∫ −640x2+210x4−10x6+(160x2−20x4) log(2)−10x2 log2(2)+(−5120x−1280x2+80x5+20x6+(1280x+320x2) log(2)+(−80x−20x2) log2(2)) log(4+x)________________________________ 16384+4096x−10752x2−2688x3+2276x4+569x5−168x6−42x7+4x8+x9+(−8192−2048x+3712x2+928x3−464x4−116x5+16x6+4x7) log(2)+(1536+384x−424x2−106x3+24x4+6x5) log2(2)+(−128−32x+16x2+4x3) log3(2)+(4+x) log4(2) dx

Optimal. Leaf size=27 \[ \frac {10 \log (4+x)}{5-\left (x-\frac {8-\log (2)}{x}\right )^2} \]

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Rubi [C] time = 15.34, antiderivative size = 4561, normalized size of antiderivative = 168.93, number of steps used = 98, number of rules used = 23, integrand size = 231, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 6688, 12, 6742, 6728, 1460, 1166, 207, 1247, 634, 618, 206, 628, 2418, 260, 2416, 2394, 2393, 2391, 2413, 706, 31, 635}

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {210 x^4-10 x^6+\left (160 x^2-20 x^4\right ) \log (2)+x^2 \left (-640-10 \log ^2(2)\right )+\left (-5120 x-1280 x^2+80 x^5+20 x^6+\left (1280 x+320 x^2\right ) \log (2)+\left (-80 x-20 x^2\right ) \log ^2(2)\right ) \log (4+x)}{16384+4096 x-10752 x^2-2688 x^3+2276 x^4+569 x^5-168 x^6-42 x^7+4 x^8+x^9+\left (-8192-2048 x+3712 x^2+928 x^3-464 x^4-116 x^5+16 x^6+4 x^7\right ) \log (2)+\left (1536+384 x-424 x^2-106 x^3+24 x^4+6 x^5\right ) \log ^2(2)+\left (-128-32 x+16 x^2+4 x^3\right ) \log ^3(2)+(4+x) \log ^4(2)} \, dx\\ &=\int \frac {10 x \left (-x \left (x^4+(-8+\log (2))^2+x^2 (-21+\log (4))\right )+2 (4+x) \left (x^4-(-8+\log (2))^2\right ) \log (4+x)\right )}{(4+x) \left (x^4+(-8+\log (2))^2+x^2 (-21+\log (4))\right )^2} \, dx\\ &=10 \int \frac {x \left (-x \left (x^4+(-8+\log (2))^2+x^2 (-21+\log (4))\right )+2 (4+x) \left (x^4-(-8+\log (2))^2\right ) \log (4+x)\right )}{(4+x) \left (x^4+(-8+\log (2))^2+x^2 (-21+\log (4))\right )^2} \, dx\\ &=10 \int \left (\frac {x^2}{(-4-x) \left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )}+\frac {2 x \left (8-x^2-\log (2)\right ) \left (-8-x^2+\log (2)\right ) \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2}\right ) \, dx\\ &=10 \int \frac {x^2}{(-4-x) \left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )} \, dx+20 \int \frac {x \left (8-x^2-\log (2)\right ) \left (-8-x^2+\log (2)\right ) \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2} \, dx\\ &=10 \int \left (\frac {(4-x) \left (-64+16 x^2+16 \log (2)-\log ^2(2)\right )}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right ) \left (16-\log ^2(2)-\log (65536)\right )}-\frac {16}{(4+x) \left (-16+\log ^2(2)+\log (65536)\right )}\right ) \, dx+20 \int \left (\frac {x \log (4+x)}{x^4+(-8+\log (2))^2-x^2 (21-\log (4))}+\frac {x \left (-2 (8-\log (2))^2+x^2 (21-\log (4))\right ) \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2}\right ) \, dx\\ &=\frac {160 \log (4+x)}{16-\log ^2(2)-\log (65536)}+20 \int \frac {x \log (4+x)}{x^4+(-8+\log (2))^2-x^2 (21-\log (4))} \, dx+20 \int \frac {x \left (-2 (8-\log (2))^2+x^2 (21-\log (4))\right ) \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2} \, dx+\frac {10 \int \frac {(4-x) \left (-64+16 x^2+16 \log (2)-\log ^2(2)\right )}{x^4+(-8+\log (2))^2+x^2 (-21+\log (4))} \, dx}{16-\log ^2(2)-\log (65536)}\\ &=\frac {160 \log (4+x)}{16-\log ^2(2)-\log (65536)}+20 \int \left (-\frac {2 x (8-\log (2))^2 \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2}+\frac {x^3 (21-\log (4)) \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2}\right ) \, dx+20 \int \left (-\frac {2 x \log (4+x)}{\sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)} \left (21-2 x^2-\log (4)+\sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)}\right )}-\frac {2 x \log (4+x)}{\sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)} \left (-21+2 x^2+\log (4)+\sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)}\right )}\right ) \, dx-\frac {10 \int \frac {x \left (-64+16 x^2+16 \log (2)-\log ^2(2)\right )}{x^4+(-8+\log (2))^2+x^2 (-21+\log (4))} \, dx}{16-\log ^2(2)-\log (65536)}+\frac {40 \int \frac {-64+16 x^2+16 \log (2)-\log ^2(2)}{x^4+(-8+\log (2))^2+x^2 (-21+\log (4))} \, dx}{16-\log ^2(2)-\log (65536)}\\ &=\frac {160 \log (4+x)}{16-\log ^2(2)-\log (65536)}-\left (40 (8-\log (2))^2\right ) \int \frac {x \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2} \, dx+(20 (21-\log (4))) \int \frac {x^3 \log (4+x)}{\left (x^4+(-8+\log (2))^2-x^2 (21-\log (4))\right )^2} \, dx-\frac {5 \operatorname {Subst}\left (\int \frac {-64+16 x+16 \log (2)-\log ^2(2)}{x^2+(-8+\log (2))^2+x (-21+\log (4))} \, dx,x,x^2\right )}{16-\log ^2(2)-\log (65536)}+\frac {\left (40 \left (8-\frac {104-\log ^2(2)}{\sqrt {185-4 \log ^2(2)+\log ^2(4)-\log (1048576)}}\right )\right ) \int \frac {1}{x^2+\frac {1}{2} (-21+\log (4))+\frac {1}{2} \sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)}} \, dx}{16-\log ^2(2)-\log (65536)}+\frac {\left (40 \left (8+\frac {104-\log ^2(2)}{\sqrt {185-4 \log ^2(2)+\log ^2(4)-\log (1048576)}}\right )\right ) \int \frac {1}{x^2+\frac {1}{2} (-21+\log (4))-\frac {1}{2} \sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)}} \, dx}{16-\log ^2(2)-\log (65536)}-\frac {40 \int \frac {x \log (4+x)}{21-2 x^2-\log (4)+\sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)}} \, dx}{\sqrt {185-4 \log ^2(2)+\log ^2(4)-\log (1048576)}}-\frac {40 \int \frac {x \log (4+x)}{-21+2 x^2+\log (4)+\sqrt {185+64 \log (2)-4 \log ^2(2)-42 \log (4)+\log ^2(4)}} \, dx}{\sqrt {185-4 \log ^2(2)+\log ^2(4)-\log (1048576)}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 1.41, size = 29, normalized size = 1.07 \begin {gather*} -\frac {10 x^2 \log (4+x)}{x^4+(-8+\log (2))^2+x^2 (-21+\log (4))} \end {gather*}

