Optimal. Leaf size=29 \[ \frac {x}{5-x}-4 \left (x+\log \left (2+\frac {5 x}{e^4 (x+\log (x))}\right )\right ) \]
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Rubi [A]
time = 0.68, antiderivative size = 40, normalized size of antiderivative = 1.38, number of steps
used = 7, number of rules used = 4, integrand size = 186, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6873, 6874,
697, 6816} \begin {gather*} -4 x+\frac {5}{5-x}+4 \log (x+\log (x))-4 \log \left (\left (5+2 e^4\right ) x+2 e^4 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rule 6816
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {500-200 x-455 x^2+200 x^3-20 x^4+e^4 \left (-190 x^2+80 x^3-8 x^4\right )+\left (-500-275 x+180 x^2-20 x^3+e^4 \left (-380 x+160 x^2-16 x^3\right )\right ) \log (x)+e^4 \left (-190+80 x-8 x^2\right ) \log ^2(x)}{(5-x)^2 \left (5 \left (1+\frac {2 e^4}{5}\right ) x^2+5 \left (1+\frac {4 e^4}{5}\right ) x \log (x)+2 e^4 \log ^2(x)\right )} \, dx\\ &=\int \left (\frac {-95+40 x-4 x^2}{(-5+x)^2}+\frac {4 (1+x)}{x (x+\log (x))}+\frac {4 \left (-2 e^4-\left (5+2 e^4\right ) x\right )}{x \left (5 \left (1+\frac {2 e^4}{5}\right ) x+2 e^4 \log (x)\right )}\right ) \, dx\\ &=4 \int \frac {1+x}{x (x+\log (x))} \, dx+4 \int \frac {-2 e^4-\left (5+2 e^4\right ) x}{x \left (5 \left (1+\frac {2 e^4}{5}\right ) x+2 e^4 \log (x)\right )} \, dx+\int \frac {-95+40 x-4 x^2}{(-5+x)^2} \, dx\\ &=4 \log (x+\log (x))-4 \log \left (\left (5+2 e^4\right ) x+2 e^4 \log (x)\right )+\int \left (-4+\frac {5}{(-5+x)^2}\right ) \, dx\\ &=\frac {5}{5-x}-4 x+4 \log (x+\log (x))-4 \log \left (\left (5+2 e^4\right ) x+2 e^4 \log (x)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 38, normalized size = 1.31 \begin {gather*} -\frac {5}{-5+x}-4 x+4 \log (x+\log (x))-4 \log \left (5 x+2 e^4 x+2 e^4 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.48, size = 41, normalized size = 1.41
method | result | size |
norman | \(\frac {-4 x^{2}+95}{x -5}+4 \ln \left (x +\ln \left (x \right )\right )-4 \ln \left (2 \,{\mathrm e}^{4} \ln \left (x \right )+2 x \,{\mathrm e}^{4}+5 x \right )\) | \(40\) |
default | \(-\frac {4 x^{2}-95}{x -5}+4 \ln \left (x +\ln \left (x \right )\right )-4 \ln \left (2 \,{\mathrm e}^{4} \ln \left (x \right )+2 x \,{\mathrm e}^{4}+5 x \right )\) | \(41\) |
risch | \(-\frac {4 x^{2}-20 x +5}{x -5}+4 \ln \left (x +\ln \left (x \right )\right )-4 \ln \left (\ln \left (x \right )+\frac {\left (2 \,{\mathrm e}^{4}+5\right ) x \,{\mathrm e}^{-4}}{2}\right )\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 47, normalized size = 1.62 \begin {gather*} -\frac {4 \, x^{2} - 20 \, x + 5}{x - 5} - 4 \, \log \left (\frac {1}{2} \, {\left (x {\left (2 \, e^{4} + 5\right )} + 2 \, e^{4} \log \left (x\right )\right )} e^{\left (-4\right )}\right ) + 4 \, \log \left (x + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 48, normalized size = 1.66 \begin {gather*} -\frac {4 \, x^{2} + 4 \, {\left (x - 5\right )} \log \left (2 \, x e^{4} + 2 \, e^{4} \log \left (x\right ) + 5 \, x\right ) - 4 \, {\left (x - 5\right )} \log \left (x + \log \left (x\right )\right ) - 20 \, x + 5}{x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 37, normalized size = 1.28 \begin {gather*} - 4 x + 4 \log {\left (x + \log {\left (x \right )} \right )} - 4 \log {\left (\frac {40 x + 16 x e^{4}}{16 e^{4}} + \log {\left (x \right )} \right )} - \frac {5}{x - 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. 77 vs.
\(2 (28) = 56\).
time = 0.43, size = 77, normalized size = 2.66 \begin {gather*} -\frac {4 \, x^{2} + 4 \, x \log \left (2 \, x e^{4} + 2 \, e^{4} \log \left (x\right ) + 5 \, x\right ) - 4 \, x \log \left (-x - \log \left (x\right )\right ) - 20 \, x - 20 \, \log \left (2 \, x e^{4} + 2 \, e^{4} \log \left (x\right ) + 5 \, x\right ) + 20 \, \log \left (-x - \log \left (x\right )\right ) + 5}{x - 5} \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 = -0.03 \begin {gather*} \int -\frac {200\,x+\ln \left (x\right )\,\left (275\,x+{\mathrm {e}}^4\,\left (16\,x^3-160\,x^2+380\,x\right )-180\,x^2+20\,x^3+500\right )+{\mathrm {e}}^4\,\left (8\,x^4-80\,x^3+190\,x^2\right )+455\,x^2-200\,x^3+20\,x^4+{\mathrm {e}}^4\,{\ln \left (x\right )}^2\,\left (8\,x^2-80\,x+190\right )-500}{\ln \left (x\right )\,\left (125\,x+{\mathrm {e}}^4\,\left (4\,x^3-40\,x^2+100\,x\right )-50\,x^2+5\,x^3\right )+{\mathrm {e}}^4\,\left (2\,x^4-20\,x^3+50\,x^2\right )+125\,x^2-50\,x^3+5\,x^4+{\mathrm {e}}^4\,{\ln \left (x\right )}^2\,\left (2\,x^2-20\,x+50\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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