Optimal. Leaf size=28 \[ (-1+x)^2 x^4 \left (\frac {1}{4} \left (1+\frac {1}{x}\right )+x+\frac {\log (2)}{x}\right )^2 \]
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Rubi [B] time = 0.03, antiderivative size = 107, normalized size of antiderivative = 3.82, number of steps used = 4, number of rules used = 1, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {12} \begin {gather*} x^8-\frac {3 x^7}{2}+\frac {9 x^6}{16}+2 x^6 \log (2)-\frac {x^5}{2}-\frac {7}{2} x^5 \log (2)+\frac {3 x^4}{8}+x^4 \log ^2(2)+\frac {3}{2} x^4 \log (2)-2 x^3 \log ^2(2)-\frac {1}{2} x^3 \log (2)+\frac {x^2}{16}+x^2 \log ^2(2)+\frac {1}{2} x^2 \log (2) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \left (x+12 x^3-20 x^4+27 x^5-84 x^6+64 x^7+\left (8 x-12 x^2+48 x^3-140 x^4+96 x^5\right ) \log (2)+\left (16 x-48 x^2+32 x^3\right ) \log ^2(2)\right ) \, dx\\ &=\frac {x^2}{16}+\frac {3 x^4}{8}-\frac {x^5}{2}+\frac {9 x^6}{16}-\frac {3 x^7}{2}+x^8+\frac {1}{8} \log (2) \int \left (8 x-12 x^2+48 x^3-140 x^4+96 x^5\right ) \, dx+\frac {1}{8} \log ^2(2) \int \left (16 x-48 x^2+32 x^3\right ) \, dx\\ &=\frac {x^2}{16}+\frac {3 x^4}{8}-\frac {x^5}{2}+\frac {9 x^6}{16}-\frac {3 x^7}{2}+x^8+\frac {1}{2} x^2 \log (2)-\frac {1}{2} x^3 \log (2)+\frac {3}{2} x^4 \log (2)-\frac {7}{2} x^5 \log (2)+2 x^6 \log (2)+x^2 \log ^2(2)-2 x^3 \log ^2(2)+x^4 \log ^2(2)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.05, size = 82, normalized size = 2.93 \begin {gather*} \frac {1}{96} x^2 \left (-144 x^5+96 x^6+6 x^4 (9+32 \log (2))+6 (1+\log (16))^2+3 x^2 \left (12+44 \log (2)+32 \log ^2(2)+\log (16)\right )-4 x \left (32 \log ^2(2)+\log (16) (3+\log (16))\right )-48 x^3 (1+\log (128))\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.59, size = 74, normalized size = 2.64 \begin {gather*} x^{8} - \frac {3}{2} \, x^{7} + \frac {9}{16} \, x^{6} - \frac {1}{2} \, x^{5} + \frac {3}{8} \, x^{4} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + \frac {1}{16} \, x^{2} + \frac {1}{2} \, {\left (4 \, x^{6} - 7 \, x^{5} + 3 \, x^{4} - x^{3} + x^{2}\right )} \log \relax (2) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 74, normalized size = 2.64 \begin {gather*} x^{8} - \frac {3}{2} \, x^{7} + \frac {9}{16} \, x^{6} - \frac {1}{2} \, x^{5} + \frac {3}{8} \, x^{4} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + \frac {1}{16} \, x^{2} + \frac {1}{2} \, {\left (4 \, x^{6} - 7 \, x^{5} + 3 \, x^{4} - x^{3} + x^{2}\right )} \log \relax (2) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 29, normalized size = 1.04
method | result | size |
gosper | \(\frac {\left (4 x^{3}+4 x \ln \relax (2)-3 x^{2}-4 \ln \relax (2)-1\right )^{2} x^{2}}{16}\) | \(29\) |
norman | \(x^{8}+\left (\frac {9}{16}+2 \ln \relax (2)\right ) x^{6}+\left (-2 \ln \relax (2)^{2}-\frac {\ln \relax (2)}{2}\right ) x^{3}+\left (-\frac {7 \ln \relax (2)}{2}-\frac {1}{2}\right ) x^{5}+\left (\ln \relax (2)^{2}+\frac {\ln \relax (2)}{2}+\frac {1}{16}\right ) x^{2}+\left (\ln \relax (2)^{2}+\frac {3 \ln \relax (2)}{2}+\frac {3}{8}\right ) x^{4}-\frac {3 x^{7}}{2}\) | \(73\) |
default | \(\frac {\ln \relax (2)^{2} \left (8 x^{4}-16 x^{3}+8 x^{2}\right )}{8}+\frac {\ln \relax (2) \left (16 x^{6}-28 x^{5}+12 x^{4}-4 x^{3}+4 x^{2}\right )}{8}+x^{8}-\frac {3 x^{7}}{2}+\frac {9 x^{6}}{16}-\frac {x^{5}}{2}+\frac {3 x^{4}}{8}+\frac {x^{2}}{16}\) | \(82\) |
risch | \(x^{4} \ln \relax (2)^{2}-2 x^{3} \ln \relax (2)^{2}+x^{2} \ln \relax (2)^{2}+2 x^{6} \ln \relax (2)-\frac {7 x^{5} \ln \relax (2)}{2}+\frac {3 x^{4} \ln \relax (2)}{2}-\frac {x^{3} \ln \relax (2)}{2}+\frac {x^{2} \ln \relax (2)}{2}+x^{8}-\frac {3 x^{7}}{2}+\frac {9 x^{6}}{16}-\frac {x^{5}}{2}+\frac {3 x^{4}}{8}+\frac {x^{2}}{16}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 74, normalized size = 2.64 \begin {gather*} x^{8} - \frac {3}{2} \, x^{7} + \frac {9}{16} \, x^{6} - \frac {1}{2} \, x^{5} + \frac {3}{8} \, x^{4} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} \log \relax (2)^{2} + \frac {1}{16} \, x^{2} + \frac {1}{2} \, {\left (4 \, x^{6} - 7 \, x^{5} + 3 \, x^{4} - x^{3} + x^{2}\right )} \log \relax (2) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 74, normalized size = 2.64 \begin {gather*} x^8-\frac {3\,x^7}{2}+\left (2\,\ln \relax (2)+\frac {9}{16}\right )\,x^6+\left (-\frac {7\,\ln \relax (2)}{2}-\frac {1}{2}\right )\,x^5+\left (\frac {3\,\ln \relax (2)}{2}+{\ln \relax (2)}^2+\frac {3}{8}\right )\,x^4+\left (-\frac {\ln \relax (2)}{2}-2\,{\ln \relax (2)}^2\right )\,x^3+\left (\frac {\ln \left (256\right )}{16}+{\ln \relax (2)}^2+\frac {1}{16}\right )\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.07, size = 88, normalized size = 3.14 \begin {gather*} x^{8} - \frac {3 x^{7}}{2} + x^{6} \left (\frac {9}{16} + 2 \log {\relax (2 )}\right ) + x^{5} \left (- \frac {7 \log {\relax (2 )}}{2} - \frac {1}{2}\right ) + x^{4} \left (\frac {3}{8} + \log {\relax (2 )}^{2} + \frac {3 \log {\relax (2 )}}{2}\right ) + x^{3} \left (- 2 \log {\relax (2 )}^{2} - \frac {\log {\relax (2 )}}{2}\right ) + x^{2} \left (\frac {1}{16} + \frac {\log {\relax (2 )}}{2} + \log {\relax (2 )}^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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