Optimal. Leaf size=26 \[ \log \left (\frac {x}{e^2+\frac {x+5 (-4+x-\log (x))^2}{x}}\right ) \]
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Rubi [A]
time = 0.31, antiderivative size = 40, normalized size of antiderivative = 1.54, number of steps
used = 5, number of rules used = 3, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6, 6874, 6816}
\begin {gather*} 2 \log (x)-\log \left (5 x^2-\left (39-e^2\right ) x+5 \log ^2(x)-10 x \log (x)+40 \log (x)+80\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 6816
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {120+\left (-29+e^2\right ) x+(70-10 x) \log (x)+10 \log ^2(x)}{80 x-39 x^2+e^2 x^2+5 x^3+\left (40 x-10 x^2\right ) \log (x)+5 x \log ^2(x)} \, dx\\ &=\int \frac {120+\left (-29+e^2\right ) x+(70-10 x) \log (x)+10 \log ^2(x)}{80 x+\left (-39+e^2\right ) x^2+5 x^3+\left (40 x-10 x^2\right ) \log (x)+5 x \log ^2(x)} \, dx\\ &=\int \left (\frac {2}{x}+\frac {-40+49 \left (1-\frac {e^2}{49}\right ) x-10 x^2-10 \log (x)+10 x \log (x)}{x \left (80-39 \left (1-\frac {e^2}{39}\right ) x+5 x^2+40 \log (x)-10 x \log (x)+5 \log ^2(x)\right )}\right ) \, dx\\ &=2 \log (x)+\int \frac {-40+49 \left (1-\frac {e^2}{49}\right ) x-10 x^2-10 \log (x)+10 x \log (x)}{x \left (80-39 \left (1-\frac {e^2}{39}\right ) x+5 x^2+40 \log (x)-10 x \log (x)+5 \log ^2(x)\right )} \, dx\\ &=2 \log (x)-\log \left (80-\left (39-e^2\right ) x+5 x^2+40 \log (x)-10 x \log (x)+5 \log ^2(x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 38, normalized size = 1.46 \begin {gather*} 2 \log (x)-\log \left (80-39 x+e^2 x+5 x^2+40 \log (x)-10 x \log (x)+5 \log ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.23, size = 38, normalized size = 1.46
method | result | size |
risch | \(2 \ln \left (x \right )-\ln \left (\ln \left (x \right )^{2}+\left (-2 x +8\right ) \ln \left (x \right )+\frac {{\mathrm e}^{2} x}{5}+x^{2}-\frac {39 x}{5}+16\right )\) | \(34\) |
default | \(2 \ln \left (x \right )-\ln \left ({\mathrm e}^{2} x +5 \ln \left (x \right )^{2}-10 x \ln \left (x \right )+5 x^{2}+40 \ln \left (x \right )-39 x +80\right )\) | \(38\) |
norman | \(2 \ln \left (x \right )-\ln \left ({\mathrm e}^{2} x +5 \ln \left (x \right )^{2}-10 x \ln \left (x \right )+5 x^{2}+40 \ln \left (x \right )-39 x +80\right )\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 31, normalized size = 1.19 \begin {gather*} -\log \left (x^{2} + \frac {1}{5} \, x {\left (e^{2} - 39\right )} - 2 \, {\left (x - 4\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 16\right ) + 2 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 35, normalized size = 1.35 \begin {gather*} -\log \left (5 \, x^{2} + x e^{2} - 10 \, {\left (x - 4\right )} \log \left (x\right ) + 5 \, \log \left (x\right )^{2} - 39 \, x + 80\right ) + 2 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 36, normalized size = 1.38 \begin {gather*} 2 \log {\left (x \right )} - \log {\left (x^{2} - \frac {39 x}{5} + \frac {x e^{2}}{5} + \left (8 - 2 x\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 16 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 38, normalized size = 1.46 \begin {gather*} -\log \left (-5 \, x^{2} - x e^{2} + 10 \, x \log \left (x\right ) - 5 \, \log \left (x\right )^{2} + 39 \, x - 40 \, \log \left (x\right ) - 80\right ) + 2 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.12, size = 34, normalized size = 1.31 \begin {gather*} 2\,\ln \left (x\right )-\ln \left (8\,\ln \left (x\right )-\frac {39\,x}{5}+\frac {x\,{\mathrm {e}}^2}{5}+{\ln \left (x\right )}^2-2\,x\,\ln \left (x\right )+x^2+16\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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