\(\int \frac {48 x+48 x^2+(-24 x^2-24 x \log (x)) \log (x+\log (x)) \log (16 \log ^2(x+\log (x)))+(x+\log (x)) \log (x+\log (x)) \log ^3(16 \log ^2(x+\log (x)))}{(x+\log (x)) \log (x+\log (x)) \log ^3(16 \log ^2(x+\log (x)))} \, dx\) [1682]

Optimal result

Integrand size = 84, antiderivative size = 19 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x-\frac {12 x^2}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \]

[Out]

x-12/ln(16*ln(x+ln(x))^2)^2*x^2

Rubi [F]

\[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx \]

[In]

Int[(48*x + 48*x^2 + (-24*x^2 - 24*x*Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2] + (x + Log[x])*Log[x +
Log[x]]*Log[16*Log[x + Log[x]]^2]^3)/((x + Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3),x]

[Out]

x + 48*Defer[Int][x/((x + Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3), x] + 48*Defer[Int][x^2/((x + L
og[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3), x] - 24*Defer[Int][x/Log[16*Log[x + Log[x]]^2]^2, x]

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x-\frac {12 x^2}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \]

[In]

Integrate[(48*x + 48*x^2 + (-24*x^2 - 24*x*Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2] + (x + Log[x])*Lo
g[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3)/((x + Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3),x]

[Out]

x - (12*x^2)/Log[16*Log[x + Log[x]]^2]^2

Maple [A] (verified)

Time = 4.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89

method	result	size
parallelrisch	\(\frac {2 \ln \left (16 \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x -24 x^{2}}{2 \ln \left (16 \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}}\)	\(36\)
risch	\(x +\frac {48 x^{2}}{\left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )^{2}}\)	\(89\)
default	\(x +\frac {24 i \ln \left (x +\ln \left (x \right )\right ) \left (x +\ln \left (x \right )\right ) x^{2}}{\left (1+x \right ) \left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )}-\frac {24 i \left (2 i x^{3}+2 i x^{2}+\pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right ) x^{2} \ln \left (x +\ln \left (x \right )\right )-2 \pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x^{2} \ln \left (x +\ln \left (x \right )\right )+\pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3} x^{2} \ln \left (x +\ln \left (x \right )\right )+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right ) x^{3} \ln \left (x +\ln \left (x \right )\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x^{3} \ln \left (x +\ln \left (x \right )\right )+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3} x^{3} \ln \left (x +\ln \left (x \right )\right )+8 i \ln \left (2\right ) \ln \left (x \right ) x^{2} \ln \left (x +\ln \left (x \right )\right )+8 i \ln \left (2\right ) x^{3} \ln \left (x +\ln \left (x \right )\right )+4 i \ln \left (x \right ) x^{2} \ln \left (x +\ln \left (x \right )\right ) \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )+4 i x^{3} \ln \left (x +\ln \left (x \right )\right ) \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )}{\left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )^{2} \left (1+x \right )}\)	\(452\)

[In]

int(((x+ln(x))*ln(x+ln(x))*ln(16*ln(x+ln(x))^2)^3+(-24*x*ln(x)-24*x^2)*ln(x+ln(x))*ln(16*ln(x+ln(x))^2)+48*x^2
+48*x)/(x+ln(x))/ln(x+ln(x))/ln(16*ln(x+ln(x))^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/2*(2*ln(16*ln(x+ln(x))^2)^2*x-24*x^2)/ln(16*ln(x+ln(x))^2)^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\frac {x \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2} - 12 \, x^{2}}{\log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2}} \]

[In]

integrate(((x+log(x))*log(x+log(x))*log(16*log(x+log(x))^2)^3+(-24*x*log(x)-24*x^2)*log(x+log(x))*log(16*log(x
+log(x))^2)+48*x^2+48*x)/(x+log(x))/log(x+log(x))/log(16*log(x+log(x))^2)^3,x, algorithm="fricas")

[Out]

(x*log(16*log(x + log(x))^2)^2 - 12*x^2)/log(16*log(x + log(x))^2)^2

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=- \frac {12 x^{2}}{\log {\left (16 \log {\left (x + \log {\left (x \right )} \right )}^{2} \right )}^{2}} + x \]

[In]

integrate(((x+ln(x))*ln(x+ln(x))*ln(16*ln(x+ln(x))**2)**3+(-24*x*ln(x)-24*x**2)*ln(x+ln(x))*ln(16*ln(x+ln(x))*
*2)+48*x**2+48*x)/(x+ln(x))/ln(x+ln(x))/ln(16*ln(x+ln(x))**2)**3,x)

[Out]

-12*x**2/log(16*log(x + log(x))**2)**2 + x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).

Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.26 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\frac {4 \, x \log \left (2\right )^{2} + 4 \, x \log \left (2\right ) \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x \log \left (\log \left (x + \log \left (x\right )\right )\right )^{2} - 3 \, x^{2}}{4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (x + \log \left (x\right )\right )\right ) + \log \left (\log \left (x + \log \left (x\right )\right )\right )^{2}} \]

[In]

integrate(((x+log(x))*log(x+log(x))*log(16*log(x+log(x))^2)^3+(-24*x*log(x)-24*x^2)*log(x+log(x))*log(16*log(x
+log(x))^2)+48*x^2+48*x)/(x+log(x))/log(x+log(x))/log(16*log(x+log(x))^2)^3,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).

