\(\int \frac {4-x+8 x \log (8 e^{-x/4} x)+(-1-2 x^2+2 x^3+2 x \log (x)) \log ^2(8 e^{-x/4} x)}{4 x \log (8 e^{-x/4} x)+(x^3+x \log (x)) \log ^2(8 e^{-x/4} x)} \, dx\) [9554]

Optimal result

Integrand size = 90, antiderivative size = 29 \[ \int \frac {4-x+8 x \log \left (8 e^{-x/4} x\right )+\left (-1-2 x^2+2 x^3+2 x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )}{4 x \log \left (8 e^{-x/4} x\right )+\left (x^3+x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )} \, dx=3+2 x-\log \left (x^2+\log (x)+\frac {4}{\log \left (8 e^{-x/4} x\right )}\right ) \]

[Out]

2*x-ln(ln(x)+4/ln(8*x/exp(1/4*x))+x^2)+3

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-1-2 x^2+2 x^3+2 x \log (x)}{x \left (x^2+\log (x)\right )}+\frac {4-x}{4 x \log \left (8 e^{-x/4} x\right )}+\frac {16+32 x^2-4 x^4+x^5-8 x^2 \log (x)+2 x^3 \log (x)-4 \log ^2(x)+x \log ^2(x)}{4 x \left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}\right ) \, dx \\ & = \frac {1}{4} \int \frac {4-x}{x \log \left (8 e^{-x/4} x\right )} \, dx+\frac {1}{4} \int \frac {16+32 x^2-4 x^4+x^5-8 x^2 \log (x)+2 x^3 \log (x)-4 \log ^2(x)+x \log ^2(x)}{x \left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx+\int \frac {-1-2 x^2+2 x^3+2 x \log (x)}{x \left (x^2+\log (x)\right )} \, dx \\ & = \log \left (\log \left (8 e^{-x/4} x\right )\right )+\frac {1}{4} \int \frac {16+32 x^2-4 x^4+x^5+2 (-4+x) x^2 \log (x)+(-4+x) \log ^2(x)}{x \left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx+\int \left (2+\frac {-1-2 x^2}{x \left (x^2+\log (x)\right )}\right ) \, dx \\ & = 2 x+\log \left (\log \left (8 e^{-x/4} x\right )\right )+\frac {1}{4} \int \left (\frac {16}{x \left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}+\frac {32 x}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}-\frac {4 x^3}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}+\frac {x^4}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}-\frac {8 x \log (x)}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}+\frac {2 x^2 \log (x)}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}+\frac {\log ^2(x)}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}-\frac {4 \log ^2(x)}{x \left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )}\right ) \, dx+\int \frac {-1-2 x^2}{x \left (x^2+\log (x)\right )} \, dx \\ & = 2 x-\log \left (x^2+\log (x)\right )+\log \left (\log \left (8 e^{-x/4} x\right )\right )+\frac {1}{4} \int \frac {x^4}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx+\frac {1}{4} \int \frac {\log ^2(x)}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx+\frac {1}{2} \int \frac {x^2 \log (x)}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx-2 \int \frac {x \log (x)}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx+4 \int \frac {1}{x \left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx+8 \int \frac {x}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx-\int \frac {x^3}{\left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx-\int \frac {\log ^2(x)}{x \left (x^2+\log (x)\right ) \left (4+x^2 \log \left (8 e^{-x/4} x\right )+\log (x) \log \left (8 e^{-x/4} x\right )\right )} \, dx \\ & = 2 x-\log \left (x^2+\log (x)\right )+\log \left (\log \left (8 e^{-x/4} x\right )\right )+\frac {1}{4} \int \frac {x^4}{\left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx+\frac {1}{4} \int \frac {\log ^2(x)}{\left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx+\frac {1}{2} \int \frac {x^2 \log (x)}{\left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx-2 \int \frac {x \log (x)}{\left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx+4 \int \frac {1}{x \left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx+8 \int \frac {x}{\left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx-\int \frac {x^3}{\left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx-\int \frac {\log ^2(x)}{x \left (x^2+\log (x)\right ) \left (4+\left (x^2+\log (x)\right ) \log \left (8 e^{-x/4} x\right )\right )} \, dx \\ \end{align*}

Mathematica [F]

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

[In]

