Optimal. Leaf size=29 \[ e^{-5+5 x \left (x+\frac {1}{6} (x+\log (x))\right )}+x+\frac {x}{4-\log (x)} \]
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Rubi [A]
time = 0.74, antiderivative size = 31, normalized size of antiderivative = 1.07, number of steps
used = 12, number of rules used = 7, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6873, 12,
6874, 2326, 2336, 2209, 2334} \begin {gather*} e^{\frac {35 x^2}{6}-5} x^{5 x/6}+x+\frac {x}{4-\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2326
Rule 2334
Rule 2336
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {126-54 \log (x)+6 \log ^2(x)+e^{\frac {1}{6} \left (-30+35 x^2+5 x \log (x)\right )} \left (80+1120 x+(40-560 x) \log (x)+(-35+70 x) \log ^2(x)+5 \log ^3(x)\right )}{6 (4-\log (x))^2} \, dx\\ &=\frac {1}{6} \int \frac {126-54 \log (x)+6 \log ^2(x)+e^{\frac {1}{6} \left (-30+35 x^2+5 x \log (x)\right )} \left (80+1120 x+(40-560 x) \log (x)+(-35+70 x) \log ^2(x)+5 \log ^3(x)\right )}{(4-\log (x))^2} \, dx\\ &=\frac {1}{6} \int \left (5 e^{-5+\frac {35 x^2}{6}} x^{5 x/6} (1+14 x+\log (x))+\frac {6 \left (21-9 \log (x)+\log ^2(x)\right )}{(-4+\log (x))^2}\right ) \, dx\\ &=\frac {5}{6} \int e^{-5+\frac {35 x^2}{6}} x^{5 x/6} (1+14 x+\log (x)) \, dx+\int \frac {21-9 \log (x)+\log ^2(x)}{(-4+\log (x))^2} \, dx\\ &=e^{-5+\frac {35 x^2}{6}} x^{5 x/6}+\int \left (1+\frac {1}{4-\log (x)}+\frac {1}{(-4+\log (x))^2}\right ) \, dx\\ &=x+e^{-5+\frac {35 x^2}{6}} x^{5 x/6}+\int \frac {1}{4-\log (x)} \, dx+\int \frac {1}{(-4+\log (x))^2} \, dx\\ &=x+e^{-5+\frac {35 x^2}{6}} x^{5 x/6}+\frac {x}{4-\log (x)}+\int \frac {1}{-4+\log (x)} \, dx+\text {Subst}\left (\int \frac {e^x}{4-x} \, dx,x,\log (x)\right )\\ &=x+e^{-5+\frac {35 x^2}{6}} x^{5 x/6}-e^4 \text {Ei}(-4+\log (x))+\frac {x}{4-\log (x)}+\text {Subst}\left (\int \frac {e^x}{-4+x} \, dx,x,\log (x)\right )\\ &=x+e^{-5+\frac {35 x^2}{6}} x^{5 x/6}+\frac {x}{4-\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 37, normalized size = 1.28 \begin {gather*} \frac {1}{6} \left (6 x+6 e^{-5+\frac {35 x^2}{6}} x^{5 x/6}-\frac {6 x}{-4+\log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs.
\(2(26)=52\).
time = 1.54, size = 55, normalized size = 1.90
method | result | size |
risch | \(x -\frac {x}{\ln \left (x \right )-4}+x^{\frac {5 x}{6}} {\mathrm e}^{-5+\frac {35 x^{2}}{6}}\) | \(26\) |
norman | \(\frac {x \ln \left (x \right )+\ln \left (x \right ) {\mathrm e}^{\frac {5 x \ln \left (x \right )}{6}+\frac {35 x^{2}}{6}-5}-5 x -4 \,{\mathrm e}^{\frac {5 x \ln \left (x \right )}{6}+\frac {35 x^{2}}{6}-5}}{\ln \left (x \right )-4}\) | \(47\) |
default | \(\frac {x \left (\ln \left (x \right )-5\right )}{\ln \left (x \right )-4}+\frac {6 \ln \left (x \right ) {\mathrm e}^{\frac {5 x \ln \left (x \right )}{6}+\frac {35 x^{2}}{6}-5}-24 \,{\mathrm e}^{\frac {5 x \ln \left (x \right )}{6}+\frac {35 x^{2}}{6}-5}}{6 \ln \left (x \right )-24}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 33, normalized size = 1.14 \begin {gather*} \frac {{\left (\log \left (x\right ) - 4\right )} e^{\left (\frac {35}{6} \, x^{2} + \frac {5}{6} \, x \log \left (x\right ) - 5\right )} + x \log \left (x\right ) - 5 \, x}{\log \left (x\right ) - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 26, normalized size = 0.90 \begin {gather*} x - \frac {x}{\log {\left (x \right )} - 4} + e^{\frac {35 x^{2}}{6} + \frac {5 x \log {\left (x \right )}}{6} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (23) = 46\).
time = 0.43, size = 54, normalized size = 1.86 \begin {gather*} \frac {x e^{5} \log \left (x\right ) - 5 \, x e^{5} + e^{\left (\frac {35}{6} \, x^{2} + \frac {5}{6} \, x \log \left (x\right )\right )} \log \left (x\right ) - 4 \, e^{\left (\frac {35}{6} \, x^{2} + \frac {5}{6} \, x \log \left (x\right )\right )}}{e^{5} \log \left (x\right ) - 4 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 32, normalized size = 1.10 \begin {gather*} x^{\frac {5\,x}{6}}\,{\mathrm {e}}^{\frac {35\,x^2}{6}-5}-\frac {5\,x-x\,\ln \left (x\right )}{\ln \left (x\right )-4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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