Optimal. Leaf size=19 \[ 2 (3-x-\log (x+\log (\log (x (1+x))))) \]
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Rubi [A]
time = 1.14, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps
used = 5, number of rules used = 4, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6820, 12, 6874,
6816} \begin {gather*} -2 x-2 \log (x+\log (\log (x (x+1)))) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6816
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 (-1-2 x-x (1+x) \log (x (1+x)) (1+x+\log (\log (x (1+x)))))}{x (1+x) \log (x (1+x)) (x+\log (\log (x (1+x))))} \, dx\\ &=2 \int \frac {-1-2 x-x (1+x) \log (x (1+x)) (1+x+\log (\log (x (1+x))))}{x (1+x) \log (x (1+x)) (x+\log (\log (x (1+x))))} \, dx\\ &=2 \int \left (-1+\frac {-1-2 x-x \log (x (1+x))-x^2 \log (x (1+x))}{x (1+x) \log (x (1+x)) (x+\log (\log (x (1+x))))}\right ) \, dx\\ &=-2 x+2 \int \frac {-1-2 x-x \log (x (1+x))-x^2 \log (x (1+x))}{x (1+x) \log (x (1+x)) (x+\log (\log (x (1+x))))} \, dx\\ &=-2 x-2 \log (x+\log (\log (x (1+x))))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 14, normalized size = 0.74 \begin {gather*} -2 (x+\log (x+\log (\log (x (1+x))))) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 x^{2}-2 x \right ) \ln \left (x^{2}+x \right ) \ln \left (\ln \left (x^{2}+x \right )\right )+\left (-2 x^{3}-4 x^{2}-2 x \right ) \ln \left (x^{2}+x \right )-4 x -2}{\left (x^{2}+x \right ) \ln \left (x^{2}+x \right ) \ln \left (\ln \left (x^{2}+x \right )\right )+\left (x^{3}+x^{2}\right ) \ln \left (x^{2}+x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 17, normalized size = 0.89 \begin {gather*} -2 \, x - 2 \, \log \left (x + \log \left (\log \left (x + 1\right ) + \log \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 16, normalized size = 0.84 \begin {gather*} -2 \, x - 2 \, \log \left (x + \log \left (\log \left (x^{2} + x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 17, normalized size = 0.89 \begin {gather*} - 2 x - 2 \log {\left (x + \log {\left (\log {\left (x^{2} + x \right )} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 16, normalized size = 0.84 \begin {gather*} -2 \, x - 2 \, \log \left (x + \log \left (\log \left (x^{2} + x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.40, size = 16, normalized size = 0.84 \begin {gather*} -2\,x-2\,\ln \left (x+\ln \left (\ln \left (x\,\left (x+1\right )\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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