\(\int \frac {(64+32 x+12 x^2+x^3) \log (x) \log (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x)+(32+16 x+2 x^2) \log ^2(\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x)}{16 x+8 x^2+x^3} \, dx\) [4903]

Optimal result

Integrand size = 93, antiderivative size = 30 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=2 \log (x) \log ^2\left (\frac {1}{3} e^{\frac {1}{4} (-2+x)-\frac {x}{4+x}} x\right ) \]

[Out]

2*ln(1/3*exp(1/4*x-1/2-x/(4+x))*x)^2*ln(x)

Rubi [F]

\[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx \]

[In]

Int[((64 + 32*x + 12*x^2 + x^3)*Log[x]*Log[(E^((-8 - 2*x + x^2)/(16 + 4*x))*x)/3] + (32 + 16*x + 2*x^2)*Log[(E
^((-8 - 2*x + x^2)/(16 + 4*x))*x)/3]^2)/(16*x + 8*x^2 + x^3),x]

[Out]

2*x + (3*x^2)/16 - 8/(4 + x) + 2*Log[x] - x*Log[x] - (x^2*Log[x])/8 - (32*Log[x])/(4 + x)^2 - (4*x*Log[x])/(4
+ x) + 4*Log[1 + x/4]*Log[x] - 2*Log[x]^2 - x*Log[x/(3*E^((8 + 2*x - x^2)/(4*(4 + x))))] + x*Log[x]*Log[x/(3*E
^((8 + 2*x - x^2)/(4*(4 + x))))] + (16*Log[x]*Log[x/(3*E^((8 + 2*x - x^2)/(4*(4 + x))))])/(4 + x) - 2*Log[4 +
x] + 4*PolyLog[2, -1/4*x] - 4*Defer[Int][Log[(E^((-8 - 2*x + x^2)/(4*(4 + x)))*x)/3]/x, x] + 4*Defer[Int][Log[
(E^((-8 - 2*x + x^2)/(4*(4 + x)))*x)/3]/(4 + x), x] + 4*Defer[Int][(Log[x]*Log[(E^((-8 - 2*x + x^2)/(4*(4 + x)
))*x)/3])/x, x] + 2*Defer[Int][Log[(E^((-8 - 2*x + x^2)/(4*(4 + x)))*x)/3]^2/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{x \left (16+8 x+x^2\right )} \, dx \\ & = \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{x (4+x)^2} \, dx \\ & = \int \left (\frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x (4+x)^2}+\frac {2 \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x}\right ) \, dx \\ & = 2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+\int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x (4+x)^2} \, dx \\ & = 2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+\int \left (\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )+\frac {4 \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x}-\frac {16 \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{(4+x)^2}\right ) \, dx \\ & = 2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx-16 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{(4+x)^2} \, dx+\int \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right ) \, dx \\ & = x \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )}{4+x}+2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+16 \int \frac {\left (-64-32 x-12 x^2-x^3\right ) \log (x)}{4 x (4+x)^3} \, dx-16 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x (4+x)} \, dx-\int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x)}{4 (4+x)^2} \, dx-\int \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right ) \, dx \\ & = -x \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+x \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )}{4+x}-\frac {1}{4} \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x)}{(4+x)^2} \, dx+2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\left (-64-32 x-12 x^2-x^3\right ) \log (x)}{x (4+x)^3} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx-16 \int \left (\frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{4 x}-\frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{4 (4+x)}\right ) \, dx+\int \frac {64+32 x+12 x^2+x^3}{4 (4+x)^2} \, dx \\ & = -x \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+x \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )}{4+x}+\frac {1}{4} \int \frac {64+32 x+12 x^2+x^3}{(4+x)^2} \, dx-\frac {1}{4} \int \left (4 \log (x)+x \log (x)+\frac {64 \log (x)}{(4+x)^2}-\frac {16 \log (x)}{4+x}\right ) \, dx+2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \left (-\frac {\log (x)}{x}+\frac {16 \log (x)}{(4+x)^3}\right ) \, dx-4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{4+x} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx \\ & = -x \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+x \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )}{4+x}+\frac {1}{4} \int \left (4+x+\frac {64}{(4+x)^2}-\frac {16}{4+x}\right ) \, dx-\frac {1}{4} \int x \log (x) \, dx+2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx-4 \int \frac {\log (x)}{x} \, dx+4 \int \frac {\log (x)}{4+x} \, dx-4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{4+x} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx-16 \int \frac {\log (x)}{(4+x)^2} \, dx+64 \int \frac {\log (x)}{(4+x)^3} \, dx-\int \log (x) \, dx \\ & = 2 x+\frac {3 x^2}{16}-\frac {16}{4+x}-x \log (x)-\frac {1}{8} x^2 \log (x)-\frac {32 \log (x)}{(4+x)^2}-\frac {4 x \log (x)}{4+x}+4 \log \left (1+\frac {x}{4}\right ) \log (x)-2 \log ^2(x)-x \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+x \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )}{4+x}-4 \log (4+x)+2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {1}{4+x} \, dx-4 \int \frac {\log \left (1+\frac {x}{4}\right )}{x} \, dx-4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{4+x} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+32 \int \frac {1}{x (4+x)^2} \, dx \\ & = 2 x+\frac {3 x^2}{16}-\frac {16}{4+x}-x \log (x)-\frac {1}{8} x^2 \log (x)-\frac {32 \log (x)}{(4+x)^2}-\frac {4 x \log (x)}{4+x}+4 \log \left (1+\frac {x}{4}\right ) \log (x)-2 \log ^2(x)-x \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+x \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )}{4+x}+4 \text {Li}_2\left (-\frac {x}{4}\right )+2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx-4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{4+x} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+32 \int \left (\frac {1}{16 x}-\frac {1}{4 (4+x)^2}-\frac {1}{16 (4+x)}\right ) \, dx \\ & = 2 x+\frac {3 x^2}{16}-\frac {8}{4+x}+2 \log (x)-x \log (x)-\frac {1}{8} x^2 \log (x)-\frac {32 \log (x)}{(4+x)^2}-\frac {4 x \log (x)}{4+x}+4 \log \left (1+\frac {x}{4}\right ) \log (x)-2 \log ^2(x)-x \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+x \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )+\frac {16 \log (x) \log \left (\frac {1}{3} e^{-\frac {8+2 x-x^2}{4 (4+x)}} x\right )}{4+x}-2 \log (4+x)+4 \text {Li}_2\left (-\frac {x}{4}\right )+2 \int \frac {\log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx-4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx+4 \int \frac {\log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{4+x} \, dx+4 \int \frac {\log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{4 (4+x)}} x\right )}{x} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(30)=60\).

Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\frac {18+\frac {9 x}{2}-\frac {9 x^2}{4}+9 (4+x) \log \left (\frac {1}{3} e^{\frac {1}{4} \left (-6+x+\frac {16}{4+x}\right )} x\right )+(4+x) \log (x) \left (-9+2 \log ^2\left (\frac {1}{3} e^{\frac {1}{4} \left (-6+x+\frac {16}{4+x}\right )} x\right )\right )}{4+x} \]

[In]

Integrate[((64 + 32*x + 12*x^2 + x^3)*Log[x]*Log[(E^((-8 - 2*x + x^2)/(16 + 4*x))*x)/3] + (32 + 16*x + 2*x^2)*
Log[(E^((-8 - 2*x + x^2)/(16 + 4*x))*x)/3]^2)/(16*x + 8*x^2 + x^3),x]

[Out]

(18 + (9*x)/2 - (9*x^2)/4 + 9*(4 + x)*Log[(E^((-6 + x + 16/(4 + x))/4)*x)/3] + (4 + x)*Log[x]*(-9 + 2*Log[(E^(
(-6 + x + 16/(4 + x))/4)*x)/3]^2))/(4 + x)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(511\) vs. \(2(24)=48\).

Time = 1.68 (sec) , antiderivative size = 512, normalized size of antiderivative = 17.07

method	result	size
parallelrisch	\(-\frac {-49152 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right ) x^{4}-393216 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right ) x^{3}-786432 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right ) x^{2}-4718592 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right )^{2} x -98304 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right )^{2} x^{3}-1179648 x^{2} \ln \left (x \right )^{2} \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )-6144 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{5}+196608 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{2} x^{3}-2097152 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3}+32768 x^{3} \ln \left (x \right )^{3}+6144 x^{5} \ln \left (x \right )+24576 x^{4} \ln \left (x \right )^{2}+24576 x^{4} \ln \left (x \right )+393216 x^{2} \ln \left (x \right )^{3}+196608 x^{3} \ln \left (x \right )^{2}+1572864 x \ln \left (x \right )^{3}+393216 x^{2} \ln \left (x \right )^{2}+2097152 \ln \left (x \right )^{3}+512 x^{6}-24576 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) x^{4}+393216 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{2} x^{2}-1572864 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3} x -32768 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3} x^{3}-393216 x^{2} \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{3}-6291456 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right ) \ln \left (x \right )^{2}+24576 \ln \left (\frac {x \,{\mathrm e}^{\frac {x^{2}-2 x -8}{4 x +16}}}{3}\right )^{2} x^{4}}{49152 \left (4+x \right )^{3}}\)	\(512\)
risch	\(\text {Expression too large to display}\)	\(944\)

