Optimal. Leaf size=25 \[ x-\log \left (\frac {36 \log ^2\left (x+9 e^8 x\right )}{x^2 \log ^2(x)}\right ) \]
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Rubi [A]
time = 0.43, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps
used = 11, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2494, 6874,
45, 2339, 29} \begin {gather*} x+2 \log (x)+2 \log (\log (x))-2 \log \left (\log \left (\left (1+9 e^8\right ) x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 45
Rule 2339
Rule 2494
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 \log (x)+(2+(2+x) \log (x)) \log \left (x+9 e^8 x\right )}{x \log (x) \log \left (\left (1+9 e^8\right ) x\right )} \, dx\\ &=\int \left (\frac {2+2 \log (x)+x \log (x)}{x \log (x)}-\frac {2}{x \log \left (\left (1+9 e^8\right ) x\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x \log \left (\left (1+9 e^8\right ) x\right )} \, dx\right )+\int \frac {2+2 \log (x)+x \log (x)}{x \log (x)} \, dx\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\left (1+9 e^8\right ) x\right )\right )\right )+\int \left (\frac {2+x}{x}+\frac {2}{x \log (x)}\right ) \, dx\\ &=-2 \log \left (\log \left (\left (1+9 e^8\right ) x\right )\right )+2 \int \frac {1}{x \log (x)} \, dx+\int \frac {2+x}{x} \, dx\\ &=-2 \log \left (\log \left (\left (1+9 e^8\right ) x\right )\right )+2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )+\int \left (1+\frac {2}{x}\right ) \, dx\\ &=x+2 \log (x)+2 \log (\log (x))-2 \log \left (\log \left (\left (1+9 e^8\right ) x\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 23, normalized size = 0.92 \begin {gather*} x+2 \log (x)+2 \log (\log (x))-2 \log \left (\log \left (x+9 e^8 x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 26, normalized size = 1.04
method | result | size |
risch | \(2 \ln \left (x \right )+x\) | \(7\) |
default | \(2 \ln \left (x \right )+x +2 \ln \left (\ln \left (x \right )\right )-2 \ln \left (\ln \left (\left (9 \,{\mathrm e}^{8}+1\right ) x \right )\right )\) | \(26\) |
norman | \(x +2 \ln \left (9 x \,{\mathrm e}^{8}+x \right )+2 \ln \left (\ln \left (x \right )\right )-2 \ln \left (\ln \left (9 x \,{\mathrm e}^{8}+x \right )\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 22, normalized size = 0.88 \begin {gather*} x + 2 \, \log \left (x\right ) - 2 \, \log \left (\log \left (9 \, x e^{8} + x\right )\right ) + 2 \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 24, normalized size = 0.96 \begin {gather*} x + 2 \, \log \left (x\right ) - 2 \, \log \left (\log \left (x\right ) + \log \left (9 \, e^{8} + 1\right )\right ) + 2 \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 27, normalized size = 1.08 \begin {gather*} x + 2 \log {\left (x \right )} - 2 \log {\left (\log {\left (x \right )} + \log {\left (1 + 9 e^{8} \right )} \right )} + 2 \log {\left (\log {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 28, normalized size = 1.12 \begin {gather*} x + 2 \, \log \left (x\right ) - 2 \, \log \left (-\log \left (x\right ) - \log \left (9 \, e^{8} + 1\right )\right ) + 2 \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.60, size = 23, normalized size = 0.92 \begin {gather*} x+2\,\ln \left (\ln \left (x\right )\right )-2\,\ln \left (\ln \left (x\,\left (9\,{\mathrm {e}}^8+1\right )\right )\right )+2\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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