Optimal. Leaf size=26 \[ \frac {x}{\log \left (3 e^{\frac {1}{2 (4-x) x}} x^2\right )} \]
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Rubi [A]
time = 0.58, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1608, 27,
6820, 6819} \begin {gather*} \frac {x}{\log \left (3 e^{\frac {1}{2 (4-x) x}} x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1608
Rule 6819
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2-33 x+16 x^2-2 x^3+\left (16 x-8 x^2+x^3\right ) \log \left (3 \exp \left (\frac {-1+\left (-8 x+2 x^2\right ) \log (x)}{-8 x+2 x^2}\right ) x\right )}{x \left (16-8 x+x^2\right ) \log ^2\left (3 \exp \left (\frac {-1+\left (-8 x+2 x^2\right ) \log (x)}{-8 x+2 x^2}\right ) x\right )} \, dx\\ &=\int \frac {2-33 x+16 x^2-2 x^3+\left (16 x-8 x^2+x^3\right ) \log \left (3 \exp \left (\frac {-1+\left (-8 x+2 x^2\right ) \log (x)}{-8 x+2 x^2}\right ) x\right )}{(-4+x)^2 x \log ^2\left (3 \exp \left (\frac {-1+\left (-8 x+2 x^2\right ) \log (x)}{-8 x+2 x^2}\right ) x\right )} \, dx\\ &=\int \frac {2-33 x+16 x^2-2 x^3+(-4+x)^2 x \log \left (3 e^{\frac {1}{8 x-2 x^2}} x^2\right )}{(4-x)^2 x \log ^2\left (3 e^{\frac {1}{(8-2 x) x}} x^2\right )} \, dx\\ &=\frac {x}{\log \left (3 e^{\frac {1}{2 (4-x) x}} x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 23, normalized size = 0.88 \begin {gather*} \frac {x}{\log \left (3 e^{\frac {1}{8 x-2 x^2}} x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.39, size = 251, normalized size = 9.65
method | result | size |
risch | \(\frac {2 i x}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-1}{2 \left (x -4\right ) x}}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-1}{2 \left (x -4\right ) x}}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-1}{2 \left (x -4\right ) x}}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-1}{2 \left (x -4\right ) x}}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-1}{2 \left (x -4\right ) x}}\right )^{2}+\pi \mathrm {csgn}\left (i x \,{\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-1}{2 \left (x -4\right ) x}}\right )^{3}+2 i \ln \left (3\right )+2 i \ln \left (x \right )+2 i \ln \left ({\mathrm e}^{\frac {2 x^{2} \ln \left (x \right )-8 x \ln \left (x \right )-1}{2 \left (x -4\right ) x}}\right )}\) | \(251\) |
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 = 0.36, size = 37, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left (x^{3} - 4 \, x^{2}\right )}}{2 \, {\left (x^{2} - 4 \, x\right )} \log \left (3\right ) + 4 \, {\left (x^{2} - 4 \, x\right )} \log \left (x\right ) - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (17) = 34\).
time = 0.36, size = 37, normalized size = 1.42 \begin {gather*} \frac {2 x^{3} - 8 x^{2}}{2 x^{2} \log {\left (3 \right )} - 8 x \log {\left (3 \right )} + \left (4 x^{2} - 16 x\right ) \log {\left (x \right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.95, size = 39, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (x^{3} - 4 \, x^{2}\right )}}{2 \, x^{2} \log \left (3\right ) + 4 \, x^{2} \log \left (x\right ) - 8 \, x \log \left (3\right ) - 16 \, x \log \left (x\right ) - 1} \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 = -0.04 \begin {gather*} \int \frac {\ln \left (3\,x\,{\mathrm {e}}^{\frac {\ln \left (x\right )\,\left (8\,x-2\,x^2\right )+1}{8\,x-2\,x^2}}\right )\,\left (x^3-8\,x^2+16\,x\right )-33\,x+16\,x^2-2\,x^3+2}{{\ln \left (3\,x\,{\mathrm {e}}^{\frac {\ln \left (x\right )\,\left (8\,x-2\,x^2\right )+1}{8\,x-2\,x^2}}\right )}^2\,\left (x^3-8\,x^2+16\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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