3.48.0 ∫ −8x−2x2+(16+12x+2x2−48x3−36x4−6x5) log(2+x)+(8+6x+x2−24x3−18x4−3x5) log2(2+x)+((−16x3−8x4) log(2+x)+(−8x3−4x4) log2(2+x)+((16+8x) log(2+x)+(8+4x) log2(2+x)) log(2x+x log(2+x) log (2+x) )) log(−x3+log(2x+x log(2+x) log (2+x) )) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ (−64x3−64x4−20x5−2x6) log(2+x)+(−32x3−32x4−10x5−x6) log2(2+x)+((64+64x+20x2+2x3) log(2+x)+(32+32x+10x2+x3) log2(2+x)) log(2x+x log(2+x) log (2+x) ) dx [4721]

Integrand size = 290, antiderivative size = 26 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x \log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{4+x} \]

Rubi [F]

\[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx \]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x (4+x)+2 (2+x) \log (2+x) \left (-4-x+12 x^3+3 x^4+4 \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )\right )+(2+x) \log ^2(2+x) \left (-4-x+12 x^3+3 x^4+4 \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )\right )}{(2+x) (4+x)^2 \log (2+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx \\ & = \int \left (-\frac {8}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {2 x}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {24 x^3}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {6 x^4}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {2 x}{(2+x) (4+x) \log (2+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {4 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {x \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {12 x^3 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {3 x^4 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {4 \log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{(4+x)^2}\right ) \, dx \\ & = -\left (2 \int \frac {x}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx\right )+2 \int \frac {x}{(2+x) (4+x) \log (2+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+3 \int \frac {x^4 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-4 \int \frac {\log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+4 \int \frac {\log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{(4+x)^2} \, dx+6 \int \frac {x^4}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-8 \int \frac {1}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+12 \int \frac {x^3 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+24 \int \frac {x^3}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-\int \frac {x \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx \\ & = -\left (2 \int \left (-\frac {4}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {1}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}\right ) \, dx\right )+2 \int \left (-\frac {1}{(2+x) \log (2+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {2}{(4+x) \log (2+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}\right ) \, dx+3 \int \left (\frac {48 \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {8 x \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {x^2 \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {256 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {256 \log (2+x)}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}\right ) \, dx-4 \int \frac {\log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+4 \int \frac {\log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{(4+x)^2} \, dx+6 \int \left (\frac {48}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {8 x}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {x^2}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {256}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {256}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}\right ) \, dx-8 \int \frac {1}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+12 \int \left (-\frac {8 \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {x \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {64 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {48 \log (2+x)}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}\right ) \, dx+24 \int \left (-\frac {8}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {x}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}-\frac {64}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {48}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}\right ) \, dx-\int \left (-\frac {4 \log (2+x)}{(4+x)^2 (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}+\frac {\log (2+x)}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx\right )-2 \int \frac {1}{(2+x) \log (2+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+3 \int \frac {x^2 \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+4 \int \frac {1}{(4+x) \log (2+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+4 \int \frac {\log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{(4+x)^2} \, dx+6 \int \frac {x^2}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+12 \int \frac {x \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+24 \int \frac {x}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-24 \int \frac {x \log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-48 \int \frac {x}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-96 \int \frac {\log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+144 \int \frac {\log (2+x)}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-192 \int \frac {1}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+288 \int \frac {1}{(2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+576 \int \frac {\log (2+x)}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-768 \int \frac {\log (2+x)}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx+1152 \int \frac {1}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-1536 \int \frac {1}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx-\int \frac {\log (2+x)}{(4+x) (2+\log (2+x)) \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\log \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )-\frac {4 \log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{4+x} \]

Maple [C] (warning: unable to verify)

Time = 0.03 (sec) , antiderivative size = 470, normalized size of antiderivative = 18.08

\[-\frac {4 \ln \left (\ln \left (x \right )-\ln \left (\ln \left (2+x \right )\right )+\ln \left (\ln \left (2+x \right )+2\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (2+x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (i \left (\ln \left (2+x \right )+2\right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )\right )}{2}-x^{3}\right )}{4+x}+\ln \left (\ln \left (\ln \left (2+x \right )+2\right )+\frac {i \left (2 i x^{3}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (2+x \right )+2\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\pi \,\operatorname {csgn}\left (i \left (\ln \left (2+x \right )+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{3}+\pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{3}-2 i \ln \left (x \right )+2 i \ln \left (\ln \left (2+x \right )\right )\right )}{2}\right )\]

Fricas [A] (veriﬁcation not implemented)

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x \log \left (-x^{3} + \log \left (\frac {x \log \left (x + 2\right ) + 2 \, x}{\log \left (x + 2\right )}\right )\right )}{x + 4} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).

Time = 19.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\log {\left (- x^{3} + \log {\left (\frac {x \log {\left (x + 2 \right )} + 2 x}{\log {\left (x + 2 \right )}} \right )} \right )} - \frac {4 \log {\left (- x^{3} + \log {\left (\frac {x \log {\left (x + 2 \right )} + 2 x}{\log {\left (x + 2 \right )}} \right )} \right )}}{x + 4} \]

Maxima [A] (veriﬁcation not implemented)

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x \log \left (-x^{3} + \log \left (x\right ) + \log \left (\log \left (x + 2\right ) + 2\right ) - \log \left (\log \left (x + 2\right )\right )\right )}{x + 4} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.72 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=-\frac {4 \, \log \left (-x^{3} + \log \left (x \log \left (x + 2\right ) + 2 \, x\right ) - \log \left (\log \left (x + 2\right )\right )\right )}{x + 4} + \log \left (x^{3} - \log \left (x \log \left (x + 2\right ) + 2 \, x\right ) + \log \left (\log \left (x + 2\right )\right )\right ) \]

Mupad [B] (veriﬁcation not implemented)

Time = 11.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x\,\ln \left (\ln \left (\frac {2\,x+x\,\ln \left (x+2\right )}{\ln \left (x+2\right )}\right )-x^3\right )}{x+4} \]

Optimal result