Optimal. Leaf size=27 \[ -\frac {2 \log (10)}{(-4+x) x}+\log ^2\left (2 e^{\frac {x^2}{25}}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps
used = 9, number of rules used = 7, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.123, Rules used = {1608, 27, 12,
6820, 75, 2631, 30} \begin {gather*} -\frac {x^4}{625}+\frac {2}{25} x^2 \log \left (2 e^{\frac {x^2}{25}}\right )+\frac {2 \log (10)}{(4-x) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 30
Rule 75
Rule 1608
Rule 2631
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-200+100 x) \log (10)+\left (64 x^3-32 x^4+4 x^5\right ) \log \left (2 e^{\frac {x^2}{25}}\right )}{x^2 \left (400-200 x+25 x^2\right )} \, dx\\ &=\int \frac {(-200+100 x) \log (10)+\left (64 x^3-32 x^4+4 x^5\right ) \log \left (2 e^{\frac {x^2}{25}}\right )}{25 (-4+x)^2 x^2} \, dx\\ &=\frac {1}{25} \int \frac {(-200+100 x) \log (10)+\left (64 x^3-32 x^4+4 x^5\right ) \log \left (2 e^{\frac {x^2}{25}}\right )}{(-4+x)^2 x^2} \, dx\\ &=\frac {1}{25} \int \left (\frac {100 (-2+x) \log (10)}{(-4+x)^2 x^2}+4 x \log \left (2 e^{\frac {x^2}{25}}\right )\right ) \, dx\\ &=\frac {4}{25} \int x \log \left (2 e^{\frac {x^2}{25}}\right ) \, dx+(4 \log (10)) \int \frac {-2+x}{(-4+x)^2 x^2} \, dx\\ &=\frac {2 \log (10)}{(4-x) x}+\frac {2}{25} x^2 \log \left (2 e^{\frac {x^2}{25}}\right )-\frac {2}{25} \int \frac {2 x^3}{25} \, dx\\ &=\frac {2 \log (10)}{(4-x) x}+\frac {2}{25} x^2 \log \left (2 e^{\frac {x^2}{25}}\right )-\frac {4 \int x^3 \, dx}{625}\\ &=-\frac {x^4}{625}+\frac {2 \log (10)}{(4-x) x}+\frac {2}{25} x^2 \log \left (2 e^{\frac {x^2}{25}}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 1.74 \begin {gather*} -\frac {x^4}{625}-\frac {\log (10)}{2 (-4+x)}+\frac {\log (10)}{2 x}+\frac {2}{25} x^2 \log \left (2 e^{\frac {x^2}{25}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.29, size = 29, normalized size = 1.07
method | result | size |
default | \(\ln \left (2 \,{\mathrm e}^{\frac {x^{2}}{25}}\right )^{2}+\frac {\ln \left (10\right )}{2 x}-\frac {\ln \left (10\right )}{2 \left (x -4\right )}\) | \(29\) |
risch | \(\frac {2 x^{2} \ln \left ({\mathrm e}^{\frac {x^{2}}{25}}\right )}{25}-\frac {-10000 x \ln \left (2\right )^{2}-200 x^{4} \ln \left (2\right )+2500 x^{2} \ln \left (2\right )^{2}+800 x^{3} \ln \left (2\right )+5000 \ln \left (5\right )+5000 \ln \left (2\right )+4 x^{6}-16 x^{5}}{2500 \left (x -4\right ) x}\) | \(73\) |
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. 170 vs.
\(2 (24) = 48\).
time = 0.27, size = 170, normalized size = 6.30 \begin {gather*} -\frac {1}{625} \, x^{4} + \frac {1}{4} \, {\left (\frac {4 \, {\left (x - 2\right )}}{x^{2} - 4 \, x} + \log \left (x - 4\right ) - \log \left (x\right )\right )} \log \left (10\right ) - \frac {1}{4} \, {\left (\frac {4}{x - 4} + \log \left (x - 4\right ) - \log \left (x\right )\right )} \log \left (10\right ) - \frac {64}{625} \, {\left (3 \, x^{2} - 80\right )} \log \left (x - 4\right ) + \frac {256}{625} \, {\left (x^{2} - 32\right )} \log \left (x - 4\right ) - \frac {64}{625} \, {\left (x^{2} - 48\right )} \log \left (x - 4\right ) + \frac {2}{25} \, {\left (x^{2} + 16 \, x - \frac {128}{x - 4} + 96 \, \log \left (x - 4\right )\right )} \log \left (2 \, e^{\left (\frac {1}{25} \, x^{2}\right )}\right ) - \frac {32}{25} \, {\left (x - \frac {16}{x - 4} + 8 \, \log \left (x - 4\right )\right )} \log \left (2 \, e^{\left (\frac {1}{25} \, x^{2}\right )}\right ) - \frac {64}{25} \, {\left (\frac {4}{x - 4} - \log \left (x - 4\right )\right )} \log \left (2 \, e^{\left (\frac {1}{25} \, x^{2}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 37, normalized size = 1.37 \begin {gather*} \frac {x^{6} - 4 \, x^{5} + 50 \, {\left (x^{4} - 4 \, x^{3}\right )} \log \left (2\right ) - 1250 \, \log \left (10\right )}{625 \, {\left (x^{2} - 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{4}}{625} + \frac {2 x^{2} \log {\left (2 \right )}}{25} - \frac {2 \log {\left (10 \right )}}{x^{2} - 4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 0.96 \begin {gather*} \frac {1}{625} \, x^{4} + \frac {2}{25} \, x^{2} \log \left (2\right ) - \frac {2 \, \log \left (10\right )}{x^{2} - 4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.12, size = 25, normalized size = 0.93 \begin {gather*} \frac {2\,x^2\,\ln \left (2\right )}{25}+\frac {x^4}{625}-\frac {2\,\ln \left (10\right )}{x\,\left (x-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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