\(\int \frac {155 x+240 x^2+120 x^3+20 x^4+(31 x+48 x^2+24 x^3+4 x^4) \log (x)+(16+31 x+24 x^2+8 x^3+x^4) \log (-16-31 x-24 x^2-8 x^3-x^4) \log (\log (-16-31 x-24 x^2-8 x^3-x^4))}{(16 x+31 x^2+24 x^3+8 x^4+x^5) \log (-16-31 x-24 x^2-8 x^3-x^4)} \, dx\) [1718]

Optimal result

Integrand size = 151, antiderivative size = 16 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=(5+\log (x)) \log \left (\log \left (x-(2+x)^4\right )\right ) \]

[Out]

ln(ln(x-(2+x)^4))*(5+ln(x))

Rubi [F]

\[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx \]

[In]

Int[(155*x + 240*x^2 + 120*x^3 + 20*x^4 + (31*x + 48*x^2 + 24*x^3 + 4*x^4)*Log[x] + (16 + 31*x + 24*x^2 + 8*x^
3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]*Log[Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]])/((16*x + 31*x^2 +
24*x^3 + 8*x^4 + x^5)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]),x]

[Out]

155*Defer[Int][1/((16 + 31*x + 24*x^2 + 8*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + 240*Defer[I
nt][x/((16 + 31*x + 24*x^2 + 8*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + 120*Defer[Int][x^2/((1
6 + 31*x + 24*x^2 + 8*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + 20*Defer[Int][x^3/((16 + 31*x +
24*x^2 + 8*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + 31*Defer[Int][Log[x]/((16 + 31*x + 24*x^2
+ 8*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + 48*Defer[Int][(x*Log[x])/((16 + 31*x + 24*x^2 +
8*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + 24*Defer[Int][(x^2*Log[x])/((16 + 31*x + 24*x^2 + 8
*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + 4*Defer[Int][(x^3*Log[x])/((16 + 31*x + 24*x^2 + 8*x
^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]), x] + Defer[Int][Log[Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]]
/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x \left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx \\ & = \int \left (\frac {155}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {240 x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {120 x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {20 x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {\left (31+48 x+24 x^2+4 x^3\right ) \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x}\right ) \, dx \\ & = 20 \int \frac {x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+120 \int \frac {x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+155 \int \frac {1}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+240 \int \frac {x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \frac {\left (31+48 x+24 x^2+4 x^3\right ) \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x} \, dx \\ & = 20 \int \frac {x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+120 \int \frac {x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+155 \int \frac {1}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+240 \int \frac {x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \left (\frac {31 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {48 x \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {24 x^2 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}+\frac {4 x^3 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )}\right ) \, dx+\int \frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x} \, dx \\ & = 4 \int \frac {x^3 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+20 \int \frac {x^3}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+24 \int \frac {x^2 \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+31 \int \frac {\log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+48 \int \frac {x \log (x)}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+120 \int \frac {x^2}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+155 \int \frac {1}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+240 \int \frac {x}{\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx+\int \frac {\log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{x} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(16)=32\).

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 3.12 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=5 \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )+\log (x) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right ) \]

[In]

Integrate[(155*x + 240*x^2 + 120*x^3 + 20*x^4 + (31*x + 48*x^2 + 24*x^3 + 4*x^4)*Log[x] + (16 + 31*x + 24*x^2
+ 8*x^3 + x^4)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]*Log[Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]])/((16*x + 31
*x^2 + 24*x^3 + 8*x^4 + x^5)*Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]),x]

[Out]

5*Log[Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]] + Log[x]*Log[Log[-16 - 31*x - 24*x^2 - 8*x^3 - x^4]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(16)=32\).

Time = 15.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.19

method	result	size
risch	\(\ln \left (x \right ) \ln \left (\ln \left (-x^{4}-8 x^{3}-24 x^{2}-31 x -16\right )\right )+5 \ln \left (\ln \left (-x^{4}-8 x^{3}-24 x^{2}-31 x -16\right )\right )\)	\(51\)

[In]

int(((x^4+8*x^3+24*x^2+31*x+16)*ln(-x^4-8*x^3-24*x^2-31*x-16)*ln(ln(-x^4-8*x^3-24*x^2-31*x-16))+(4*x^4+24*x^3+
48*x^2+31*x)*ln(x)+20*x^4+120*x^3+240*x^2+155*x)/(x^5+8*x^4+24*x^3+31*x^2+16*x)/ln(-x^4-8*x^3-24*x^2-31*x-16),
x,method=_RETURNVERBOSE)

[Out]

ln(x)*ln(ln(-x^4-8*x^3-24*x^2-31*x-16))+5*ln(ln(-x^4-8*x^3-24*x^2-31*x-16))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx={\left (\log \left (x\right ) + 5\right )} \log \left (\log \left (-x^{4} - 8 \, x^{3} - 24 \, x^{2} - 31 \, x - 16\right )\right ) \]

[In]

integrate(((x^4+8*x^3+24*x^2+31*x+16)*log(-x^4-8*x^3-24*x^2-31*x-16)*log(log(-x^4-8*x^3-24*x^2-31*x-16))+(4*x^
4+24*x^3+48*x^2+31*x)*log(x)+20*x^4+120*x^3+240*x^2+155*x)/(x^5+8*x^4+24*x^3+31*x^2+16*x)/log(-x^4-8*x^3-24*x^
2-31*x-16),x, algorithm="fricas")

[Out]

(log(x) + 5)*log(log(-x^4 - 8*x^3 - 24*x^2 - 31*x - 16))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (14) = 28\).

