Optimal. Leaf size=22 \[ x \left (-1+5^{-x} \left (x \left (-1+(-2+\log (5))^2\right )\right )^x\right ) \]
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Rubi [F]
time = 0.69, 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 \left (-1+5^{-x} \left (3 x-4 x \log (5)+x \log ^2(5)\right )^x \left (1+x+x \log \left (\frac {1}{5} \left (3 x-4 x \log (5)+x \log ^2(5)\right )\right )\right )\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-x+\int 5^{-x} \left (3 x-4 x \log (5)+x \log ^2(5)\right )^x \left (1+x+x \log \left (\frac {1}{5} \left (3 x-4 x \log (5)+x \log ^2(5)\right )\right )\right ) \, dx\\ &=-x+\int 5^{-x} \left (x (3-4 \log (5))+x \log ^2(5)\right )^x \left (1+x+x \log \left (\frac {1}{5} \left (3 x-4 x \log (5)+x \log ^2(5)\right )\right )\right ) \, dx\\ &=-x+\int 5^{-x} \left (x \left (3-4 \log (5)+\log ^2(5)\right )\right )^x \left (1+x+x \log \left (\frac {1}{5} \left (3 x-4 x \log (5)+x \log ^2(5)\right )\right )\right ) \, dx\\ &=-x+\int (x (1-\log (5)))^x \left (\frac {1}{5} (3-\log (5))\right )^x \left (1+x+x \log \left (\frac {1}{5} \left (3 x-4 x \log (5)+x \log ^2(5)\right )\right )\right ) \, dx\\ &=-x+\int \left ((x (1-\log (5)))^x \left (\frac {1}{5} (3-\log (5))\right )^x+\frac {(x (1-\log (5)))^{1+x} \left (\frac {1}{5} (3-\log (5))\right )^x}{1-\log (5)}+\frac {(x (1-\log (5)))^{1+x} \left (\frac {1}{5} (3-\log (5))\right )^x \log \left (\frac {1}{5} x \left (3-4 \log (5)+\log ^2(5)\right )\right )}{1-\log (5)}\right ) \, dx\\ &=-x+\frac {\int (x (1-\log (5)))^{1+x} \left (\frac {1}{5} (3-\log (5))\right )^x \, dx}{1-\log (5)}+\frac {\int (x (1-\log (5)))^{1+x} \left (\frac {1}{5} (3-\log (5))\right )^x \log \left (\frac {1}{5} x \left (3-4 \log (5)+\log ^2(5)\right )\right ) \, dx}{1-\log (5)}+\int (x (1-\log (5)))^x \left (\frac {1}{5} (3-\log (5))\right )^x \, dx\\ &=-x+\frac {\int (x (1-\log (5)))^{1+x} \left (\frac {1}{5} (3-\log (5))\right )^x \, dx}{1-\log (5)}-\frac {\int \frac {\int (x (1-\log (5)))^{1+x} \left (\frac {1}{5} (3-\log (5))\right )^x \, dx}{x} \, dx}{1-\log (5)}+\frac {\log \left (\frac {1}{5} x \left (3-4 \log (5)+\log ^2(5)\right )\right ) \int (x (1-\log (5)))^{1+x} \left (\frac {1}{5} (3-\log (5))\right )^x \, dx}{1-\log (5)}+\int (x (1-\log (5)))^x \left (\frac {1}{5} (3-\log (5))\right )^x \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 38, normalized size = 1.73 \begin {gather*} -x+\frac {5^{-x} \left (x \left (3-4 \log (5)+\log ^2(5)\right )\right )^{1+x}}{3-4 \log (5)+\log ^2(5)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 27, normalized size = 1.23
method | result | size |
risch | \(-x +\left (\frac {x \ln \left (5\right )^{2}}{5}-\frac {4 x \ln \left (5\right )}{5}+\frac {3 x}{5}\right )^{x} x\) | \(25\) |
default | \(-x +{\mathrm e}^{x \ln \left (\frac {x \ln \left (5\right )^{2}}{5}-\frac {4 x \ln \left (5\right )}{5}+\frac {3 x}{5}\right )} x\) | \(27\) |
norman | \(-x +{\mathrm e}^{x \ln \left (\frac {x \ln \left (5\right )^{2}}{5}-\frac {4 x \ln \left (5\right )}{5}+\frac {3 x}{5}\right )} x\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 31, normalized size = 1.41 \begin {gather*} x e^{\left (-x \log \left (5\right ) + x \log \left (x\right ) + x \log \left (\log \left (5\right ) - 1\right ) + x \log \left (\log \left (5\right ) - 3\right )\right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.59, size = 24, normalized size = 1.09 \begin {gather*} {\left (\frac {1}{5} \, x \log \left (5\right )^{2} - \frac {4}{5} \, x \log \left (5\right ) + \frac {3}{5} \, x\right )}^{x} x - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 29, normalized size = 1.32 \begin {gather*} x e^{x \log {\left (- \frac {4 x \log {\left (5 \right )}}{5} + \frac {x \log {\left (5 \right )}^{2}}{5} + \frac {3 x}{5} \right )}} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [B]
time = 3.11, size = 22, normalized size = 1.00 \begin {gather*} x\,\left ({\left (\frac {3\,x}{5}-\frac {4\,x\,\ln \left (5\right )}{5}+\frac {x\,{\ln \left (5\right )}^2}{5}\right )}^x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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