Optimal. Leaf size=25 \[ \frac {4}{-4+x+\frac {5}{x+\frac {2}{\log (2)}-\log (-x)}} \]
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Rubi [F] time = 4.81, 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 {-16 x-16 x^2 \log (2)+\left (-20+20 x-4 x^3\right ) \log ^2(2)+\left (16 x \log (2)+8 x^2 \log ^2(2)\right ) \log (-x)-4 x \log ^2(2) \log ^2(-x)}{64 x-32 x^2+4 x^3+\left (-80 x+84 x^2-32 x^3+4 x^4\right ) \log (2)+\left (25 x-40 x^2+26 x^3-8 x^4+x^5\right ) \log ^2(2)+\left (\left (-64 x+32 x^2-4 x^3\right ) \log (2)+\left (40 x-42 x^2+16 x^3-2 x^4\right ) \log ^2(2)\right ) \log (-x)+\left (16 x-8 x^2+x^3\right ) \log ^2(2) \log ^2(-x)} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-5 \log ^2(2)-x^3 \log ^2(2)-x \left (4-5 \log ^2(2)\right )-x^2 \log (16)+2 x \log (2) (2+x \log (2)) \log (-x)-x \log ^2(2) \log ^2(-x)\right )}{x \left (x (2-4 \log (2))-8 \left (1-\frac {5 \log (2)}{8}\right )+x^2 \log (2)+(-x \log (2)+\log (16)) \log (-x)\right )^2} \, dx\\ &=4 \int \frac {-5 \log ^2(2)-x^3 \log ^2(2)-x \left (4-5 \log ^2(2)\right )-x^2 \log (16)+2 x \log (2) (2+x \log (2)) \log (-x)-x \log ^2(2) \log ^2(-x)}{x \left (x (2-4 \log (2))-8 \left (1-\frac {5 \log (2)}{8}\right )+x^2 \log (2)+(-x \log (2)+\log (16)) \log (-x)\right )^2} \, dx\\ &=4 \int \left (-\frac {\log ^2(2)}{(x \log (2)-\log (16))^2}+\frac {-5 \log ^2(2) \log ^2(16)-x^3 \left (11 \log ^4(2)-8 \log ^3(2) \log (16)-\log (4) \log ^2(16)+\log ^2(2) \log (16) (8+\log (16))\right )+x^2 \left (35 \log ^4(2)+32 \log ^2(2) \log (16)-\log ^3(16)-4 \log ^3(2) (16+5 \log (16))\right )-x \left (25 \log ^4(2)-32 \log (2) \log (16)+4 \log ^2(16)-10 \log ^3(2) (8+\log (16))+\log ^2(2) \left (64+20 \log (16)-5 \log ^2(16)\right )\right )}{x (x \log (2)-\log (16))^2 \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2}+\frac {2 \log ^2(2) \log (32)}{(x \log (2)-\log (16))^2 \left (-8 \left (1-\frac {5 \log (2)}{8}\right )+x^2 \log (2)+2 x (1-\log (4))-x \log (2) \log (-x)+\log (16) \log (-x)\right )}\right ) \, dx\\ &=\frac {4 \log (2)}{x \log (2)-\log (16)}+4 \int \frac {-5 \log ^2(2) \log ^2(16)-x^3 \left (11 \log ^4(2)-8 \log ^3(2) \log (16)-\log (4) \log ^2(16)+\log ^2(2) \log (16) (8+\log (16))\right )+x^2 \left (35 \log ^4(2)+32 \log ^2(2) \log (16)-\log ^3(16)-4 \log ^3(2) (16+5 \log (16))\right )-x \left (25 \log ^4(2)-32 \log (2) \log (16)+4 \log ^2(16)-10 \log ^3(2) (8+\log (16))+\log ^2(2) \left (64+20 \log (16)-5 \log ^2(16)\right )\right )}{x (x \log (2)-\log (16))^2 \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2} \, dx+\left (8 \log ^2(2) \log (32)\right ) \int \frac {1}{(x \log (2)-\log (16))^2 \left (-8 \left (1-\frac {5 \log (2)}{8}\right )+x^2 \log (2)+2 x (1-\log (4))-x \log (2) \log (-x)+\log (16) \log (-x)\right )} \, dx\\ &=\frac {4 \log (2)}{x \log (2)-\log (16)}+4 \int \left (-\frac {5 \log ^2(2)}{x \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2}+\frac {-11 \log ^4(2)+8 \log ^3(2) \log (16)+\log (4) \log ^2(16)-\log ^2(2) \log (16) (8+\log (16))}{\log ^2(2) \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2}+\frac {40 \log ^5(2)-\log (2) \log ^3(16)+\log ^4(16)+16 \log ^3(2) \log (16) (2+\log (16))-2 \log ^2(2) \log ^2(16) (8+\log (16))-2 \log ^4(2) (32+21 \log (16))}{\log ^2(2) (x \log (2)-\log (16)) \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2}+\frac {-25 \log ^6(2)-\log (2) \log ^4(16)+\log (4) \log ^4(16)+40 \log ^5(2) (2+\log (16))+8 \log ^3(2) \log (16) (2+\log (16))^2-\log ^2(2) \log ^2(16) \left (4+8 \log (16)+\log ^2(16)\right )-2 \log ^4(2) \left (32+42 \log (16)+13 \log ^2(16)\right )}{\log ^2(2) (x \log (2)-\log (16))^2 \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2}\right ) \, dx+\left (8 \log ^2(2) \log (32)\right ) \int \frac {1}{(x \log (2)-\log (16))^2 \left (-8 \left (1-\frac {5 \log (2)}{8}\right )+x^2 \log (2)+2 x (1-\log (4))-x \log (2) \log (-x)+\log (16) \log (-x)\right )} \, dx\\ &=\frac {4 \log (2)}{x \log (2)-\log (16)}-\left (20 \log ^2(2)\right ) \int \frac {1}{x \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2} \, dx+\frac {\left (4 \left (-11 \log ^4(2)+8 \log ^3(2) \log (16)+\log (4) \log ^2(16)-\log ^2(2) \log (16) (8+\log (16))\right )\right ) \int \frac {1}{\left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2} \, dx}{\log ^2(2)}+\frac {\left (4 \left (40 \log ^5(2)-\log (2) \log ^3(16)+\log ^4(16)+16 \log ^3(2) \log (16) (2+\log (16))-2 \log ^2(2) \log ^2(16) (8+\log (16))-2 \log ^4(2) (32+21 \log (16))\right )\right ) \int \frac {1}{(x \log (2)-\log (16)) \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2} \, dx}{\log ^2(2)}-\frac {\left (4 \left (25 \log ^6(2)+\log (2) \log ^4(16)-\log (4) \log ^4(16)-40 \log ^5(2) (2+\log (16))-8 \log ^3(2) \log (16) (2+\log (16))^2+\log ^2(2) \log ^2(16) \left (4+8 \log (16)+\log ^2(16)\right )+\log ^4(2) \left (64+84 \log (16)+26 \log ^2(16)\right )\right )\right ) \int \frac {1}{(x \log (2)-\log (16))^2 \left (8 \left (1-\frac {5 \log (2)}{8}\right )-x^2 \log (2)-2 x (1-\log (4))+x \log (2) \log (-x)-\log (16) \log (-x)\right )^2} \, dx}{\log ^2(2)}+\left (8 \log ^2(2) \log (32)\right ) \int \frac {1}{(x \log (2)-\log (16))^2 \left (-8 \left (1-\frac {5 \log (2)}{8}\right )+x^2 \log (2)+2 x (1-\log (4))-x \log (2) \log (-x)+\log (16) \log (-x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.23, size = 228, normalized size = 9.12 \begin {gather*} -\frac {4 \left (-x^4 \log ^3(2)+x^3 \log (2) \left (5 \log ^2(2)-\log (4)+\log (2) \log (16)\right )+x^2 \left (-15 \log ^3(2)-\log ^2(2) (-2+\log (16))+\log ^2(16)\right )-\log (2) \log (16) \log (32)+\log (16) \log (256)+\log ^2(2) \log (1048576)+x \log (2) \left (-5 \log ^2(2)+\log (4)+\log (2) (-40+\log (2097152))\right )+\left (x^3 \log ^3(2)-x \log (4) \log (16)-\log (2) \log ^2(16)-x^2 \log ^2(2) \log (512)+x \log ^2(2) (8+\log (524288))\right ) \log (-x)\right )}{\left (x^3 \log ^2(2)-\log ^2(16)-x^2 \left (\log ^2(2)+\log (4) \log (16)\right )+x (-\log (2) \log (32)+\log (16) \log (64))\right ) \left (-8+x (2-4 \log (2))+x^2 \log (2)+\log (32)+(-x \log (2)+\log (16)) \log (-x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 45, normalized size = 1.80 \begin {gather*} -\frac {4 \, {\left (x \log \relax (2) - \log \relax (2) \log \left (-x\right ) + 2\right )}}{{\left (x - 4\right )} \log \relax (2) \log \left (-x\right ) - {\left (x^{2} - 4 \, x + 5\right )} \log \relax (2) - 2 \, x + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 74, normalized size = 2.96 \begin {gather*} -\frac {20 \, \log \relax (2)}{x^{3} \log \relax (2) - x^{2} \log \relax (2) \log \left (-x\right ) - 8 \, x^{2} \log \relax (2) + 8 \, x \log \relax (2) \log \left (-x\right ) + 2 \, x^{2} + 21 \, x \log \relax (2) - 16 \, \log \relax (2) \log \left (-x\right ) - 16 \, x - 20 \, \log \relax (2) + 32} + \frac {4}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 56, normalized size = 2.24
method | result | size |
norman | \(\frac {4 x \ln \relax (2)-4 \ln \relax (2) \ln \left (-x \right )+8}{x^{2} \ln \relax (2)-\ln \left (-x \right ) \ln \relax (2) x -4 x \ln \relax (2)+4 \ln \relax (2) \ln \left (-x \right )+5 \ln \relax (2)+2 x -8}\) | \(56\) |
risch | \(\frac {4}{x -4}-\frac {20 \ln \relax (2)}{\left (x -4\right ) \left (x^{2} \ln \relax (2)-\ln \left (-x \right ) \ln \relax (2) x -4 x \ln \relax (2)+4 \ln \relax (2) \ln \left (-x \right )+5 \ln \relax (2)+2 x -8\right )}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 54, normalized size = 2.16 \begin {gather*} \frac {4 \, {\left (x \log \relax (2) - \log \relax (2) \log \left (-x\right ) + 2\right )}}{x^{2} \log \relax (2) - 2 \, x {\left (2 \, \log \relax (2) - 1\right )} - {\left (x \log \relax (2) - 4 \, \log \relax (2)\right )} \log \left (-x\right ) + 5 \, \log \relax (2) - 8} \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 {16\,x-\ln \left (-x\right )\,\left (8\,{\ln \relax (2)}^2\,x^2+16\,\ln \relax (2)\,x\right )+{\ln \relax (2)}^2\,\left (4\,x^3-20\,x+20\right )+16\,x^2\,\ln \relax (2)+4\,x\,{\ln \left (-x\right )}^2\,{\ln \relax (2)}^2}{64\,x+\ln \left (-x\right )\,\left ({\ln \relax (2)}^2\,\left (-2\,x^4+16\,x^3-42\,x^2+40\,x\right )-\ln \relax (2)\,\left (4\,x^3-32\,x^2+64\,x\right )\right )-\ln \relax (2)\,\left (-4\,x^4+32\,x^3-84\,x^2+80\,x\right )-32\,x^2+4\,x^3+{\ln \relax (2)}^2\,\left (x^5-8\,x^4+26\,x^3-40\,x^2+25\,x\right )+{\ln \left (-x\right )}^2\,{\ln \relax (2)}^2\,\left (x^3-8\,x^2+16\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.40, size = 70, normalized size = 2.80 \begin {gather*} \frac {20 \log {\relax (2 )}}{- x^{3} \log {\relax (2 )} - 2 x^{2} + 8 x^{2} \log {\relax (2 )} - 21 x \log {\relax (2 )} + 16 x + \left (x^{2} \log {\relax (2 )} - 8 x \log {\relax (2 )} + 16 \log {\relax (2 )}\right ) \log {\left (- x \right )} - 32 + 20 \log {\relax (2 )}} + \frac {4}{x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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