Optimal. Leaf size=23 \[ 4 \log \left ((1+x) \left (-x+\frac {x \log (3 x)}{\log (1+x)}\right )\right ) \]
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Rubi [A] time = 0.89, antiderivative size = 32, normalized size of antiderivative = 1.39, number of steps used = 9, number of rules used = 7, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.092, Rules used = {6741, 6742, 72, 6684, 2390, 2302, 29} \begin {gather*} 4 \log (x)+4 \log (x+1)+4 \log (\log (3 x)-\log (x+1))-4 \log (\log (x+1)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 72
Rule 2302
Rule 2390
Rule 6684
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(4+4 x) \log (1+x)+(-4-8 x) \log ^2(1+x)+\log (3 x) (-4 x+(4+8 x) \log (1+x))}{x (1+x) (\log (3 x)-\log (1+x)) \log (1+x)} \, dx\\ &=\int \left (\frac {4 (1+2 x)}{x (1+x)}+\frac {4}{x (1+x) (\log (3 x)-\log (1+x))}-\frac {4}{(1+x) \log (1+x)}\right ) \, dx\\ &=4 \int \frac {1+2 x}{x (1+x)} \, dx+4 \int \frac {1}{x (1+x) (\log (3 x)-\log (1+x))} \, dx-4 \int \frac {1}{(1+x) \log (1+x)} \, dx\\ &=4 \log (\log (3 x)-\log (1+x))+4 \int \left (\frac {1}{x}+\frac {1}{1+x}\right ) \, dx-4 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,1+x\right )\\ &=4 \log (x)+4 \log (1+x)+4 \log (\log (3 x)-\log (1+x))-4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (1+x)\right )\\ &=4 \log (x)+4 \log (1+x)+4 \log (\log (3 x)-\log (1+x))-4 \log (\log (1+x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.22, size = 28, normalized size = 1.22 \begin {gather*} 4 (\log (x)+\log (1+x)+\log (\log (3 x)-\log (1+x))-\log (\log (1+x))) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 30, normalized size = 1.30 \begin {gather*} 4 \, \log \left (x^{2} + x\right ) + 4 \, \log \left (\log \left (3 \, x\right ) - \log \left (x + 1\right )\right ) - 4 \, \log \left (\log \left (x + 1\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 32, normalized size = 1.39 \begin {gather*} 4 \, \log \left (x + 1\right ) + 4 \, \log \relax (x) + 4 \, \log \left (-\log \left (3 \, x\right ) + \log \left (x + 1\right )\right ) - 4 \, \log \left (\log \left (x + 1\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 1.35
method | result | size |
risch | \(4 \ln \left (x^{2}+x \right )-4 \ln \left (\ln \left (x +1\right )\right )+4 \ln \left (\ln \left (x +1\right )-\ln \left (3 x \right )\right )\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 34, normalized size = 1.48 \begin {gather*} 4 \, \log \left (x + 1\right ) + 4 \, \log \relax (x) + 4 \, \log \left (-\log \relax (3) + \log \left (x + 1\right ) - \log \relax (x)\right ) - 4 \, \log \left (\log \left (x + 1\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.45, size = 30, normalized size = 1.30 \begin {gather*} 4\,\ln \left (x\,\left (x+1\right )\right )-4\,\ln \left (\ln \left (x+1\right )\right )+4\,\ln \left (\ln \left (3\,x\right )-\ln \left (x+1\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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