Time = 0.62 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.19 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\log {\left (x \right )} \log {\left (\log {\left (- x^{4} - 8 x^{3} - 24 x^{2} - 31 x - 16 \right )} \right )} + 5 \log {\left (\log {\left (- x^{4} - 8 x^{3} - 24 x^{2} - 31 x - 16 \right )} \right )} \]

[In]

integrate(((x**4+8*x**3+24*x**2+31*x+16)*ln(-x**4-8*x**3-24*x**2-31*x-16)*ln(ln(-x**4-8*x**3-24*x**2-31*x-16))
+(4*x**4+24*x**3+48*x**2+31*x)*ln(x)+20*x**4+120*x**3+240*x**2+155*x)/(x**5+8*x**4+24*x**3+31*x**2+16*x)/ln(-x
**4-8*x**3-24*x**2-31*x-16),x)

[Out]

log(x)*log(log(-x**4 - 8*x**3 - 24*x**2 - 31*x - 16)) + 5*log(log(-x**4 - 8*x**3 - 24*x**2 - 31*x - 16))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx={\left (\log \left (x\right ) + 5\right )} \log \left (\log \left (-x^{4} - 8 \, x^{3} - 24 \, x^{2} - 31 \, x - 16\right )\right ) \]

[In]

integrate(((x^4+8*x^3+24*x^2+31*x+16)*log(-x^4-8*x^3-24*x^2-31*x-16)*log(log(-x^4-8*x^3-24*x^2-31*x-16))+(4*x^
4+24*x^3+48*x^2+31*x)*log(x)+20*x^4+120*x^3+240*x^2+155*x)/(x^5+8*x^4+24*x^3+31*x^2+16*x)/log(-x^4-8*x^3-24*x^
2-31*x-16),x, algorithm="maxima")

[Out]

(log(x) + 5)*log(log(-x^4 - 8*x^3 - 24*x^2 - 31*x - 16))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 52, normalized size of antiderivative = 3.25 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\log \left (x\right ) \log \left (\log \left (-x^{4} - 8 \, x^{3} - 24 \, x^{2} - 31 \, x - 16\right )\right ) + 5 \, \log \left (\pi - i \, \log \left (x^{4} + 8 \, x^{3} + 24 \, x^{2} + 31 \, x + 16\right )\right ) \]

[In]

integrate(((x^4+8*x^3+24*x^2+31*x+16)*log(-x^4-8*x^3-24*x^2-31*x-16)*log(log(-x^4-8*x^3-24*x^2-31*x-16))+(4*x^
4+24*x^3+48*x^2+31*x)*log(x)+20*x^4+120*x^3+240*x^2+155*x)/(x^5+8*x^4+24*x^3+31*x^2+16*x)/log(-x^4-8*x^3-24*x^
2-31*x-16),x, algorithm="giac")

[Out]

log(x)*log(log(-x^4 - 8*x^3 - 24*x^2 - 31*x - 16)) + 5*log(pi - I*log(x^4 + 8*x^3 + 24*x^2 + 31*x + 16))

Mupad [B] (verification not implemented)

Time = 9.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {155 x+240 x^2+120 x^3+20 x^4+\left (31 x+48 x^2+24 x^3+4 x^4\right ) \log (x)+\left (16+31 x+24 x^2+8 x^3+x^4\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right ) \log \left (\log \left (-16-31 x-24 x^2-8 x^3-x^4\right )\right )}{\left (16 x+31 x^2+24 x^3+8 x^4+x^5\right ) \log \left (-16-31 x-24 x^2-8 x^3-x^4\right )} \, dx=\ln \left (\ln \left (-x^4-8\,x^3-24\,x^2-31\,x-16\right )\right )\,\left (\ln \left (x\right )+5\right ) \]

[In]

int((155*x + log(x)*(31*x + 48*x^2 + 24*x^3 + 4*x^4) + 240*x^2 + 120*x^3 + 20*x^4 + log(- 31*x - 24*x^2 - 8*x^
3 - x^4 - 16)*log(log(- 31*x - 24*x^2 - 8*x^3 - x^4 - 16))*(31*x + 24*x^2 + 8*x^3 + x^4 + 16))/(log(- 31*x - 2
4*x^2 - 8*x^3 - x^4 - 16)*(16*x + 31*x^2 + 24*x^3 + 8*x^4 + x^5)),x)

[Out]

log(log(- 31*x - 24*x^2 - 8*x^3 - x^4 - 16))*(log(x) + 5)