Optimal. Leaf size=20 \[ \log ^{e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 5.77, 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 {e^{-\frac {5-e^x+x+\log (4)}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \left (x+\left (5+e^x (-1+x)+\log (4)\right ) \log (x) \log (\log (x))\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \left (x+\left (5+e^x (-1+x)+\log (4)\right ) \log (x) \log (\log (x))\right )}{x^2} \, dx\\ &=\int \left (\frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} (-1+x) \log ^{e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \log (\log (x))}{x^2}+\frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \left (x+5 \left (1+\frac {2 \log (2)}{5}\right ) \log (x) \log (\log (x))\right )}{x^2}\right ) \, dx\\ &=\int \frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} (-1+x) \log ^{e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \log (\log (x))}{x^2} \, dx+\int \frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \left (x+5 \left (1+\frac {2 \log (2)}{5}\right ) \log (x) \log (\log (x))\right )}{x^2} \, dx\\ &=\int \frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} (-1+x) \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x^2} \, dx+\int \frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) (x+(5+\log (4)) \log (x) \log (\log (x)))}{x^2} \, dx\\ &=\int \left (-\frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x^2}+\frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x}\right ) \, dx+\int \left (\frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x)}{x}+\frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} (5+\log (4)) \log ^{e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \log (\log (x))}{x^2}\right ) \, dx\\ &=(5+\log (4)) \int \frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-\frac {5-e^x+x+\log (4)}{x}}}(x) \log (\log (x))}{x^2} \, dx+\int \frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x)}{x} \, dx-\int \frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x^2} \, dx+\int \frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x} \, dx\\ &=(5+\log (4)) \int \frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x^2} \, dx+\int \frac {e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{-1+e^{-\frac {5-e^x+x+\log (4)}{x}}}(x)}{x} \, dx-\int \frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x^2} \, dx+\int \frac {e^{-1+\frac {e^x}{x}+x-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}} \log ^{e^{-1+\frac {e^x}{x}-\frac {5 \left (1+\frac {2 \log (2)}{5}\right )}{x}}}(x) \log (\log (x))}{x} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.41, size = 25, normalized size = 1.25 \begin {gather*} \log ^{4^{-1/x} e^{\frac {-5+e^x-x}{x}}}(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 20, normalized size = 1.00
method | result | size |
risch | \(\ln \left (x \right )^{{\mathrm e}^{\frac {{\mathrm e}^{x}-2 \ln \left (2\right )-5-x}{x}}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.59, size = 24, normalized size = 1.20 \begin {gather*} \log \left (x\right )^{e^{\left (\frac {e^{x}}{x} - \frac {2 \, \log \left (2\right )}{x} - \frac {5}{x} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 20, normalized size = 1.00 \begin {gather*} \log \left (x\right )^{e^{\left (-\frac {x - e^{x} + 2 \, \log \left (2\right ) + 5}{x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 179.04, size = 20, normalized size = 1.00 \begin {gather*} e^{e^{- \frac {x - e^{x} + 2 \log {\left (2 \right )} + 5}{x}} \log {\left (\log {\left (x \right )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.40, size = 28, normalized size = 1.40 \begin {gather*} {\ln \left (x\right )}^{\frac {{\mathrm {e}}^{-1}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{-\frac {5}{x}}}{2^{2/x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________