Optimal. Leaf size=27 \[ -2+x+\left (-x+\frac {2}{x \left (\frac {1}{2}-x^2+\log (x)\right )}\right )^2 \]
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Rubi [F] time = 1.81, 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 {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}+\left (-64+32 x^2+6 x^3-52 x^4-24 x^5-48 x^6+24 x^7+48 x^8\right ) \log (x)+\left (12 x^3+24 x^4-24 x^5-48 x^6\right ) \log ^2(x)+\left (8 x^3+16 x^4\right ) \log ^3(x)}{x^3-6 x^5+12 x^7-8 x^9+\left (6 x^3-24 x^5+24 x^7\right ) \log (x)+\left (12 x^3-24 x^5\right ) \log ^2(x)+8 x^3 \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-96+208 x^2+x^3-62 x^4-6 x^5+52 x^6+12 x^7+24 x^8-8 x^9-16 x^{10}-\left (64-32 x^2-6 x^3+52 x^4+24 x^5+48 x^6-24 x^7-48 x^8\right ) \log (x)-12 x^3 \left (-1-2 x+2 x^2+4 x^3\right ) \log ^2(x)+8 x^3 (1+2 x) \log ^3(x)}{x^3 \left (1-2 x^2+2 \log (x)\right )^3} \, dx\\ &=\int \left (1+2 x-\frac {64 \left (-1+2 x^2\right )}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3}-\frac {16 \left (2-x^2+2 x^4\right )}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2}\right ) \, dx\\ &=x+x^2-16 \int \frac {2-x^2+2 x^4}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2} \, dx-64 \int \frac {-1+2 x^2}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3} \, dx\\ &=x+x^2-16 \int \left (\frac {2}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2}-\frac {1}{x \left (-1+2 x^2-2 \log (x)\right )^2}+\frac {2 x}{\left (-1+2 x^2-2 \log (x)\right )^2}\right ) \, dx-64 \int \left (-\frac {1}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3}+\frac {2}{x \left (-1+2 x^2-2 \log (x)\right )^3}\right ) \, dx\\ &=x+x^2+16 \int \frac {1}{x \left (-1+2 x^2-2 \log (x)\right )^2} \, dx-32 \int \frac {1}{x^3 \left (-1+2 x^2-2 \log (x)\right )^2} \, dx-32 \int \frac {x}{\left (-1+2 x^2-2 \log (x)\right )^2} \, dx+64 \int \frac {1}{x^3 \left (-1+2 x^2-2 \log (x)\right )^3} \, dx-128 \int \frac {1}{x \left (-1+2 x^2-2 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 38, normalized size = 1.41 \begin {gather*} x+x^2+\frac {16}{x^2 \left (1-2 x^2+2 \log (x)\right )^2}-\frac {8}{1-2 x^2+2 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 119, normalized size = 4.41 \begin {gather*} \frac {4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} - 4 \, x^{5} + 17 \, x^{4} + x^{3} + 4 \, {\left (x^{4} + x^{3}\right )} \log \relax (x)^{2} - 8 \, x^{2} - 4 \, {\left (2 \, x^{6} + 2 \, x^{5} - x^{4} - x^{3} + 4 \, x^{2}\right )} \log \relax (x) + 16}{4 \, x^{6} - 4 \, x^{4} + 4 \, x^{2} \log \relax (x)^{2} + x^{2} - 4 \, {\left (2 \, x^{4} - x^{2}\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 65, normalized size = 2.41 \begin {gather*} x^{2} + x + \frac {8 \, {\left (2 \, x^{4} - 2 \, x^{2} \log \relax (x) - x^{2} + 2\right )}}{4 \, x^{6} - 8 \, x^{4} \log \relax (x) - 4 \, x^{4} + 4 \, x^{2} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x) + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 1.59
method | result | size |
risch | \(x^{2}+x +\frac {16 x^{4}-16 x^{2} \ln \relax (x )-8 x^{2}+16}{x^{2} \left (2 x^{2}-2 \ln \relax (x )-1\right )^{2}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 119, normalized size = 4.41 \begin {gather*} \frac {4 \, x^{8} + 4 \, x^{7} - 4 \, x^{6} - 4 \, x^{5} + 17 \, x^{4} + x^{3} + 4 \, {\left (x^{4} + x^{3}\right )} \log \relax (x)^{2} - 8 \, x^{2} - 4 \, {\left (2 \, x^{6} + 2 \, x^{5} - x^{4} - x^{3} + 4 \, x^{2}\right )} \log \relax (x) + 16}{4 \, x^{6} - 4 \, x^{4} + 4 \, x^{2} \log \relax (x)^{2} + x^{2} - 4 \, {\left (2 \, x^{4} - x^{2}\right )} \log \relax (x)} \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 {\ln \relax (x)\,\left (48\,x^8+24\,x^7-48\,x^6-24\,x^5-52\,x^4+6\,x^3+32\,x^2-64\right )+{\ln \relax (x)}^3\,\left (16\,x^4+8\,x^3\right )+208\,x^2+x^3-62\,x^4-6\,x^5+52\,x^6+12\,x^7+24\,x^8-8\,x^9-16\,x^{10}+{\ln \relax (x)}^2\,\left (-48\,x^6-24\,x^5+24\,x^4+12\,x^3\right )-96}{\ln \relax (x)\,\left (24\,x^7-24\,x^5+6\,x^3\right )+{\ln \relax (x)}^2\,\left (12\,x^3-24\,x^5\right )+8\,x^3\,{\ln \relax (x)}^3+x^3-6\,x^5+12\,x^7-8\,x^9} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 61, normalized size = 2.26 \begin {gather*} x^{2} + x + \frac {16 x^{4} - 16 x^{2} \log {\relax (x )} - 8 x^{2} + 16}{4 x^{6} - 4 x^{4} + 4 x^{2} \log {\relax (x )}^{2} + x^{2} + \left (- 8 x^{4} + 4 x^{2}\right ) \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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