Optimal. Leaf size=22 \[ 5 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} \]
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Rubi [F]
time = 0.98, 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 {\exp \left (\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}\right ) (8-2 x-4 \log (5 x))}{\left (48-24 x+3 x^2\right ) \log ^3(5 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 {\exp \left (\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}\right ) (8-2 x-4 \log (5 x))}{3 (-4+x)^2 \log ^3(5 x)} \, dx\\ &=\frac {1}{3} \int \frac {\exp \left (\frac {x+(24-6 x+(-12+3 x) \log (5)) \log ^2(5 x)}{(-12+3 x) \log ^2(5 x)}\right ) (8-2 x-4 \log (5 x))}{(-4+x)^2 \log ^3(5 x)} \, dx\\ &=\frac {1}{3} \int \frac {10 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} (4-x-2 \log (5 x))}{(4-x)^2 \log ^3(5 x)} \, dx\\ &=\frac {10}{3} \int \frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} (4-x-2 \log (5 x))}{(4-x)^2 \log ^3(5 x)} \, dx\\ &=\frac {10}{3} \int \left (-\frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x) \log ^3(5 x)}-\frac {2 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x)^2 \log ^2(5 x)}\right ) \, dx\\ &=-\left (\frac {10}{3} \int \frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x) \log ^3(5 x)} \, dx\right )-\frac {20}{3} \int \frac {e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}}}{(-4+x)^2 \log ^2(5 x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.09, size = 22, normalized size = 1.00 \begin {gather*} 5 e^{-2+\frac {x}{3 (-4+x) \log ^2(5 x)}} \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. \(52\) vs.
\(2(19)=38\).
time = 2.46, size = 53, normalized size = 2.41
method | result | size |
risch | \(5^{\frac {x}{x -4}} \left (\frac {1}{625}\right )^{\frac {1}{x -4}} {\mathrm e}^{-\frac {6 x \ln \left (5 x \right )^{2}-24 \ln \left (5 x \right )^{2}-x}{3 \left (x -4\right ) \ln \left (5 x \right )^{2}}}\) | \(53\) |
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. 62 vs.
\(2 (19) = 38\).
time = 0.66, size = 62, normalized size = 2.82 \begin {gather*} 5 \, e^{\left (\frac {4}{3 \, {\left (x \log \left (5\right )^{2} + {\left (x - 4\right )} \log \left (x\right )^{2} - 4 \, \log \left (5\right )^{2} + 2 \, {\left (x \log \left (5\right ) - 4 \, \log \left (5\right )\right )} \log \left (x\right )\right )}} + \frac {1}{3 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )}} - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 35, normalized size = 1.59 \begin {gather*} e^{\left (\frac {3 \, {\left ({\left (x - 4\right )} \log \left (5\right ) - 2 \, x + 8\right )} \log \left (5 \, x\right )^{2} + x}{3 \, {\left (x - 4\right )} \log \left (5 \, x\right )^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 34, normalized size = 1.55 \begin {gather*} e^{\frac {x + \left (- 6 x + \left (3 x - 12\right ) \log {\left (5 \right )} + 24\right ) \log {\left (5 x \right )}^{2}}{\left (3 x - 12\right ) \log {\left (5 x \right )}^{2}}} \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. 137 vs.
\(2 (19) = 38\).
time = 1.01, size = 137, normalized size = 6.23 \begin {gather*} e^{\left (\frac {x \log \left (5\right ) \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} - \frac {2 \, x \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} - \frac {4 \, \log \left (5\right ) \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} + \frac {8 \, \log \left (5 \, x\right )^{2}}{x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}} + \frac {x}{3 \, {\left (x \log \left (5 \, x\right )^{2} - 4 \, \log \left (5 \, x\right )^{2}\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.00, size = 401, normalized size = 18.23 \begin {gather*} \frac {5^{\frac {{\ln \left (x\right )}^2}{{\ln \left (x\right )}^2+2\,\ln \left (5\right )\,\ln \left (x\right )+{\ln \left (5\right )}^2}}\,{\mathrm {e}}^{\frac {3\,x\,{\ln \left (5\right )}^3}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {6\,x\,{\ln \left (5\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {24\,{\ln \left (x\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {x}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {6\,x\,{\ln \left (x\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{-\frac {12\,{\ln \left (5\right )}^3}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {24\,{\ln \left (5\right )}^2}{3\,x\,{\ln \left (x\right )}^2-12\,{\ln \left (x\right )}^2+3\,x\,{\ln \left (5\right )}^2-24\,\ln \left (5\right )\,\ln \left (x\right )-12\,{\ln \left (5\right )}^2+6\,x\,\ln \left (5\right )\,\ln \left (x\right )}}}{x^{\frac {2\,\left (2\,\ln \left (5\right )-{\ln \left (5\right )}^2\right )}{{\ln \left (x\right )}^2+2\,\ln \left (5\right )\,\ln \left (x\right )+{\ln \left (5\right )}^2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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