Integrate[(4 - x + 8*x*Log[(8*x)/E^(x/4)] + (-1 - 2*x^2 + 2*x^3 + 2*x*Log[x])*Log[(8*x)/E^(x/4)]^2)/(4*x*Log[(
8*x)/E^(x/4)] + (x^3 + x*Log[x])*Log[(8*x)/E^(x/4)]^2),x]

[Out]

Integrate[(4 - x + 8*x*Log[(8*x)/E^(x/4)] + (-1 - 2*x^2 + 2*x^3 + 2*x*Log[x])*Log[(8*x)/E^(x/4)]^2)/(4*x*Log[(
8*x)/E^(x/4)] + (x^3 + x*Log[x])*Log[(8*x)/E^(x/4)]^2), x]

Maple [A] (verified)

Time = 1.80 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66

method	result	size
parallelrisch	\(2 x +\ln \left (\ln \left (8 x \,{\mathrm e}^{-\frac {x}{4}}\right )\right )-\ln \left (\ln \left (8 x \,{\mathrm e}^{-\frac {x}{4}}\right ) x^{2}+\ln \left (x \right ) \ln \left (8 x \,{\mathrm e}^{-\frac {x}{4}}\right )+4\right )\)	\(48\)
risch	\(2 x -\ln \left (\ln \left (x \right )+x^{2}\right )+\ln \left (\ln \left ({\mathrm e}^{\frac {x}{4}}\right )+\frac {i \left (-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{2}+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \right )+\pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{3}-\pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{2} \operatorname {csgn}\left (i x \right )+6 i \ln \left (2\right )+2 i \ln \left (x \right )\right )}{2}\right )-\ln \left (\ln \left ({\mathrm e}^{\frac {x}{4}}\right )+\frac {i \left (-\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{2}+\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \right )+\pi \,x^{2} \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{3}-\pi \,x^{2} \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{2} \operatorname {csgn}\left (i x \right )-\ln \left (x \right ) \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{2}+\ln \left (x \right ) \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right ) \operatorname {csgn}\left (i x \right )+\ln \left (x \right ) \pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{3}-\ln \left (x \right ) \pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-\frac {x}{4}}\right )^{2} \operatorname {csgn}\left (i x \right )+6 i \ln \left (2\right ) x^{2}+2 i x^{2} \ln \left (x \right )+2 i \ln \left (x \right )^{2}+6 i \ln \left (2\right ) \ln \left (x \right )+8 i\right )}{2 \ln \left (x \right )+2 x^{2}}\right )\)	\(342\)

[In]

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

[Out]

2*x+ln(ln(8*x/exp(1/4*x)))-ln(ln(8*x/exp(1/4*x))*x^2+ln(8*x/exp(1/4*x))*ln(x)+4)

Fricas [B] (verification not implemented)

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

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {4-x+8 x \log \left (8 e^{-x/4} x\right )+\left (-1-2 x^2+2 x^3+2 x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )}{4 x \log \left (8 e^{-x/4} x\right )+\left (x^3+x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )} \, dx=2 \, x - \log \left (-x^{3} + 12 \, x^{2} \log \left (2\right ) + {\left (4 \, x^{2} - x + 12 \, \log \left (2\right )\right )} \log \left (x\right ) + 4 \, \log \left (x\right )^{2} + 16\right ) + \log \left (-x + 12 \, \log \left (2\right ) + 4 \, \log \left (x\right )\right ) \]

[In]

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

[Out]

2*x - log(-x^3 + 12*x^2*log(2) + (4*x^2 - x + 12*log(2))*log(x) + 4*log(x)^2 + 16) + log(-x + 12*log(2) + 4*lo
g(x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 0.73 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {4-x+8 x \log \left (8 e^{-x/4} x\right )+\left (-1-2 x^2+2 x^3+2 x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )}{4 x \log \left (8 e^{-x/4} x\right )+\left (x^3+x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )} \, dx=2 x + \log {\left (- \frac {x}{4} + \log {\left (x \right )} + 3 \log {\left (2 \right )} \right )} - \log {\left (- \frac {x^{3}}{4} + 3 x^{2} \log {\left (2 \right )} + \left (x^{2} - \frac {x}{4} + 3 \log {\left (2 \right )}\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 4 \right )} \]

[In]

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

[Out]

2*x + log(-x/4 + log(x) + 3*log(2)) - log(-x**3/4 + 3*x**2*log(2) + (x**2 - x/4 + 3*log(2))*log(x) + log(x)**2
+ 4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).

Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {4-x+8 x \log \left (8 e^{-x/4} x\right )+\left (-1-2 x^2+2 x^3+2 x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )}{4 x \log \left (8 e^{-x/4} x\right )+\left (x^3+x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )} \, dx=2 \, x - \log \left (-\frac {1}{4} \, x^{3} + 3 \, x^{2} \log \left (2\right ) + \frac {1}{4} \, {\left (4 \, x^{2} - x + 12 \, \log \left (2\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 4\right ) + \log \left (-\frac {1}{4} \, x + 3 \, \log \left (2\right ) + \log \left (x\right )\right ) \]

[In]

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

[Out]

2*x - log(-1/4*x^3 + 3*x^2*log(2) + 1/4*(4*x^2 - x + 12*log(2))*log(x) + log(x)^2 + 4) + log(-1/4*x + 3*log(2)
+ log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).

Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {4-x+8 x \log \left (8 e^{-x/4} x\right )+\left (-1-2 x^2+2 x^3+2 x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )}{4 x \log \left (8 e^{-x/4} x\right )+\left (x^3+x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )} \, dx=2 \, x - \log \left (-x^{3} + 12 \, x^{2} \log \left (2\right ) + 4 \, x^{2} \log \left (x\right ) - x \log \left (x\right ) + 12 \, \log \left (2\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2} + 16\right ) + \log \left (-x + 12 \, \log \left (2\right ) + 4 \, \log \left (x\right )\right ) \]

[In]

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

[Out]

2*x - log(-x^3 + 12*x^2*log(2) + 4*x^2*log(x) - x*log(x) + 12*log(2)*log(x) + 4*log(x)^2 + 16) + log(-x + 12*l
og(2) + 4*log(x))

Mupad [B] (verification not implemented)

Time = 17.60 (sec) , antiderivative size = 491, normalized size of antiderivative = 16.93 \[ \int \frac {4-x+8 x \log \left (8 e^{-x/4} x\right )+\left (-1-2 x^2+2 x^3+2 x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )}{4 x \log \left (8 e^{-x/4} x\right )+\left (x^3+x \log (x)\right ) \log ^2\left (8 e^{-x/4} x\right )} \, dx=2\,x-\ln \left (\frac {16\,x+4\,x\,{\ln \left (x\right )}^2-17\,x^2\,\ln \left (x\right )+4\,x^3\,\ln \left (x\right )-16\,{\ln \left (x\right )}^2-48\,x^2\,\ln \left (2\right )+12\,x^3\,\ln \left (2\right )-48\,\ln \left (2\right )\,\ln \left (x\right )+4\,x\,\ln \left (x\right )+4\,x^3-x^4+12\,x\,\ln \left (2\right )\,\ln \left (x\right )-64}{x}\right )+\ln \left (x\,\left (x-4\right )\right )-\ln \left (9\,x^3\,{\ln \left (2\right )}^2-36\,x^2\,{\ln \left (2\right )}^2-8\,x+3\,x\,\ln \left (2\right )+\frac {9\,x\,{\ln \left (2\right )}^2}{2}+\frac {45\,x^2\,\ln \left (2\right )}{4}+3\,x^3\,\ln \left (2\right )+\frac {45\,x^4\,\ln \left (2\right )}{2}-6\,x^5\,\ln \left (2\right )-18\,{\ln \left (2\right )}^2+\frac {515\,x^2}{8}-\frac {287\,x^3}{32}+30\,x^4-\frac {23\,x^5}{16}-\frac {7\,x^6}{2}+x^7+32\right )+\ln \left (\frac {1}{x^2}\right )+\ln \left (32\,x-384\,\ln \left (2\right )-128\,\ln \left (x\right )-\frac {261\,x^2\,{\ln \left (2\right )}^2}{2}+432\,x^2\,{\ln \left (2\right )}^3-72\,x^3\,{\ln \left (2\right )}^2-108\,x^3\,{\ln \left (2\right )}^3-261\,x^4\,{\ln \left (2\right )}^2+72\,x^5\,{\ln \left (2\right )}^2+72\,{\ln \left (2\right )}^2\,\ln \left (x\right )-\frac {515\,x^2\,\ln \left (x\right )}{2}+\frac {287\,x^3\,\ln \left (x\right )}{8}-120\,x^4\,\ln \left (x\right )+\frac {23\,x^5\,\ln \left (x\right )}{4}+14\,x^6\,\ln \left (x\right )-4\,x^7\,\ln \left (x\right )+96\,x\,\ln \left (2\right )-54\,x\,{\ln \left (2\right )}^2-\frac {1539\,x^2\,\ln \left (2\right )}{2}-54\,x\,{\ln \left (2\right )}^3+\frac {951\,x^3\,\ln \left (2\right )}{8}-357\,x^4\,\ln \left (2\right )+\frac {159\,x^5\,\ln \left (2\right )}{4}+36\,x^6\,\ln \left (2\right )-12\,x^7\,\ln \left (2\right )+32\,x\,\ln \left (x\right )+216\,{\ln \left (2\right )}^3-8\,x^2+\frac {515\,x^3}{8}-\frac {287\,x^4}{32}+30\,x^5-\frac {23\,x^6}{16}-\frac {7\,x^7}{2}+x^8-12\,x\,\ln \left (2\right )\,\ln \left (x\right )-18\,x\,{\ln \left (2\right )}^2\,\ln \left (x\right )-45\,x^2\,\ln \left (2\right )\,\ln \left (x\right )-12\,x^3\,\ln \left (2\right )\,\ln \left (x\right )-90\,x^4\,\ln \left (2\right )\,\ln \left (x\right )+24\,x^5\,\ln \left (2\right )\,\ln \left (x\right )+144\,x^2\,{\ln \left (2\right )}^2\,\ln \left (x\right )-36\,x^3\,{\ln \left (2\right )}^2\,\ln \left (x\right )\right ) \]

