Optimal. Leaf size=28 \[ x-\frac {1}{5} \log \left (-e^{e^{e^{4 x}}} \left (4-e^x\right )+\log (x)\right ) \]
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Rubi [F]
time = 27.98, 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 {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \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 {1-e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )-5 x \log (x)}{5 x \left (4 e^{e^{e^{4 x}}}-e^{e^{e^{4 x}}+x}-\log (x)\right )} \, dx\\ &=\frac {1}{5} \int \frac {1-e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )-5 x \log (x)}{x \left (4 e^{e^{e^{4 x}}}-e^{e^{e^{4 x}}+x}-\log (x)\right )} \, dx\\ &=\frac {1}{5} \int \left (-4 e^{e^{4 x}+4 x}+4 e^{-e^{e^{4 x}}+e^{4 x}+3 x} \log (x)+4 e^{-2 e^{e^{4 x}}+e^{4 x}+2 x} \left (4 e^{e^{e^{4 x}}}-\log (x)\right ) \log (x)+4 e^{-3 e^{e^{4 x}}+e^{4 x}+x} \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2 \log (x)+4 e^{-4 e^{e^{4 x}}} \left (e^{4 e^{e^{4 x}}}+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-e^{e^{4 x}} \log ^4(x)\right )-\frac {e^{-4 e^{e^{4 x}}} \left (e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-4 e^{e^{4 x}} x \log ^5(x)\right )}{x \left (-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {e^{-4 e^{e^{4 x}}} \left (e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-4 e^{e^{4 x}} x \log ^5(x)\right )}{x \left (-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)\right )} \, dx\right )-\frac {4}{5} \int e^{e^{4 x}+4 x} \, dx+\frac {4}{5} \int e^{-e^{e^{4 x}}+e^{4 x}+3 x} \log (x) \, dx+\frac {4}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}+2 x} \left (4 e^{e^{e^{4 x}}}-\log (x)\right ) \log (x) \, dx+\frac {4}{5} \int e^{-3 e^{e^{4 x}}+e^{4 x}+x} \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2 \log (x) \, dx+\frac {4}{5} \int e^{-4 e^{e^{4 x}}} \left (e^{4 e^{e^{4 x}}}+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-e^{e^{4 x}} \log ^4(x)\right ) \, dx\\ &=-\left (\frac {1}{5} \int \left (\frac {4 e^{e^{e^{4 x}}}}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)}+\frac {1}{x \left (-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)\right )}-\frac {\log (x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)}-\frac {1024 e^{e^{4 x}} \log (x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)}+\frac {1024 e^{-e^{e^{4 x}}+e^{4 x}} \log ^2(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)}-\frac {384 e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^3(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)}+\frac {64 e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^4(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)}-\frac {4 e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^5(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)}\right ) \, dx\right )-\frac {1}{5} \text {Subst}\left (\int e^x \, dx,x,e^{4 x}\right )+\frac {4}{5} \int \left (4 e^{-e^{e^{4 x}}+e^{4 x}+2 x} \log (x)-e^{-2 e^{e^{4 x}}+e^{4 x}+2 x} \log ^2(x)\right ) \, dx+\frac {4}{5} \int \left (16 e^{-e^{e^{4 x}}+e^{4 x}+x} \log (x)-8 e^{-2 e^{e^{4 x}}+e^{4 x}+x} \log ^2(x)+e^{-3 e^{e^{4 x}}+e^{4 x}+x} \log ^3(x)\right ) \, dx+\frac {4}{5} \int \left (1+64 e^{-e^{e^{4 x}}+e^{4 x}} \log (x)-48 e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+12 e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^3(x)-e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^4(x)\right ) \, dx-\frac {4}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )}{x} \, dx+\frac {1}{5} (4 \log (x)) \text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )\\ &=-\frac {1}{5} e^{e^{4 x}}+\frac {4 x}{5}-\frac {1}{5} \int \frac {1}{x \left (-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)\right )} \, dx+\frac {1}{5} \int \frac {\log (x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)} \, dx-\frac {4}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}+2 x} \log ^2(x) \, dx+\frac {4}{5} \int e^{-3 e^{e^{4 x}}+e^{4 x}+x} \log ^3(x) \, dx-\frac {4}{5} \int e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^4(x) \, dx-\frac {4}{5} \int \frac {e^{e^{e^{4 x}}}}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)} \, dx+\frac {4}{5} \int \frac {e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^5(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)} \, dx-\frac {4}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )}{x} \, dx+\frac {16}{5} \int e^{-e^{e^{4 x}}+e^{4 x}+2 x} \log (x) \, dx-\frac {32}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}+x} \log ^2(x) \, dx+\frac {48}{5} \int e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^3(x) \, dx+\frac {64}{5} \int e^{-e^{e^{4 x}}+e^{4 x}+x} \log (x) \, dx-\frac {64}{5} \int \frac {e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^4(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)} \, dx-\frac {192}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^2(x) \, dx+\frac {256}{5} \int e^{-e^{e^{4 x}}+e^{4 x}} \log (x) \, dx+\frac {384}{5} \int \frac {e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^3(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)} \, dx+\frac {1024}{5} \int \frac {e^{e^{4 x}} \log (x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)} \, dx-\frac {1024}{5} \int \frac {e^{-e^{e^{4 x}}+e^{4 x}} \log ^2(x)}{-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)} \, dx+\frac {1}{5} (4 \log (x)) \text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )\\ &=-\frac {1}{5} e^{e^{4 x}}+\frac {4 x}{5}-\frac {1}{5} \int \frac {1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )} \, dx+\frac {1}{5} \int \frac {\log (x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {4}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}+2 x} \log ^2(x) \, dx+\frac {4}{5} \int e^{-3 e^{e^{4 x}}+e^{4 x}+x} \log ^3(x) \, dx-\frac {4}{5} \int e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^4(x) \, dx-\frac {4}{5} \int \frac {e^{e^{e^{4 x}}}}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx+\frac {4}{5} \int \frac {e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^5(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {4}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )}{x} \, dx-\frac {16}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^2}+x^2} \, dx,x,e^{2 x}\right )}{2 x} \, dx-\frac {32}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}+x} \log ^2(x) \, dx+\frac {48}{5} \int e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^3(x) \, dx-\frac {64}{5} \int \frac {e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^4(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {64}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^4}+x^4} \, dx,x,e^x\right )}{x} \, dx-\frac {192}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^2(x) \, dx-\frac {256}{5} \int \frac {\text {Subst}\left (\int \frac {e^{-e^x+x}}{x} \, dx,x,e^{4 x}\right )}{4 x} \, dx+\frac {384}{5} \int \frac {e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^3(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx+\frac {1024}{5} \int \frac {e^{e^{4 x}} \log (x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {1024}{5} \int \frac {e^{-e^{e^{4 x}}+e^{4 x}} \log ^2(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx+\frac {1}{5} (4 \log (x)) \text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )+\frac {1}{5} (8 \log (x)) \text {Subst}\left (\int e^{-e^{x^2}+x^2} \, dx,x,e^{2 x}\right )+\frac {1}{5} (64 \log (x)) \text {Subst}\left (\int e^{-e^{x^4}+x^4} \, dx,x,e^x\right )+\frac {1}{5} (64 \log (x)) \text {Subst}\left (\int \frac {e^{-e^x+x}}{x} \, dx,x,e^{4 x}\right )\\ &=-\frac {1}{5} e^{e^{4 x}}+\frac {4 x}{5}-\frac {1}{5} \int \frac {1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )} \, dx+\frac {1}{5} \int \frac {\log (x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {4}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}+2 x} \log ^2(x) \, dx+\frac {4}{5} \int e^{-3 e^{e^{4 x}}+e^{4 x}+x} \log ^3(x) \, dx-\frac {4}{5} \int e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^4(x) \, dx-\frac {4}{5} \int \frac {e^{e^{e^{4 x}}}}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx+\frac {4}{5} \int \frac {e^{-4 e^{e^{4 x}}+e^{4 x}} \log ^5(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {4}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )}{x} \, dx-\frac {8}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^2}+x^2} \, dx,x,e^{2 x}\right )}{x} \, dx-\frac {32}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}+x} \log ^2(x) \, dx+\frac {48}{5} \int e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^3(x) \, dx-\frac {64}{5} \int \frac {e^{-3 e^{e^{4 x}}+e^{4 x}} \log ^4(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {64}{5} \int \frac {\text {Subst}\left (\int e^{-e^{x^4}+x^4} \, dx,x,e^x\right )}{x} \, dx-\frac {64}{5} \int \frac {\text {Subst}\left (\int \frac {e^{-e^x+x}}{x} \, dx,x,e^{4 x}\right )}{x} \, dx-\frac {192}{5} \int e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^2(x) \, dx+\frac {384}{5} \int \frac {e^{-2 e^{e^{4 x}}+e^{4 x}} \log ^3(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx+\frac {1024}{5} \int \frac {e^{e^{4 x}} \log (x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx-\frac {1024}{5} \int \frac {e^{-e^{e^{4 x}}+e^{4 x}} \log ^2(x)}{e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)} \, dx+\frac {1}{5} (4 \log (x)) \text {Subst}\left (\int e^{-e^{x^4}+x^4} x^2 \, dx,x,e^x\right )+\frac {1}{5} (8 \log (x)) \text {Subst}\left (\int e^{-e^{x^2}+x^2} \, dx,x,e^{2 x}\right )+\frac {1}{5} (64 \log (x)) \text {Subst}\left (\int e^{-e^{x^4}+x^4} \, dx,x,e^x\right )+\frac {1}{5} (64 \log (x)) \text {Subst}\left (\int \frac {e^{-e^x+x}}{x} \, dx,x,e^{4 x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.23, size = 36, normalized size = 1.29 \begin {gather*} \frac {1}{5} \left (5 x-\log \left (-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 29, normalized size = 1.04
method | result | size |
risch | \(x -\frac {\ln \left ({\mathrm e}^{x}-4\right )}{5}-\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4 x}}}+\frac {\ln \left (x \right )}{{\mathrm e}^{x}-4}\right )}{5}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 33, normalized size = 1.18 \begin {gather*} x - \frac {1}{5} \, \log \left (\frac {{\left (e^{x} - 4\right )} e^{\left (e^{\left (e^{\left (4 \, x\right )}\right )}\right )} + \log \left (x\right )}{e^{x} - 4}\right ) - \frac {1}{5} \, \log \left (e^{x} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 33, normalized size = 1.18 \begin {gather*} x - \frac {1}{5} \, \log \left (\frac {{\left (e^{x} - 4\right )} e^{\left (e^{\left (e^{\left (4 \, x\right )}\right )}\right )} + \log \left (x\right )}{e^{x} - 4}\right ) - \frac {1}{5} \, \log \left (e^{x} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.58, size = 29, normalized size = 1.04 \begin {gather*} x - \frac {\log {\left (e^{x} - 4 \right )}}{5} - \frac {\log {\left (e^{e^{e^{4 x}}} + \frac {\log {\left (x \right )}}{e^{x} - 4} \right )}}{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 = 1.28, size = 39, normalized size = 1.39 \begin {gather*} x-\frac {\ln \left (\frac {\ln \left (x\right )-4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}}+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}}\,{\mathrm {e}}^x}{{\mathrm {e}}^x-4}\right )}{5}-\frac {\ln \left ({\mathrm {e}}^x-4\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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