Optimal. Leaf size=31 \[ x+\left (5+e^{e^x}\right ) \left (25-\log \left (1-\frac {x}{2+e^{2+x}}+\log (3)\right )\right ) \]
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Rubi [F]
time = 7.32, 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 {14-2 x+4 \log (3)+e^{4+2 x} (1+\log (3))+e^{2+x} (9-6 x+4 \log (3))+e^{e^x} \left (2+e^{4+3 x} (25+25 \log (3))+e^x (100-50 x+100 \log (3))+e^{2+x} \left (1-x+e^x (100-25 x+100 \log (3))\right )\right )+e^{e^x} \left (e^{2+2 x} (-4+x-4 \log (3))+e^x (-4+2 x-4 \log (3))+e^{4+3 x} (-1-\log (3))\right ) \log \left (\frac {2-x+2 \log (3)+e^{2+x} (1+\log (3))}{2+e^{2+x}}\right )}{4-2 x+4 \log (3)+e^{4+2 x} (1+\log (3))+e^{2+x} (4-x+4 \log (3))} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x-14 \left (1+\frac {2 \log (3)}{7}\right )-e^{4+2 x} (1+\log (3))-e^{2+x} (9-6 x+4 \log (3))-e^{e^x} \left (2+e^{4+3 x} (25+25 \log (3))+e^x (100-50 x+100 \log (3))+e^{2+x} \left (1-x+e^x (100-25 x+100 \log (3))\right )\right )-e^{e^x} \left (e^{2+2 x} (-4+x-4 \log (3))+e^x (-4+2 x-4 \log (3))+e^{4+3 x} (-1-\log (3))\right ) \log \left (\frac {2-x+2 \log (3)+e^{2+x} (1+\log (3))}{2+e^{2+x}}\right )}{\left (2+e^{2+x}\right ) \left (x-2 (1+\log (3))-e^{2+x} (1+\log (3))\right )} \, dx\\ &=\int \left (1-\frac {2 \left (5+e^{e^x}\right )}{2+e^{2+x}}+\frac {\left (5+e^{e^x}\right ) (-3+x-\log (9))}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))}-e^{e^x+x} \left (-25+\log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )\right )\right ) \, dx\\ &=x-2 \int \frac {5+e^{e^x}}{2+e^{2+x}} \, dx+\int \frac {\left (5+e^{e^x}\right ) (-3+x-\log (9))}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx-\int e^{e^x+x} \left (-25+\log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )\right ) \, dx\\ &=x-2 \text {Subst}\left (\int \frac {5+e^x}{x \left (2+e^2 x\right )} \, dx,x,e^x\right )+\int \left (\frac {\left (5+e^{e^x}\right ) x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))}+\frac {3 \left (-5-e^{e^x}\right ) \left (1+\frac {2 \log (3)}{3}\right )}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))}\right ) \, dx-\int \left (-25 e^{e^x+x}+e^{e^x+x} \log \left (\frac {-x+2 (1+\log (3))+e^{2+x} (1+\log (3))}{2+e^{2+x}}\right )\right ) \, dx\\ &=x-2 \text {Subst}\left (\int \left (\frac {5}{x \left (2+e^2 x\right )}+\frac {e^x}{x \left (2+e^2 x\right )}\right ) \, dx,x,e^x\right )+25 \int e^{e^x+x} \, dx+(3+\log (9)) \int \frac {-5-e^{e^x}}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+\int \frac {\left (5+e^{e^x}\right ) x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx-\int e^{e^x+x} \log \left (\frac {-x+2 (1+\log (3))+e^{2+x} (1+\log (3))}{2+e^{2+x}}\right ) \, dx\\ &=x-e^{e^x} \log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )-2 \text {Subst}\left (\int \frac {e^x}{x \left (2+e^2 x\right )} \, dx,x,e^x\right )-10 \text {Subst}\left (\int \frac {1}{x \left (2+e^2 x\right )} \, dx,x,e^x\right )+25 \text {Subst}\left (\int e^x \, dx,x,e^x\right )+(3+\log (9)) \int \left (\frac {5}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))}+\frac {e^{e^x}}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))}\right ) \, dx+\int \frac {e^{e^x} \left (2-e^{2+x} (-1+x)\right )}{\left (2+e^{2+x}\right ) \left (x-2 (1+\log (3))-e^{2+x} (1+\log (3))\right )} \, dx+\int \left (\frac {5 x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))}+\frac {e^{e^x} x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))}\right ) \, dx\\ &=25 e^{e^x}+x-e^{e^x} \log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )-2 \text {Subst}\left (\int \left (\frac {e^x}{2 x}-\frac {e^{2+x}}{2 \left (2+e^2 x\right )}\right ) \, dx,x,e^x\right )+5 \int \frac {x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx-5 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\left (5 e^2\right ) \text {Subst}\left (\int \frac {1}{2+e^2 x} \, dx,x,e^x\right )+(3+\log (9)) \int \frac {e^{e^x}}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+(5 (3+\log (9))) \int \frac {1}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+\int \left (\frac {2 e^{e^x}}{2+e^{2+x}}+\frac {e^{e^x} (3-x+\log (9))}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))}\right ) \, dx\\ &=25 e^{e^x}-4 x+5 \log \left (2+e^{2+x}\right )-e^{e^x} \log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )+2 \int \frac {e^{e^x}}{2+e^{2+x}} \, dx+5 \int \frac {x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+(5 (3+\log (9))) \int \frac {1}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} (3-x+\log (9))}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx-\text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {e^{2+x}}{2+e^2 x} \, dx,x,e^x\right )\\ &=25 e^{e^x}-4 x-\text {Ei}\left (e^x\right )+e^{-\frac {2}{e^2}} \text {Ei}\left (\frac {2+e^{2+x}}{e^2}\right )+5 \log \left (2+e^{2+x}\right )-e^{e^x} \log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )+2 \text {Subst}\left (\int \frac {e^x}{x \left (2+e^2 x\right )} \, dx,x,e^x\right )+5 \int \frac {x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+(5 (3+\log (9))) \int \frac {1}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+\int \left (\frac {3 e^{e^x} \left (1+\frac {2 \log (3)}{3}\right )}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))}+\frac {e^{e^x} x}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))}\right ) \, dx\\ &=25 e^{e^x}-4 x-\text {Ei}\left (e^x\right )+e^{-\frac {2}{e^2}} \text {Ei}\left (\frac {2+e^{2+x}}{e^2}\right )+5 \log \left (2+e^{2+x}\right )-e^{e^x} \log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )+2 \text {Subst}\left (\int \left (\frac {e^x}{2 x}-\frac {e^{2+x}}{2 \left (2+e^2 x\right )}\right ) \, dx,x,e^x\right )+5 \int \frac {x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+(5 (3+\log (9))) \int \frac {1}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx\\ &=25 e^{e^x}-4 x-\text {Ei}\left (e^x\right )+e^{-\frac {2}{e^2}} \text {Ei}\left (\frac {2+e^{2+x}}{e^2}\right )+5 \log \left (2+e^{2+x}\right )-e^{e^x} \log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )+5 \int \frac {x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+(5 (3+\log (9))) \int \frac {1}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+\text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {e^{2+x}}{2+e^2 x} \, dx,x,e^x\right )\\ &=25 e^{e^x}-4 x+5 \log \left (2+e^{2+x}\right )-e^{e^x} \log \left (\frac {2-x+e^{2+x} (1+\log (3))+\log (9)}{2+e^{2+x}}\right )+5 \int \frac {x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+(3+\log (9)) \int \frac {e^{e^x}}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+(5 (3+\log (9))) \int \frac {1}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{x-2 (1+\log (3))-e^{2+x} (1+\log (3))} \, dx+\int \frac {e^{e^x} x}{-x+2 (1+\log (3))+e^{2+x} (1+\log (3))} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 3.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {14-2 x+4 \log (3)+e^{4+2 x} (1+\log (3))+e^{2+x} (9-6 x+4 \log (3))+e^{e^x} \left (2+e^{4+3 x} (25+25 \log (3))+e^x (100-50 x+100 \log (3))+e^{2+x} \left (1-x+e^x (100-25 x+100 \log (3))\right )\right )+e^{e^x} \left (e^{2+2 x} (-4+x-4 \log (3))+e^x (-4+2 x-4 \log (3))+e^{4+3 x} (-1-\log (3))\right ) \log \left (\frac {2-x+2 \log (3)+e^{2+x} (1+\log (3))}{2+e^{2+x}}\right )}{4-2 x+4 \log (3)+e^{4+2 x} (1+\log (3))+e^{2+x} (4-x+4 \log (3))} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.45, size = 313, normalized size = 10.10
method | result | size |
