∫ ex(36x−60x2+42x3−24x4+6x5+(−12+48x2−48x3+12x4) log(2))_______________________________________________ x4+(−4x2+4x3) log(2)+(4−8x+4x2) log2(2) dx

Optimal. Leaf size=30 \[ \frac {3 e^x x \left (-4+\frac {3}{x}+x\right )}{\frac {x^2}{-2+2 x}+\log (2)} \]

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Rubi [C] time = 6.51, antiderivative size = 1114, normalized size of antiderivative = 37.13, number of steps used = 30, number of rules used = 9, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6741, 12, 6688, 6742, 2176, 2194, 6728, 2178, 2177}

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 e^x (1-x) \left (-x^4+x^3 (3-\log (4))+x (6-\log (4))-\log (4)-2 x^2 (2-\log (8))\right )}{x^4+4 x^3 \log (2)-4 x^2 (1-\log (2)) \log (2)+4 \log ^2(2)-8 x \log ^2(2)} \, dx\\ &=6 \int \frac {e^x (1-x) \left (-x^4+x^3 (3-\log (4))+x (6-\log (4))-\log (4)-2 x^2 (2-\log (8))\right )}{x^4+4 x^3 \log (2)-4 x^2 (1-\log (2)) \log (2)+4 \log ^2(2)-8 x \log ^2(2)} \, dx\\ &=6 \int \frac {e^x (1-x) \left (-x^4+x^3 (3-\log (4))+x (6-\log (4))-\log (4)-2 x^2 (2-\log (8))\right )}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx\\ &=6 \int \left (e^x x-4 e^x \left (1+\frac {\log (2)}{2}\right )+\frac {e^x \left (-10-14 \log (4)-5 \log ^2(4)+\log (8) \log (16)+\log (64)+x \left (7+\log ^2(4)+\log (4096)\right )\right )}{x^2-\log (4)+x \log (4)}+\frac {e^x \left (x \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )-\log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right )}{\left (x^2-\log (4)+x \log (4)\right )^2}\right ) \, dx\\ &=6 \int e^x x \, dx+6 \int \frac {e^x \left (-10-14 \log (4)-5 \log ^2(4)+\log (8) \log (16)+\log (64)+x \left (7+\log ^2(4)+\log (4096)\right )\right )}{x^2-\log (4)+x \log (4)} \, dx+6 \int \frac {e^x \left (x \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )-\log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right )}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx-(12 (2+\log (2))) \int e^x \, dx\\ &=6 e^x x-12 e^x (2+\log (2))-6 \int e^x \, dx+6 \int \left (\frac {e^x \left (7+\log ^2(4)+\log (4096)-\frac {-20-35 \log (4)-10 \log ^2(4)-\log ^3(4)+2 \log (8) \log (16)+2 \log (64)-\log (4) \log (4096)}{\sqrt {\log (4) (4+\log (4))}}\right )}{2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}}+\frac {e^x \left (7+\log ^2(4)+\log (4096)+\frac {-20-35 \log (4)-10 \log ^2(4)-\log ^3(4)+2 \log (8) \log (16)+2 \log (64)-\log (4) \log (4096)}{\sqrt {\log (4) (4+\log (4))}}\right )}{2 x+\log (4)-\sqrt {\log (4) (4+\log (4))}}\right ) \, dx+6 \int \left (\frac {e^x x \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )}{\left (x^2-\log (4)+x \log (4)\right )^2}-\frac {e^x \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )}{\left (x^2-\log (4)+x \log (4)\right )^2}\right ) \, dx\\ &=-6 e^x+6 e^x x-12 e^x (2+\log (2))+\left (6 \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \frac {e^x x}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx+\left (6 \left (7+\log ^2(4)+\log (4096)-\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )\right ) \int \frac {e^x}{2 x+\log (4)-\sqrt {\log (4) (4+\log (4))}} \, dx+\left (6 \left (7+\log ^2(4)+\log (4096)+\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )\right ) \int \frac {e^x}{2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}} \, dx-\left (6 \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \frac {e^x}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx\\ &=-6 e^x+6 e^x x-12 e^x (2+\log (2))+\frac {3}{2} e^{\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \text {Ei}\left (\frac {1}{2} \left (2 x+\log (4)-\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (7+\log ^2(4)+\log (4096)-\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )+\frac {3}{2} e^{-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \text {Ei}\left (\frac {1}{2} \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (7+\log ^2(4)+\log (4096)+\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )+\left (6 \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \left (\frac {2 e^x \left (-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}{\log (4) (4+\log (4)) \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}-\frac {2 e^x}{\sqrt {\log (4)} (4+\log (4))^{3/2} \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}+\frac {2 e^x \left (-\log (4)-\sqrt {\log (4) (4+\log (4))}\right )}{\log (4) (4+\log (4)) \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}-\frac {2 e^x}{\sqrt {\log (4)} (4+\log (4))^{3/2} \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}\right ) \, dx-\left (6 \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \left (\frac {4 e^x}{\log (4) (4+\log (4)) \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}+\frac {4 e^x}{(\log (4) (4+\log (4)))^{3/2} \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}+\frac {4 e^x}{\log (4) (4+\log (4)) \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}+\frac {4 e^x}{(\log (4) (4+\log (4)))^{3/2} \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C] time = 4.86, size = 629, normalized size = 20.97 \begin {gather*} \frac {3 e^{-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \left (4 e^{x+\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \left (-3 \log ^{\frac {7}{2}}(4) \sqrt {4+\log (4)}-12 \sqrt {\log (4) (4+\log (4))}-5 x^2 \left (\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))}\right )+x^3 \left (\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))}\right )-x \left ((-13+24 \log (2)) \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}-3 \log ^{\frac {7}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))} (-7+\log (8))+\log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)} (-12+\log (64))\right )+\log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)} (-9+\log (64))+3 \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (-1+\log (64))\right )+3 e^{\sqrt {\log (4) (4+\log (4))}} \text {Ei}\left (x+\log (2)-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}\right ) \left (x^2-\log (4)+x \log (4)\right ) \left (5 \log ^3(4)-3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)+\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))-\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )-3 \text {Ei}\left (x+\frac {1}{2} \left (\log (4)+\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (x^2 \left (5 \log ^3(4)+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)-\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))+\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )+x \log (4) \left (5 \log ^3(4)+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)-\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))+\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )+\log ^2(4) \left (-5 \log ^2(4)-3 \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+\log (64) \sqrt {\log ^2(4)+\log (256)}+\log (4) \log (1024)\right )\right )\right )}{2 \sqrt {\log (4)} (4+\log (4))^{3/2} \left (x^2-\log (4)+x \log (4)\right )} \end {gather*}

