Optimal. Leaf size=29 \[ 4+2 \left (3+\frac {3}{x}\right )^{\frac {e^{e^4 x} (5-x)}{x}} x \]
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Rubi [F]
time = 7.23, 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 {\left (\frac {3+3 x}{x}\right )^{\frac {e^{e^4 x} (5-x)}{x}} \left (\left (2 x+2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )+e^{e^4 x} \log \left (\frac {3+3 x}{x}\right ) \left (-10+2 x+\left (-10-10 x+e^4 \left (10 x+8 x^2-2 x^3\right )\right ) \log \left (\frac {3+3 x}{x}\right )\right )\right )}{\left (x+x^2\right ) \log \left (\frac {3+3 x}{x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (\frac {3+3 x}{x}\right )^{\frac {e^{e^4 x} (5-x)}{x}} \left (\left (2 x+2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )+e^{e^4 x} \log \left (\frac {3+3 x}{x}\right ) \left (-10+2 x+\left (-10-10 x+e^4 \left (10 x+8 x^2-2 x^3\right )\right ) \log \left (\frac {3+3 x}{x}\right )\right )\right )}{x (1+x) \log \left (\frac {3+3 x}{x}\right )} \, dx\\ &=\int \frac {\left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \left (2 e^{e^4 x} (-5+x)+2 x (1+x)-2 e^{e^4 x} \left (5-5 \left (-1+e^4\right ) x-4 e^4 x^2+e^4 x^3\right ) \log \left (3+\frac {3}{x}\right )\right )}{x (1+x)} \, dx\\ &=\int \left (2 \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}+\frac {2 e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \left (-5+x-5 \log \left (3+\frac {3}{x}\right )-5 \left (1-e^4\right ) x \log \left (3+\frac {3}{x}\right )+4 e^4 x^2 \log \left (3+\frac {3}{x}\right )-e^4 x^3 \log \left (3+\frac {3}{x}\right )\right )}{x (1+x)}\right ) \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx+2 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \left (-5+x-5 \log \left (3+\frac {3}{x}\right )-5 \left (1-e^4\right ) x \log \left (3+\frac {3}{x}\right )+4 e^4 x^2 \log \left (3+\frac {3}{x}\right )-e^4 x^3 \log \left (3+\frac {3}{x}\right )\right )}{x (1+x)} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx+2 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \left (-5+x-\left (5-5 \left (-1+e^4\right ) x-4 e^4 x^2+e^4 x^3\right ) \log \left (3+\frac {3}{x}\right )\right )}{x (1+x)} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx+2 \int \left (\frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} (-5+x)}{x (1+x)}-\frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \left (5-5 e^4 x+e^4 x^2\right ) \log \left (3+\frac {3}{x}\right )}{x}\right ) \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx+2 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} (-5+x)}{x (1+x)} \, dx-2 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \left (5-5 e^4 x+e^4 x^2\right ) \log \left (3+\frac {3}{x}\right )}{x} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx+2 \int \left (-\frac {5 e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x}+\frac {6 e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{1+x}\right ) \, dx+2 \int \frac {5 \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-5 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx-\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{x (1+x)} \, dx-\left (2 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx+\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx+2 \int \left (\frac {5 \left (\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\right )}{x (1+x)}-\frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{x (1+x)}\right ) \, dx-10 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx+12 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{1+x} \, dx-\left (2 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx+\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-2 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{x (1+x)} \, dx-10 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx+10 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx}{x (1+x)} \, dx+12 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{1+x} \, dx-\left (2 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx+\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-2 \int \left (\frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{x}-\frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{1+x}\right ) \, dx-10 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx+10 \int \left (\frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx}{x (1+x)}-\frac {\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx}{x (1+x)}\right ) \, dx+12 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{1+x} \, dx-\left (2 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx+\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-2 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{x} \, dx+2 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{1+x} \, dx-10 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx+10 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx}{x (1+x)} \, dx-10 \int \frac {\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx}{x (1+x)} \, dx+12 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{1+x} \, dx-\left (2 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx+\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-2 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{x} \, dx+2 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{1+x} \, dx-10 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx+10 \int \left (\frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx}{x}-\frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx}{1+x}\right ) \, dx-10 \int \left (\frac {\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx}{x}-\frac {\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx}{1+x}\right ) \, dx+12 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{1+x} \, dx-\left (2 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx+\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\\ &=2 \int \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-2 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{x} \, dx+2 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx}{1+x} \, dx-10 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx+10 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx}{x} \, dx-10 \int \frac {\int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx}{1+x} \, dx-10 \int \frac {\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx}{x} \, dx+10 \int \frac {\int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx}{1+x} \, dx+12 \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{1+x} \, dx-\left (2 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \, dx+\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int e^{4+e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} \, dx-\left (10 \log \left (3+\frac {3}{x}\right )\right ) \int \frac {e^{e^4 x} \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}}}{x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 26, normalized size = 0.90 \begin {gather*} 2 \left (3+\frac {3}{x}\right )^{-\frac {e^{e^4 x} (-5+x)}{x}} x \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.39, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (\left (-2 x^{3}+8 x^{2}+10 x \right ) {\mathrm e}^{4}-10 x -10\right ) \ln \left (\frac {3 x +3}{x}\right )+2 x -10\right ) {\mathrm e}^{\ln \left (\ln \left (\frac {3 x +3}{x}\right )\right )+x \,{\mathrm e}^{4}}+\left (2 x^{2}+2 x \right ) \ln \left (\frac {3 x +3}{x}\right )\right ) {\mathrm e}^{\frac {\left (5-x \right ) {\mathrm e}^{\ln \left (\ln \left (\frac {3 x +3}{x}\right )\right )+x \,{\mathrm e}^{4}}}{x}}}{\left (x^{2}+x \right ) \ln \left (\frac {3 x +3}{x}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (29) = 58\).
time = 0.70, size = 71, normalized size = 2.45 \begin {gather*} 2 \, x e^{\left (-e^{\left (x e^{4}\right )} \log \left (3\right ) - e^{\left (x e^{4}\right )} \log \left (x + 1\right ) + e^{\left (x e^{4}\right )} \log \left (x\right ) + \frac {5 \, e^{\left (x e^{4}\right )} \log \left (3\right )}{x} + \frac {5 \, e^{\left (x e^{4}\right )} \log \left (x + 1\right )}{x} - \frac {5 \, e^{\left (x e^{4}\right )} \log \left (x\right )}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 28, normalized size = 0.97 \begin {gather*} 2 \, x e^{\left (-\frac {{\left (x - 5\right )} e^{\left (x e^{4} + \log \left (\log \left (\frac {3 \, {\left (x + 1\right )}}{x}\right )\right )\right )}}{x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 116.54, size = 24, normalized size = 0.83 \begin {gather*} 2 x e^{\frac {\left (5 - x\right ) e^{x e^{4}} \log {\left (\frac {3 x + 3}{x} \right )}}{x}} \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 = 1.93, size = 29, normalized size = 1.00 \begin {gather*} 2\,x\,{\left (\frac {3}{x}+3\right )}^{\frac {5\,{\mathrm {e}}^{x\,{\mathrm {e}}^4}}{x}-{\mathrm {e}}^{x\,{\mathrm {e}}^4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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