risch | \(-{\mathrm e}^{{\mathrm e}^{x}} \ln \left (\left (\ln \left (3\right ) {\mathrm e}^{x}+{\mathrm e}^{x}\right ) {\mathrm e}^{2}+2 \ln \left (3\right )-x +2\right )+{\mathrm e}^{{\mathrm e}^{x}} \ln \left ({\mathrm e}^{2+x}+2\right )+x +5 \ln \left ({\mathrm e}^{x}+2 \,{\mathrm e}^{-2}\right )-5 \ln \left ({\mathrm e}^{x}+\frac {\left (2 \ln \left (3\right )-x +2\right ) {\mathrm e}^{-2}}{\ln \left (3\right )+1}\right )+\frac {i {\mathrm e}^{{\mathrm e}^{x}} \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2+x}+2}\right ) \mathrm {csgn}\left (i \left (\left (\ln \left (3\right ) {\mathrm e}^{x}+{\mathrm e}^{x}\right ) {\mathrm e}^{2}+2 \ln \left (3\right )-x +2\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (3\right ) {\mathrm e}^{x}+{\mathrm e}^{x}\right ) {\mathrm e}^{2}+2 \ln \left (3\right )-x +2\right )}{{\mathrm e}^{2+x}+2}\right )}{2}-\frac {i {\mathrm e}^{{\mathrm e}^{x}} \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2+x}+2}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (3\right ) {\mathrm e}^{x}+{\mathrm e}^{x}\right ) {\mathrm e}^{2}+2 \ln \left (3\right )-x +2\right )}{{\mathrm e}^{2+x}+2}\right )^{2}}{2}-\frac {i {\mathrm e}^{{\mathrm e}^{x}} \pi \,\mathrm {csgn}\left (i \left (\left (\ln \left (3\right ) {\mathrm e}^{x}+{\mathrm e}^{x}\right ) {\mathrm e}^{2}+2 \ln \left (3\right )-x +2\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (3\right ) {\mathrm e}^{x}+{\mathrm e}^{x}\right ) {\mathrm e}^{2}+2 \ln \left (3\right )-x +2\right )}{{\mathrm e}^{2+x}+2}\right )^{2}}{2}+\frac {i {\mathrm e}^{{\mathrm e}^{x}} \pi \mathrm {csgn}\left (\frac {i \left (\left (\ln \left (3\right ) {\mathrm e}^{x}+{\mathrm e}^{x}\right ) {\mathrm e}^{2}+2 \ln \left (3\right )-x +2\right )}{{\mathrm e}^{2+x}+2}\right )^{3}}{2}+25 \,{\mathrm e}^{{\mathrm e}^{x}}\) | \(313\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 48, normalized size = 1.55 \begin {gather*} -{\left (e^{\left (e^{x}\right )} + 5\right )} \log \left ({\left (e^{2} \log \left (3\right ) + e^{2}\right )} e^{x} - x + 2 \, \log \left (3\right ) + 2\right ) + {\left (e^{\left (e^{x}\right )} + 5\right )} \log \left (e^{\left (x + 2\right )} + 2\right ) + x + 25 \, e^{\left (e^{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 42, normalized size = 1.35 \begin {gather*} -{\left (e^{\left (e^{x}\right )} + 5\right )} \log \left (\frac {{\left (\log \left (3\right ) + 1\right )} e^{\left (x + 2\right )} - x + 2 \, \log \left (3\right ) + 2}{e^{\left (x + 2\right )} + 2}\right ) + x + 25 \, e^{\left (e^{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (27) = 54\).
time = 0.58, size = 97, normalized size = 3.13 \begin {gather*} {\left (x e^{x} - e^{\left (x + e^{x}\right )} \log \left (e^{\left (x + 2\right )} \log \left (3\right ) - x + e^{\left (x + 2\right )} + 2 \, \log \left (3\right ) + 2\right ) - 5 \, e^{x} \log \left (-e^{\left (x + 2\right )} \log \left (3\right ) + x - e^{\left (x + 2\right )} - 2 \, \log \left (3\right ) - 2\right ) + e^{\left (x + e^{x}\right )} \log \left (e^{\left (x + 2\right )} + 2\right ) + 5 \, e^{x} \log \left (-e^{\left (x + 2\right )} - 2\right ) + 25 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\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 = -0.03 \begin {gather*} \int \frac {4\,\ln \left (3\right )-2\,x+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{x+2}\,\left ({\mathrm {e}}^x\,\left (100\,\ln \left (3\right )-25\,x+100\right )-x+1\right )+{\mathrm {e}}^{3\,x+4}\,\left (25\,\ln \left (3\right )+25\right )+{\mathrm {e}}^x\,\left (100\,\ln \left (3\right )-50\,x+100\right )+2\right )+{\mathrm {e}}^{x+2}\,\left (4\,\ln \left (3\right )-6\,x+9\right )+{\mathrm {e}}^{2\,x+4}\,\left (\ln \left (3\right )+1\right )-\ln \left (\frac {2\,\ln \left (3\right )-x+{\mathrm {e}}^{x+2}\,\left (\ln \left (3\right )+1\right )+2}{{\mathrm {e}}^{x+2}+2}\right )\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^x\,\left (4\,\ln \left (3\right )-2\,x+4\right )+{\mathrm {e}}^{3\,x+4}\,\left (\ln \left (3\right )+1\right )+{\mathrm {e}}^{2\,x+2}\,\left (4\,\ln \left (3\right )-x+4\right )\right )+14}{4\,\ln \left (3\right )-2\,x+{\mathrm {e}}^{x+2}\,\left (4\,\ln \left (3\right )-x+4\right )+{\mathrm {e}}^{2\,x+4}\,\left (\ln \left (3\right )+1\right )+4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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