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fricas [A] time = 0.56, size = 30, normalized size = 1.00 \begin {gather*} \frac {6 \, {\left (x^{3} - 5 \, x^{2} + 7 \, x - 3\right )} e^{x}}{x^{2} + 2 \, {\left (x - 1\right )} \log \relax (2)} \end {gather*}

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giac [A] time = 0.20, size = 40, normalized size = 1.33 \begin {gather*} \frac {6 \, {\left (x^{3} e^{x} - 5 \, x^{2} e^{x} + 7 \, x e^{x} - 3 \, e^{x}\right )}}{x^{2} + 2 \, x \log \relax (2) - 2 \, \log \relax (2)} \end {gather*}

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method	result	size

gosper	\(\frac {6 \left (x^{3}-5 x^{2}+7 x -3\right ) {\mathrm e}^{x}}{2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)}\)	\(33\)
risch	\(\frac {6 \left (x^{3}-5 x^{2}+7 x -3\right ) {\mathrm e}^{x}}{2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)}\)	\(33\)
norman	\(\frac {42 \,{\mathrm e}^{x} x -30 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x^{3}-18 \,{\mathrm e}^{x}}{2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)}\)	\(41\)
default	\(-24 \,{\mathrm e}^{x}+\frac {48 \,{\mathrm e}^{x} \ln \relax (2) \left (2 x \ln \relax (2)^{2}-2 \ln \relax (2)^{2}+4 x \ln \relax (2)-3 \ln \relax (2)+x \right )}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+12 \,{\mathrm e}^{x} \ln \relax (2)+\frac {48 \ln \relax (2)^{4} {\mathrm e}^{x}}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+\frac {168 \ln \relax (2)^{3} {\mathrm e}^{x}}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+\frac {144 \ln \relax (2)^{2} {\mathrm e}^{x}}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+\frac {6 \ln \relax (2) {\mathrm e}^{x}}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+\frac {6 \,{\mathrm e}^{x} x}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+\frac {60 \,{\mathrm e}^{x} \left (x \ln \relax (2)-\ln \relax (2)+x \right )}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+\frac {18 \,{\mathrm e}^{x} \left (x -2\right )}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}-\frac {48 \,{\mathrm e}^{x} \ln \relax (2)^{4} x}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}-6 \,{\mathrm e}^{x} \left (-x +1+4 \ln \relax (2)\right )+\frac {42 \,{\mathrm e}^{x} \ln \relax (2) \left (2 x \ln \relax (2)-2 \ln \relax (2)+3 x -2\right )}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}-\frac {216 \,{\mathrm e}^{x} \ln \relax (2)^{2} x}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}+\frac {12 \,{\mathrm e}^{x} \ln \relax (2)^{2} \left (4 x \ln \relax (2)^{2}-4 \ln \relax (2)^{2}+10 x \ln \relax (2)-8 \ln \relax (2)+5 x -2\right )}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}-\frac {48 \,{\mathrm e}^{x} \ln \relax (2) x}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}-\frac {192 \,{\mathrm e}^{x} \ln \relax (2)^{3} x}{\left (\ln \relax (2)+2\right ) \left (2 x \ln \relax (2)+x^{2}-2 \ln \relax (2)\right )}\)	\(494\)

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maxima [A] time = 0.49, size = 32, normalized size = 1.07 \begin {gather*} \frac {6 \, {\left (x^{3} - 5 \, x^{2} + 7 \, x - 3\right )} e^{x}}{x^{2} + 2 \, x \log \relax (2) - 2 \, \log \relax (2)} \end {gather*}

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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (36\,x+\ln \relax (2)\,\left (12\,x^4-48\,x^3+48\,x^2-12\right )-60\,x^2+42\,x^3-24\,x^4+6\,x^5\right )}{{\ln \relax (2)}^2\,\left (4\,x^2-8\,x+4\right )-\ln \relax (2)\,\left (4\,x^2-4\,x^3\right )+x^4} \,d x \end {gather*}

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sympy [A] time = 0.19, size = 32, normalized size = 1.07 \begin {gather*} \frac {\left (6 x^{3} - 30 x^{2} + 42 x - 18\right ) e^{x}}{x^{2} + 2 x \log {\relax (2 )} - 2 \log {\relax (2 )}} \end {gather*}

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3.59.90 \(\int \frac {e^x (36 x-60 x^2+42 x^3-24 x^4+6 x^5+(-12+48 x^2-48 x^3+12 x^4) \log (2))}{x^4+(-4 x^2+4 x^3) \log (2)+(4-8 x+4 x^2) \log ^2(2)} \, dx\)