[In]

int(((2*x^2+16*x+32)*ln(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12*x^2+32*x+64)*ln(x)*ln(1/3*x*exp((x^2-2*x-8)
/(4*x+16))))/(x^3+8*x^2+16*x),x,method=_RETURNVERBOSE)

[Out]

-1/49152*(24576*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^2*x^4-6144*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*x^5+19660
8*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^2*x^3-24576*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*x^4+393216*ln(1/3*x*ex
p(1/4*(x^2-2*x-8)/(4+x)))^2*x^2-2097152*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^3-49152*ln(1/3*x*exp(1/4*(x^2-2*x
-8)/(4+x)))*ln(x)*x^4-393216*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*ln(x)*x^3-786432*ln(1/3*x*exp(1/4*(x^2-2*x-8
)/(4+x)))*ln(x)*x^2+32768*x^3*ln(x)^3+6144*x^5*ln(x)+24576*x^4*ln(x)^2+24576*x^4*ln(x)+393216*x^2*ln(x)^3+1966
08*x^3*ln(x)^2+1572864*x*ln(x)^3+393216*x^2*ln(x)^2+2097152*ln(x)^3+512*x^6-6291456*ln(1/3*x*exp(1/4*(x^2-2*x-
8)/(4+x)))*ln(x)^2-1572864*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^3*x-32768*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))
^3*x^3-393216*x^2*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))^3-4718592*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*ln(x)^2*
x-98304*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x)))*ln(x)^2*x^3-1179648*x^2*ln(x)^2*ln(1/3*x*exp(1/4*(x^2-2*x-8)/(4+x
))))/(4+x)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (24) = 48\).

Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.67 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\frac {16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{3} + 8 \, {\left (x^{3} + 2 \, x^{2} - 4 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (3\right ) - 16 \, x - 32\right )} \log \left (x\right )^{2} + {\left (x^{4} - 4 \, x^{3} + 16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (3\right )^{2} - 12 \, x^{2} - 8 \, {\left (x^{3} + 2 \, x^{2} - 16 \, x - 32\right )} \log \left (3\right ) + 32 \, x + 64\right )} \log \left (x\right )}{8 \, {\left (x^{2} + 8 \, x + 16\right )}} \]

[In]

integrate(((2*x^2+16*x+32)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12*x^2+32*x+64)*log(x)*log(1/3*x*exp((x
^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x),x, algorithm="fricas")

[Out]

1/8*(16*(x^2 + 8*x + 16)*log(x)^3 + 8*(x^3 + 2*x^2 - 4*(x^2 + 8*x + 16)*log(3) - 16*x - 32)*log(x)^2 + (x^4 -
4*x^3 + 16*(x^2 + 8*x + 16)*log(3)^2 - 12*x^2 - 8*(x^3 + 2*x^2 - 16*x - 32)*log(3) + 32*x + 64)*log(x))/(x^2 +
8*x + 16)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (24) = 48\).

Time = 0.42 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.80 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=2 \log {\left (x \right )}^{3} + \left (2 \log {\left (3 \right )}^{2} + 6 \log {\left (3 \right )} + \frac {17}{2}\right ) \log {\left (x \right )} + \frac {\left (x^{4} - 8 x^{3} \log {\left (3 \right )} - 4 x^{3} - 80 x^{2} - 64 x^{2} \log {\left (3 \right )} - 512 x - 256 x \log {\left (3 \right )} - 1024 - 512 \log {\left (3 \right )}\right ) \log {\left (x \right )}}{8 x^{2} + 64 x + 128} + \frac {\left (x^{2} - 4 x \log {\left (3 \right )} - 2 x - 16 \log {\left (3 \right )} - 8\right ) \log {\left (x \right )}^{2}}{x + 4} \]

[In]

integrate(((2*x**2+16*x+32)*ln(1/3*x*exp((x**2-2*x-8)/(4*x+16)))**2+(x**3+12*x**2+32*x+64)*ln(x)*ln(1/3*x*exp(
(x**2-2*x-8)/(4*x+16))))/(x**3+8*x**2+16*x),x)

[Out]

2*log(x)**3 + (2*log(3)**2 + 6*log(3) + 17/2)*log(x) + (x**4 - 8*x**3*log(3) - 4*x**3 - 80*x**2 - 64*x**2*log(
3) - 512*x - 256*x*log(3) - 1024 - 512*log(3))*log(x)/(8*x**2 + 64*x + 128) + (x**2 - 4*x*log(3) - 2*x - 16*lo
g(3) - 8)*log(x)**2/(x + 4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (24) = 48\).

Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.10 \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\frac {16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{3} + 8 \, {\left (x^{3} - 2 \, x^{2} {\left (2 \, \log \left (3\right ) - 1\right )} - 16 \, x {\left (2 \, \log \left (3\right ) + 1\right )} - 64 \, \log \left (3\right ) - 32\right )} \log \left (x\right )^{2} + {\left (x^{4} - 4 \, x^{3} {\left (2 \, \log \left (3\right ) + 1\right )} + 4 \, {\left (4 \, \log \left (3\right )^{2} - 4 \, \log \left (3\right ) - 3\right )} x^{2} + 32 \, {\left (4 \, \log \left (3\right )^{2} + 4 \, \log \left (3\right ) + 1\right )} x + 256 \, \log \left (3\right )^{2} + 256 \, \log \left (3\right ) + 64\right )} \log \left (x\right )}{8 \, {\left (x^{2} + 8 \, x + 16\right )}} \]

[In]

integrate(((2*x^2+16*x+32)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12*x^2+32*x+64)*log(x)*log(1/3*x*exp((x
^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x),x, algorithm="maxima")

[Out]

1/8*(16*(x^2 + 8*x + 16)*log(x)^3 + 8*(x^3 - 2*x^2*(2*log(3) - 1) - 16*x*(2*log(3) + 1) - 64*log(3) - 32)*log(
x)^2 + (x^4 - 4*x^3*(2*log(3) + 1) + 4*(4*log(3)^2 - 4*log(3) - 3)*x^2 + 32*(4*log(3)^2 + 4*log(3) + 1)*x + 25
6*log(3)^2 + 256*log(3) + 64)*log(x))/(x^2 + 8*x + 16)

Giac [F]

\[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\int { \frac {2 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (\frac {1}{3} \, x e^{\left (\frac {x^{2} - 2 \, x - 8}{4 \, {\left (x + 4\right )}}\right )}\right )^{2} + {\left (x^{3} + 12 \, x^{2} + 32 \, x + 64\right )} \log \left (\frac {1}{3} \, x e^{\left (\frac {x^{2} - 2 \, x - 8}{4 \, {\left (x + 4\right )}}\right )}\right ) \log \left (x\right )}{x^{3} + 8 \, x^{2} + 16 \, x} \,d x } \]

[In]

integrate(((2*x^2+16*x+32)*log(1/3*x*exp((x^2-2*x-8)/(4*x+16)))^2+(x^3+12*x^2+32*x+64)*log(x)*log(1/3*x*exp((x
^2-2*x-8)/(4*x+16))))/(x^3+8*x^2+16*x),x, algorithm="giac")

[Out]

integrate((2*(x^2 + 8*x + 16)*log(1/3*x*e^(1/4*(x^2 - 2*x - 8)/(x + 4)))^2 + (x^3 + 12*x^2 + 32*x + 64)*log(1/
3*x*e^(1/4*(x^2 - 2*x - 8)/(x + 4)))*log(x))/(x^3 + 8*x^2 + 16*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (64+32 x+12 x^2+x^3\right ) \log (x) \log \left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )+\left (32+16 x+2 x^2\right ) \log ^2\left (\frac {1}{3} e^{\frac {-8-2 x+x^2}{16+4 x}} x\right )}{16 x+8 x^2+x^3} \, dx=\int \frac {\left (2\,x^2+16\,x+32\right )\,{\ln \left (\frac {x\,{\mathrm {e}}^{-\frac {-x^2+2\,x+8}{4\,x+16}}}{3}\right )}^2+\ln \left (x\right )\,\left (x^3+12\,x^2+32\,x+64\right )\,\ln \left (\frac {x\,{\mathrm {e}}^{-\frac {-x^2+2\,x+8}{4\,x+16}}}{3}\right )}{x^3+8\,x^2+16\,x} \,d x \]

[In]

int((log((x*exp(-(2*x - x^2 + 8)/(4*x + 16)))/3)^2*(16*x + 2*x^2 + 32) + log((x*exp(-(2*x - x^2 + 8)/(4*x + 16
)))/3)*log(x)*(32*x + 12*x^2 + x^3 + 64))/(16*x + 8*x^2 + x^3),x)

[Out]

int((log((x*exp(-(2*x - x^2 + 8)/(4*x + 16)))/3)^2*(16*x + 2*x^2 + 32) + log((x*exp(-(2*x - x^2 + 8)/(4*x + 16
)))/3)*log(x)*(32*x + 12*x^2 + x^3 + 64))/(16*x + 8*x^2 + x^3), x)