Optimal. Leaf size=34 \[ e^{\frac {-x+\frac {-1+x}{2+x}-\frac {\left (3+e^x\right ) \log (\log (x))}{5 x}}{x}} \]
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Rubi [F]
time = 34.03, 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 (\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}\right ) \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (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 {\exp \left (\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}\right ) \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{x^3 \left (20+20 x+5 x^2\right ) \log (x)} \, dx\\ &=\int \frac {\exp \left (\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}\right ) \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{5 x^3 (2+x)^2 \log (x)} \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}\right ) \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{x^3 (2+x)^2 \log (x)} \, dx\\ &=\frac {1}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) \left (-\left (\left (3+e^x\right ) (2+x)^2\right )+\log (x) \left (-5 x \left (-2-2 x+x^2\right )-\left (-6+e^x (-2+x)\right ) (2+x)^2 \log (\log (x))\right )\right )}{x^3 (2+x)^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {e^{x-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) (1-2 \log (x) \log (\log (x))+x \log (x) \log (\log (x)))}{x^3}+\frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) \left (-12-12 x-3 x^2+10 x \log (x)+10 x^2 \log (x)-5 x^3 \log (x)+24 \log (x) \log (\log (x))+24 x \log (x) \log (\log (x))+6 x^2 \log (x) \log (\log (x))\right )}{x^3 (2+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{x-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) (1-2 \log (x) \log (\log (x))+x \log (x) \log (\log (x)))}{x^3} \, dx\right )+\frac {1}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) \left (-12-12 x-3 x^2+10 x \log (x)+10 x^2 \log (x)-5 x^3 \log (x)+24 \log (x) \log (\log (x))+24 x \log (x) \log (\log (x))+6 x^2 \log (x) \log (\log (x))\right )}{x^3 (2+x)^2} \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) (1-2 \log (x) \log (\log (x))+x \log (x) \log (\log (x)))}{x^3} \, dx\right )+\frac {1}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) \left (-3 (2+x)^2+\log (x) \left (-5 x \left (-2-2 x+x^2\right )+6 (2+x)^2 \log (\log (x))\right )\right )}{x^3 (2+x)^2} \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) (1+(-2+x) \log (x) \log (\log (x)))}{x^3} \, dx\right )+\frac {1}{5} \int \left (\frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) \left (-12-12 x-3 x^2+10 x \log (x)+10 x^2 \log (x)-5 x^3 \log (x)\right )}{x^3 (2+x)^2}+\frac {6 e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{1-\frac {3+e^x+5 x^2}{5 x^2}}(x) \log (\log (x))}{x^3}\right ) \, dx\\ &=\frac {1}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) \left (-12-12 x-3 x^2+10 x \log (x)+10 x^2 \log (x)-5 x^3 \log (x)\right )}{x^3 (2+x)^2} \, dx-\frac {1}{5} \int \left (\frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3}+\frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} (-2+x) \log ^{1-\frac {3+e^x+5 x^2}{5 x^2}}(x) \log (\log (x))}{x^3}\right ) \, dx+\frac {6}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{1-\frac {3+e^x+5 x^2}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx\right )+\frac {1}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x) \left (-3 (2+x)^2-5 x \left (-2-2 x+x^2\right ) \log (x)\right )}{x^3 (2+x)^2} \, dx-\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} (-2+x) \log ^{1-\frac {3+e^x+5 x^2}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx+\frac {6}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx\right )+\frac {1}{5} \int \left (-\frac {3 e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3}-\frac {5 e^{-\frac {1+x+x^2}{x (2+x)}} \left (-2-2 x+x^2\right ) \log ^{1-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^2 (2+x)^2}\right ) \, dx-\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} (-2+x) \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx+\frac {6}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx\right )-\frac {1}{5} \int \left (-\frac {2 e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3}+\frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^2}\right ) \, dx-\frac {3}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx+\frac {6}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx-\int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \left (-2-2 x+x^2\right ) \log ^{1-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^2 (2+x)^2} \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx\right )-\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^2} \, dx+\frac {2}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx-\frac {3}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx+\frac {6}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx-\int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \left (-2-2 x+x^2\right ) \log ^{-\frac {3+e^x}{5 x^2}}(x)}{x^2 (2+x)^2} \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx\right )-\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^2} \, dx+\frac {2}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx-\frac {3}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx+\frac {6}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx-\int \left (-\frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x)}{2 x^2}+\frac {3 e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x)}{2 (2+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx\right )-\frac {1}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^2} \, dx+\frac {2}{5} \int \frac {e^{\frac {(-1+x) (1+x)^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx+\frac {1}{2} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x)}{x^2} \, dx-\frac {3}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x+5 x^2}{5 x^2}}(x)}{x^3} \, dx+\frac {6}{5} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \log (\log (x))}{x^3} \, dx-\frac {3}{2} \int \frac {e^{-\frac {1+x+x^2}{x (2+x)}} \log ^{-\frac {3+e^x}{5 x^2}}(x)}{(2+x)^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.19, size = 35, normalized size = 1.03 \begin {gather*} e^{-\frac {1+x+x^2}{2 x+x^2}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 51, normalized size = 1.50
method | result | size |
risch | \({\mathrm e}^{-\frac {{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right ) x +5 x^{3}+2 \,{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right )+3 x \ln \left (\ln \left (x \right )\right )+5 x^{2}+6 \ln \left (\ln \left (x \right )\right )+5 x}{5 x^{2} \left (2+x \right )}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 33, normalized size = 0.97 \begin {gather*} e^{\left (-\frac {e^{x} \log \left (\log \left (x\right )\right )}{5 \, x^{2}} + \frac {3}{2 \, {\left (x + 2\right )}} - \frac {1}{2 \, x} - \frac {3 \, \log \left (\log \left (x\right )\right )}{5 \, x^{2}} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 43, normalized size = 1.26 \begin {gather*} e^{\left (-\frac {5 \, x^{3} + 5 \, x^{2} + {\left ({\left (x + 2\right )} e^{x} + 3 \, x + 6\right )} \log \left (\log \left (x\right )\right ) + 5 \, x}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.99, size = 42, normalized size = 1.24 \begin {gather*} e^{\frac {- 5 x^{3} - 5 x^{2} - 5 x + \left (- 3 x + \left (- x - 2\right ) e^{x} - 6\right ) \log {\left (\log {\left (x \right )} \right )}}{5 x^{3} + 10 x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs.
\(2 (31) = 62\).
time = 3.58, size = 118, normalized size = 3.47 \begin {gather*} e^{\left (-\frac {x^{3}}{x^{3} + 2 \, x^{2}} - \frac {x e^{x} \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}} - \frac {x^{2}}{x^{3} + 2 \, x^{2}} - \frac {3 \, x \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}} - \frac {2 \, e^{x} \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}} - \frac {x}{x^{3} + 2 \, x^{2}} - \frac {6 \, \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 97, normalized size = 2.85 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {5\,x}{5\,x^3+10\,x^2}}\,{\mathrm {e}}^{-\frac {5\,x^2}{5\,x^3+10\,x^2}}\,{\mathrm {e}}^{-\frac {5\,x^3}{5\,x^3+10\,x^2}}}{{\ln \left (x\right )}^{\frac {{\mathrm {e}}^x+3}{5\,\left (x^2+2\,x\right )}+\frac {2\,\left ({\mathrm {e}}^x+3\right )}{5\,\left (x^3+2\,x^2\right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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