Optimal. Leaf size=23 \[ 2+\log \left (\frac {5}{(4+x)^2}-\log ^2\left (3 x+x^2\right )\right ) \]
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Rubi [F] time = 37.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10 x (3+x)-2 (4+x)^3 (3+2 x) \log (x (3+x))}{x \left (12+7 x+x^2\right ) \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx\\ &=\int \left (\frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{6 x \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )}-\frac {2 \left (15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))\right )}{3 (3+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )}+\frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{2 (4+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )}\right ) \, dx\\ &=\frac {1}{6} \int \frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{x \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )} \, dx+\frac {1}{2} \int \frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{(4+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )} \, dx-\frac {2}{3} \int \frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{(3+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )} \, dx\\ &=\frac {1}{6} \int \frac {-5 x (3+x)-(4+x)^3 (3+2 x) \log (x (3+x))}{x \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx+\frac {1}{2} \int \frac {-5 x (3+x)-(4+x)^3 (3+2 x) \log (x (3+x))}{(4+x) \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx-\frac {2}{3} \int \frac {-5 x (3+x)-(4+x)^3 (3+2 x) \log (x (3+x))}{(3+x) \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.88, size = 44, normalized size = 1.91 \begin {gather*} -2 \log (4+x)+\log \left (5-16 \log ^2(x (3+x))-8 x \log ^2(x (3+x))-x^2 \log ^2(x (3+x))\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 33, normalized size = 1.43 \begin {gather*} \log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x^{2} + 3 \, x\right )^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 49, normalized size = 2.13 \begin {gather*} \log \left (x^{2} \log \left (x^{2} + 3 \, x\right )^{2} + 8 \, x \log \left (x^{2} + 3 \, x\right )^{2} + 16 \, \log \left (x^{2} + 3 \, x\right )^{2} - 5\right ) - 2 \, \log \left (x + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 25, normalized size = 1.09
method | result | size |
risch | \(\ln \left (\ln \left (x^{2}+3 x \right )^{2}-\frac {5}{x^{2}+8 x +16}\right )\) | \(25\) |
norman | \(-2 \ln \left (4+x \right )+\ln \left (\ln \left (x^{2}+3 x \right )^{2} x^{2}+8 \ln \left (x^{2}+3 x \right )^{2} x +16 \ln \left (x^{2}+3 x \right )^{2}-5\right )\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 58, normalized size = 2.52 \begin {gather*} \log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right )^{2} + 2 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right ) \log \relax (x) + {\left (x^{2} + 8 \, x + 16\right )} \log \relax (x)^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \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 = -0.04 \begin {gather*} -\int \frac {30\,x+\ln \left (x^2+3\,x\right )\,\left (4\,x^4+54\,x^3+264\,x^2+544\,x+384\right )+10\,x^2}{60\,x+35\,x^2+5\,x^3-{\ln \left (x^2+3\,x\right )}^2\,\left (x^5+15\,x^4+84\,x^3+208\,x^2+192\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 20, normalized size = 0.87 \begin {gather*} \log {\left (\log {\left (x^{2} + 3 x \right )}^{2} - \frac {5}{x^{2} + 8 x + 16} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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