[In]

int((8*x*log(8*x*exp(-x/4)) - x + log(8*x*exp(-x/4))^2*(2*x*log(x) - 2*x^2 + 2*x^3 - 1) + 4)/(log(8*x*exp(-x/4
))^2*(x*log(x) + x^3) + 4*x*log(8*x*exp(-x/4))),x)

[Out]

2*x - log((16*x + 4*x*log(x)^2 - 17*x^2*log(x) + 4*x^3*log(x) - 16*log(x)^2 - 48*x^2*log(2) + 12*x^3*log(2) -
48*log(2)*log(x) + 4*x*log(x) + 4*x^3 - x^4 + 12*x*log(2)*log(x) - 64)/x) + log(x*(x - 4)) - log(9*x^3*log(2)^
2 - 36*x^2*log(2)^2 - 8*x + 3*x*log(2) + (9*x*log(2)^2)/2 + (45*x^2*log(2))/4 + 3*x^3*log(2) + (45*x^4*log(2))
/2 - 6*x^5*log(2) - 18*log(2)^2 + (515*x^2)/8 - (287*x^3)/32 + 30*x^4 - (23*x^5)/16 - (7*x^6)/2 + x^7 + 32) +
log(1/x^2) + log(32*x - 384*log(2) - 128*log(x) - (261*x^2*log(2)^2)/2 + 432*x^2*log(2)^3 - 72*x^3*log(2)^2 -
108*x^3*log(2)^3 - 261*x^4*log(2)^2 + 72*x^5*log(2)^2 + 72*log(2)^2*log(x) - (515*x^2*log(x))/2 + (287*x^3*log
(x))/8 - 120*x^4*log(x) + (23*x^5*log(x))/4 + 14*x^6*log(x) - 4*x^7*log(x) + 96*x*log(2) - 54*x*log(2)^2 - (15
39*x^2*log(2))/2 - 54*x*log(2)^3 + (951*x^3*log(2))/8 - 357*x^4*log(2) + (159*x^5*log(2))/4 + 36*x^6*log(2) -
12*x^7*log(2) + 32*x*log(x) + 216*log(2)^3 - 8*x^2 + (515*x^3)/8 - (287*x^4)/32 + 30*x^5 - (23*x^6)/16 - (7*x^
7)/2 + x^8 - 12*x*log(2)*log(x) - 18*x*log(2)^2*log(x) - 45*x^2*log(2)*log(x) - 12*x^3*log(2)*log(x) - 90*x^4*
log(2)*log(x) + 24*x^5*log(2)*log(x) + 144*x^2*log(2)^2*log(x) - 36*x^3*log(2)^2*log(x))