2}+\frac {24 \int \frac {e^{\frac {36}{e^2 x^2}} \left (-1296+216 x-144 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3} \, dx}{e^2}-\frac {1152 \int \frac {e^{\frac {144}{e^2 x^2}}}{x^3} \, dx}{e^2}-\frac {10368 \int \frac {e^{\frac {108}{e^2 x^2}}}{x^3} \, dx}{e^2}\\ &=48 e^{\frac {108}{e^2 x^2}}+4 e^{\frac {144}{e^2 x^2}}+24 e^{\frac {36}{e^2 x^2}} \left (18-3 x+2 x^2\right )+\left (18-3 x+2 x^2\right )^2+4 e^{\frac {72}{e^2 x^2}} \left (54-3 x+2 x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 36, normalized size = 1.33 \begin {gather*} \left (18+12 e^{\frac {36}{e^2 x^2}}+2 e^{\frac {72}{e^2 x^2}}-3 x+2 x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 79, normalized size = 2.93 \begin {gather*} 4 \, x^{4} - 12 \, x^{3} + 81 \, x^{2} + 4 \, {\left (2 \, x^{2} - 3 \, x + 54\right )} e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right )} + 24 \, {\left (2 \, x^{2} - 3 \, x + 18\right )} e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right )} - 108 \, x + 4 \, e^{\left (\frac {144 \, e^{\left (-2\right )}}{x^{2}}\right )} + 48 \, e^{\left (\frac {108 \, e^{\left (-2\right )}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 134, normalized size = 4.96 \begin {gather*} {\left (4 \, x^{4} e^{2} - 12 \, x^{3} e^{2} + 81 \, x^{2} e^{2} + 8 \, x^{2} e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 48 \, x^{2} e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} - 108 \, x e^{2} - 12 \, x e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} - 72 \, x e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 4 \, e^{\left (\frac {144 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 48 \, e^{\left (\frac {108 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 216 \, e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 432 \, e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 108, normalized size = 4.00
method | result | size |
risch | \(4 x^{4}-12 x^{3}+81 x^{2}-108 x +4 \,{\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}}+48 \,{\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}}+\left (8 x^{2} {\mathrm e}^{2}-12 \,{\mathrm e}^{2} x +216 \,{\mathrm e}^{2}\right ) {\mathrm e}^{\frac {-2 x^{2}+72 \,{\mathrm e}^{-2}}{x^{2}}}+\left (48 x^{2} {\mathrm e}^{2}-72 \,{\mathrm e}^{2} x +432 \,{\mathrm e}^{2}\right ) {\mathrm e}^{\frac {-2 x^{2}+36 \,{\mathrm e}^{-2}}{x^{2}}}\) | \(108\) |
default | \({\mathrm e}^{-2} \left (72 i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e} \erf \left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )+432 i \sqrt {\pi }\, {\mathrm e} \erf \left (\frac {6 i {\mathrm e}^{-1}}{x}\right )-108 \,{\mathrm e}^{2} x +432 \,{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}+216 \,{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}+48 \,{\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}+4 \,{\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}+4 x^{4} {\mathrm e}^{2}-12 x^{3} {\mathrm e}^{2}+81 x^{2} {\mathrm e}^{2}-1728 \expIntegralEi \left (1, -\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )-576 \expIntegralEi \left (1, -\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )-96 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}} x^{2}}{2}-18 \,{\mathrm e}^{-2} \expIntegralEi \left (1, -\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )+72 \,{\mathrm e}^{2} \left (-{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}} x -6 i {\mathrm e}^{-1} \sqrt {\pi }\, \erf \left (\frac {6 i {\mathrm e}^{-1}}{x}\right )\right )-16 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-36 \,{\mathrm e}^{-2} \expIntegralEi \left (1, -\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )+12 \,{\mathrm e}^{2} \left (-{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}} x -6 i {\mathrm e}^{-2} \sqrt {\pi }\, \sqrt {2}\, {\mathrm e} \erf \left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )\right )\right )\) | \(337\) |
derivativedivides | \(-{\mathrm e}^{-2} \left (-72 i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e} \erf \left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )-432 i \sqrt {\pi }\, {\mathrm e} \erf \left (\frac {6 i {\mathrm e}^{-1}}{x}\right )+108 \,{\mathrm e}^{2} x -432 \,{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}-216 \,{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}-48 \,{\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}-4 \,{\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2}-4 x^{4} {\mathrm e}^{2}+12 x^{3} {\mathrm e}^{2}-81 x^{2} {\mathrm e}^{2}+1728 \expIntegralEi \left (1, -\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )+576 \expIntegralEi \left (1, -\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )+96 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}} x^{2}}{2}-18 \,{\mathrm e}^{-2} \expIntegralEi \left (1, -\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )-72 \,{\mathrm e}^{2} \left (-{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}} x -6 i {\mathrm e}^{-1} \sqrt {\pi }\, \erf \left (\frac {6 i {\mathrm e}^{-1}}{x}\right )\right )+16 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-36 \,{\mathrm e}^{-2} \expIntegralEi \left (1, -\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )-12 \,{\mathrm e}^{2} \left (-{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}} x -6 i {\mathrm e}^{-2} \sqrt {\pi }\, \sqrt {2}\, {\mathrm e} \erf \left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )\right )\right )\) | \(338\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.45, size = 237, normalized size = 8.78 \begin {gather*} {\left (4 \, x^{4} e^{2} - 12 \, x^{3} e^{2} - 36 \, \sqrt {2} x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}} e^{2} \Gamma \left (-\frac {1}{2}, -\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right ) - 216 \, x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}} e^{2} \Gamma \left (-\frac {1}{2}, -\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right ) + 81 \, x^{2} e^{2} - 108 \, x e^{2} - \frac {72 \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (6 \, \sqrt {2} \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}\right ) - 1\right )}}{x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}} - \frac {432 \, \sqrt {\pi } {\left (\operatorname {erf}\left (6 \, \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}\right ) - 1\right )}}{x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}} + 576 \, {\rm Ei}\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right ) + 1728 \, {\rm Ei}\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right ) + 4 \, e^{\left (\frac {144 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 48 \, e^{\left (\frac {108 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 216 \, e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 432 \, e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} - 1728 \, \Gamma \left (-1, -\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right ) - 576 \, \Gamma \left (-1, -\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right )\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.43, size = 107, normalized size = 3.96 \begin {gather*} 432\,{\mathrm {e}}^{\frac {36\,{\mathrm {e}}^{-2}}{x^2}}-108\,x+216\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^{-2}}{x^2}}+48\,{\mathrm {e}}^{\frac {108\,{\mathrm {e}}^{-2}}{x^2}}+4\,{\mathrm {e}}^{\frac {144\,{\mathrm {e}}^{-2}}{x^2}}+48\,x^2\,{\mathrm {e}}^{\frac {36\,{\mathrm {e}}^{-2}}{x^2}}+8\,x^2\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^{-2}}{x^2}}+81\,x^2-12\,x^3+4\,x^4-72\,x\,{\mathrm {e}}^{\frac {36\,{\mathrm {e}}^{-2}}{x^2}}-12\,x\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^{-2}}{x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.35, size = 82, normalized size = 3.04 \begin {gather*} 4 x^{4} - 12 x^{3} + 81 x^{2} - 108 x + \left (8 x^{2} - 12 x + 216\right ) e^{\frac {72}{x^{2} e^{2}}} + \left (48 x^{2} - 72 x + 432\right ) e^{\frac {36}{x^{2} e^{2}}} + 4 e^{\frac {144}{x^{2} e^{2}}} + 48 e^{\frac {108}{x^{2} e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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