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fricas [A] time = 0.69, size = 34, normalized size = 1.26 \begin {gather*} -\frac {10 \, x^{2} \log \left (x + 4\right )}{x^{4} - 21 \, x^{2} + 2 \, {\left (x^{2} - 8\right )} \log \relax (2) + \log \relax (2)^{2} + 64} \end {gather*}

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giac [A] time = 0.53, size = 36, normalized size = 1.33 \begin {gather*} -\frac {10 \, x^{2} \log \left (x + 4\right )}{x^{4} + 2 \, x^{2} \log \relax (2) - 21 \, x^{2} + \log \relax (2)^{2} - 16 \, \log \relax (2) + 64} \end {gather*}

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method	result	size

norman	\(-\frac {10 x^{2} \ln \left (4+x \right )}{x^{4}+2 x^{2} \ln \relax (2)+\ln \relax (2)^{2}-21 x^{2}-16 \ln \relax (2)+64}\)	\(37\)
risch	\(-\frac {10 x^{2} \ln \left (4+x \right )}{x^{4}+2 x^{2} \ln \relax (2)+\ln \relax (2)^{2}-21 x^{2}-16 \ln \relax (2)+64}\)	\(37\)
derivativedivides	\(-\frac {10 \ln \left (4+x \right ) \left (4+x \right ) \left (\ln \relax (2)^{2} \left (4+x \right )-16 \left (4+x \right )^{3}-8 \ln \relax (2)^{2}-16 \left (4+x \right ) \ln \relax (2)+256 \left (4+x \right )^{2}+128 \ln \relax (2)-3328-1216 x \right )}{\left (\ln \relax (2)^{2}+16 \ln \relax (2)-16\right ) \left (\left (4+x \right )^{4}+2 \ln \relax (2) \left (4+x \right )^{2}-16 \left (4+x \right )^{3}+\ln \relax (2)^{2}-16 \left (4+x \right ) \ln \relax (2)+75 \left (4+x \right )^{2}+16 \ln \relax (2)-368-88 x \right )}-\frac {160 \ln \left (4+x \right )}{\ln \relax (2)^{2}+16 \ln \relax (2)-16}\)	\(135\)
default	\(-\frac {10 \ln \left (4+x \right ) \left (4+x \right ) \left (\ln \relax (2)^{2} \left (4+x \right )-16 \left (4+x \right )^{3}-8 \ln \relax (2)^{2}-16 \left (4+x \right ) \ln \relax (2)+256 \left (4+x \right )^{2}+128 \ln \relax (2)-3328-1216 x \right )}{\left (\ln \relax (2)^{2}+16 \ln \relax (2)-16\right ) \left (\left (4+x \right )^{4}+2 \ln \relax (2) \left (4+x \right )^{2}-16 \left (4+x \right )^{3}+\ln \relax (2)^{2}-16 \left (4+x \right ) \ln \relax (2)+75 \left (4+x \right )^{2}+16 \ln \relax (2)-368-88 x \right )}-\frac {160 \ln \left (4+x \right )}{\ln \relax (2)^{2}+16 \ln \relax (2)-16}\)	\(135\)

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maxima [A] time = 0.47, size = 34, normalized size = 1.26 \begin {gather*} -\frac {10 \, x^{2} \log \left (x + 4\right )}{x^{4} + x^{2} {\left (2 \, \log \relax (2) - 21\right )} + \log \relax (2)^{2} - 16 \, \log \relax (2) + 64} \end {gather*}

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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

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sympy [B] time = 0.45, size = 39, normalized size = 1.44 \begin {gather*} - \frac {10 x^{2} \log {\left (x + 4 \right )}}{x^{4} - 21 x^{2} + 2 x^{2} \log {\relax (2 )} - 16 \log {\relax (2 )} + \log {\relax (2 )}^{2} + 64} \end {gather*}

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