Optimal. Leaf size=22 \[ \frac {x}{9-x+\frac {1}{4} e^{e^{2 x}} \log (x)} \]
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Rubi [F]
time = 1.40, 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 {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{1296-288 x+16 x^2+e^{e^{2 x}} (72-8 x) \log (x)+e^{2 e^{2 x}} \log ^2(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 {144+e^{e^{2 x}} \left (-4+\left (4-8 e^{2 x} x\right ) \log (x)\right )}{\left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx\\ &=\int \left (\frac {4 \left (36-e^{e^{2 x}}+e^{e^{2 x}} \log (x)\right )}{\left (-36+4 x-e^{e^{2 x}} \log (x)\right )^2}-\frac {8 e^{e^{2 x}+2 x} x \log (x)}{\left (36-4 x+e^{e^{2 x}} \log (x)\right )^2}\right ) \, dx\\ &=4 \int \frac {36-e^{e^{2 x}}+e^{e^{2 x}} \log (x)}{\left (-36+4 x-e^{e^{2 x}} \log (x)\right )^2} \, dx-8 \int \frac {e^{e^{2 x}+2 x} x \log (x)}{\left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx\\ &=4 \int \left (\frac {-1+\log (x)}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )}+\frac {4 (9-x+x \log (x))}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )^2}\right ) \, dx-8 \int \frac {e^{e^{2 x}+2 x} x \log (x)}{\left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx\\ &=4 \int \frac {-1+\log (x)}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )} \, dx-8 \int \frac {e^{e^{2 x}+2 x} x \log (x)}{\left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx+16 \int \frac {9-x+x \log (x)}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx\\ &=4 \int \left (-\frac {1}{-36+4 x-e^{e^{2 x}} \log (x)}-\frac {1}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )}\right ) \, dx-8 \int \frac {e^{e^{2 x}+2 x} x \log (x)}{\left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx+16 \int \left (\frac {x}{\left (-36+4 x-e^{e^{2 x}} \log (x)\right )^2}-\frac {x}{\log (x) \left (-36+4 x-e^{e^{2 x}} \log (x)\right )^2}+\frac {9}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {1}{-36+4 x-e^{e^{2 x}} \log (x)} \, dx\right )-4 \int \frac {1}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )} \, dx-8 \int \frac {e^{e^{2 x}+2 x} x \log (x)}{\left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx+16 \int \frac {x}{\left (-36+4 x-e^{e^{2 x}} \log (x)\right )^2} \, dx-16 \int \frac {x}{\log (x) \left (-36+4 x-e^{e^{2 x}} \log (x)\right )^2} \, dx+144 \int \frac {1}{\log (x) \left (36-4 x+e^{e^{2 x}} \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 20, normalized size = 0.91 \begin {gather*} \frac {4 x}{36-4 x+e^{e^{2 x}} \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 20, normalized size = 0.91
method | result | size |
risch | \(-\frac {4 x}{-\ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{2 x}}+4 x -36}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 18, normalized size = 0.82 \begin {gather*} \frac {4 \, x}{e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (x\right ) - 4 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 18, normalized size = 0.82 \begin {gather*} \frac {4 \, x}{e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (x\right ) - 4 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 17, normalized size = 0.77 \begin {gather*} \frac {4 x}{- 4 x + e^{e^{2 x}} \log {\left (x \right )} + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 18, normalized size = 0.82 \begin {gather*} \frac {4 \, x}{e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (x\right ) - 4 \, x + 36} \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 = -0.05 \begin {gather*} \int -\frac {{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\left (\ln \left (x\right )\,\left (8\,x\,{\mathrm {e}}^{2\,x}-4\right )+4\right )-144}{{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}}\,{\ln \left (x\right )}^2-288\,x+16\,x^2-{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\ln \left (x\right )\,\left (8\,x-72\right )+1296} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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