Optimal. Leaf size=29 \[ x+x \log \left (-2 x+\log (x)-\frac {1+\log (x)}{-3+\log ^2\left (x+x^2\right )}\right ) \]
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Rubi [F] time = 91.58, 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 {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (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 {15-21 x-36 x^2+(12+12 x) \log (x)+(2+4 x+(2+4 x) \log (x)) \log \left (x+x^2\right )+\left (-8+16 x+24 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (1-3 x-4 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )+\left (3-15 x-18 x^2+(12+12 x) \log (x)+\left (-1+11 x+12 x^2+(-7-7 x) \log (x)\right ) \log ^2\left (x+x^2\right )+\left (-2 x-2 x^2+(1+x) \log (x)\right ) \log ^4\left (x+x^2\right )\right ) \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2\left (x+x^2\right )}{-3+\log ^2\left (x+x^2\right )}\right )}{(1+x) \left (3-\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx\\ &=\int \left (-\frac {15}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {21 x}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {36 x^2}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}-\frac {2 (1+2 x) (1+\log (x)) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}-\frac {(-8+24 x-7 \log (x)) \log ^2(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {(-1+4 x-\log (x)) \log ^4(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}+\frac {12 \log (x)}{\left (-3+\log ^2(x (1+x))\right ) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )}+\log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x (1+x))}{-3+\log ^2(x (1+x))}\right )\right ) \, dx\\ &=-\left (2 \int \frac {(1+2 x) (1+\log (x)) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx\right )+12 \int \frac {\log (x)}{\left (-3+\log ^2(x (1+x))\right ) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )} \, dx-15 \int \frac {1}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+21 \int \frac {x}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+36 \int \frac {x^2}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx-\int \frac {(-8+24 x-7 \log (x)) \log ^2(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+\int \frac {(-1+4 x-\log (x)) \log ^4(x (1+x))}{\left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )} \, dx+\int \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x (1+x))}{-3+\log ^2(x (1+x))}\right ) \, dx\\ &=x \log \left (\frac {1-6 x+4 \log (x)+(2 x-\log (x)) \log ^2(x (1+x))}{3-\log ^2(x (1+x))}\right )-2 \int \left (\frac {(1+2 x) \log (x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right )}-\frac {(1+2 x) (2 x-\log (x)) \log (x (1+x))}{(1+x) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx+12 \int \left (-\frac {\log (x)}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}+\frac {\log (x) (-2 x+\log (x))}{(1+\log (x)) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )}\right ) \, dx-15 \int \left (\frac {1}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}+\frac {-2 x+\log (x)}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx+21 \int \left (\frac {x}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}-\frac {x (2 x-\log (x))}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx+36 \int \left (\frac {x^2}{(1+x) (1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}-\frac {x^2 (2 x-\log (x))}{(1+x) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx-\int \frac {6 \left (-2+x+3 x^2\right )-2 (1+2 x) (1+\log (x)) \log (x (1+x))+\left (7-5 x-12 x^2\right ) \log ^2(x (1+x))+\left (-1+x+2 x^2\right ) \log ^4(x (1+x))}{(1+x) \left (-3+\log ^2(x (1+x))\right ) \left (1-6 x+2 x \log ^2(x (1+x))-\log (x) \left (-4+\log ^2(x (1+x))\right )\right )} \, dx+\int \left (\frac {-1+4 x-\log (x)}{2 x-\log (x)}+\frac {9 (-1+4 x-\log (x))}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}-\frac {(-1+6 x-4 \log (x))^2 (-1+4 x-\log (x))}{(2 x-\log (x)) (1+\log (x)) \left (1-6 x+4 \log (x)+2 x \log ^2(x (1+x))-\log (x) \log ^2(x (1+x))\right )}\right ) \, dx-\int \left (\frac {3 (-8+24 x-7 \log (x))}{(1+\log (x)) \left (-3+\log ^2(x (1+x))\right )}+\frac {(-8+24 x-7 \log (x)) (-1+6 x-4 \log (x))}{(1+\log (x)) \left (-1+6 x-4 \log (x)-2 x \log ^2(x (1+x))+\log (x) \log ^2(x (1+x))\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.31, size = 42, normalized size = 1.45 \begin {gather*} x+x \log \left (\frac {-1+6 x-4 \log (x)+(-2 x+\log (x)) \log ^2(x (1+x))}{-3+\log ^2(x (1+x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 45, normalized size = 1.55 \begin {gather*} x \log \left (-\frac {{\left (2 \, x - \log \relax (x)\right )} \log \left (x^{2} + x\right )^{2} - 6 \, x + 4 \, \log \relax (x) + 1}{\log \left (x^{2} + x\right )^{2} - 3}\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 11.56, size = 86, normalized size = 2.97 \begin {gather*} x \log \left (-2 \, x \log \left (x + 1\right )^{2} - 4 \, x \log \left (x + 1\right ) \log \relax (x) + \log \left (x + 1\right )^{2} \log \relax (x) - 2 \, x \log \relax (x)^{2} + 2 \, \log \left (x + 1\right ) \log \relax (x)^{2} + \log \relax (x)^{3} + 6 \, x - 4 \, \log \relax (x) - 1\right ) - x \log \left (\log \left (x + 1\right )^{2} + 2 \, \log \left (x + 1\right ) \log \relax (x) + \log \relax (x)^{2} - 3\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.02, size = 10690, normalized size = 368.62
method | result | size |
risch | \(\text {Expression too large to display}\) | \(10690\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.19, size = 81, normalized size = 2.79 \begin {gather*} x \log \left (-2 \, {\left (x - \log \left (x + 1\right )\right )} \log \relax (x)^{2} + \log \relax (x)^{3} - 2 \, {\left (\log \left (x + 1\right )^{2} - 3\right )} x - {\left (4 \, x \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} + 4\right )} \log \relax (x) - 1\right ) - x \log \left (\log \left (x + 1\right )^{2} + 2 \, \log \left (x + 1\right ) \log \relax (x) + \log \relax (x)^{2} - 3\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 45, normalized size = 1.55 \begin {gather*} x\,\left (\ln \left (-\frac {\left (2\,x-\ln \relax (x)\right )\,{\ln \left (x^2+x\right )}^2-6\,x+4\,\ln \relax (x)+1}{{\ln \left (x^2+x\right )}^2-3}\right )+1\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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