Time = 18.49 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x - \frac {12 \, {\left (x^{3} + x^{2}\right )}}{x \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2} + \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2}} \]

[In]

integrate(((x+log(x))*log(x+log(x))*log(16*log(x+log(x))^2)^3+(-24*x*log(x)-24*x^2)*log(x+log(x))*log(16*log(x
+log(x))^2)+48*x^2+48*x)/(x+log(x))/log(x+log(x))/log(16*log(x+log(x))^2)^3,x, algorithm="giac")

[Out]

x - 12*(x^3 + x^2)/(x*log(16*log(x + log(x))^2)^2 + log(16*log(x + log(x))^2)^2)

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 739, normalized size of antiderivative = 38.89 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x+\ln \left (x+\ln \left (x\right )\right )\,\left (\frac {6\,x^2+6\,x}{x+1}-\frac {3\,x^2+3\,x}{x+1}+\frac {6\,x^3+12\,x^2+3\,x}{x+1}-\frac {9\,x^3+15\,x^2+6\,x}{x+1}+\ln \left (x\right )\,\left (\frac {3\,x^2+6\,x}{x+1}-\frac {6\,x^2+6\,x}{x+1}\right )\right )-\frac {12\,x^2-\frac {6\,x^2\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )\,\left (x+\ln \left (x\right )\right )}{x+1}}{{\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )}^2}-\frac {\frac {6\,x^2\,\ln \left (x+\ln \left (x\right )\right )\,\left (x+\ln \left (x\right )\right )}{x+1}-\frac {3\,x^2\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )\,\left (x+\ln \left (x\right )\right )\,\left (2\,x+\ln \left (x+\ln \left (x\right )\right )+2\,x^2\,\ln \left (x+\ln \left (x\right )\right )+2\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (x\right )+x^2+4\,x\,\ln \left (x+\ln \left (x\right )\right )+x\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (x\right )+1\right )}{{\left (x+1\right )}^3}}{\ln \left (16\,{\ln \left (x+\ln \left (x\right )\right )}^2\right )}-{\ln \left (x+\ln \left (x\right )\right )}^2\,\left (\frac {60\,x^5+264\,x^4+432\,x^3+312\,x^2+84\,x}{x^3+3\,x^2+3\,x+1}-\frac {54\,x^5+243\,x^4+\frac {831\,x^3}{2}+\frac {639\,x^2}{2}+\frac {213\,x}{2}+\frac {21}{2}}{x^3+3\,x^2+3\,x+1}-\frac {6\,x^2+18\,x+\frac {50}{3}}{x^3+3\,x^2+3\,x+1}-{\ln \left (x\right )}^2\,\left (\frac {3\,x^2+9\,x+3}{x^3+3\,x^2+3\,x+1}-3\right )+\ln \left (x\right )\,\left (\frac {3\,\left (\frac {11\,x^3}{3}+13\,x^2+16\,x+\frac {22}{3}\right )}{x^3+3\,x^2+3\,x+1}-\frac {6\,x^2+18\,x+14}{x^3+3\,x^2+3\,x+1}+\frac {36\,x^4+108\,x^3+108\,x^2+36\,x}{x^3+3\,x^2+3\,x+1}-\frac {27\,x^4+119\,x^3+192\,x^2+120\,x+26}{x^3+3\,x^2+3\,x+1}+18\right )+\frac {18\,\left (\frac {11\,x^3}{6}+\frac {13\,x^2}{2}+8\,x+\frac {11}{3}\right )}{x^3+3\,x^2+3\,x+1}+\frac {3\,\left (\frac {121\,x^3}{18}+\frac {155\,x^2}{6}+\frac {209\,x}{6}+\frac {109}{6}\right )}{x^3+3\,x^2+3\,x+1}-\frac {36\,x^4+108\,x^3+108\,x^2+36\,x}{x^3+3\,x^2+3\,x+1}+\frac {108\,x^4+324\,x^3+324\,x^2+108\,x}{x^3+3\,x^2+3\,x+1}-\frac {81\,x^4+\frac {1181\,x^3}{3}+730\,x^2+613\,x+\frac {613}{3}}{x^3+3\,x^2+3\,x+1}+111\right ) \]

[In]

int((48*x + 48*x^2 - log(x + log(x))*log(16*log(x + log(x))^2)*(24*x*log(x) + 24*x^2) + log(x + log(x))*log(16
*log(x + log(x))^2)^3*(x + log(x)))/(log(x + log(x))*log(16*log(x + log(x))^2)^3*(x + log(x))),x)

[Out]

x + log(x + log(x))*((6*x + 6*x^2)/(x + 1) - (3*x + 3*x^2)/(x + 1) + (3*x + 12*x^2 + 6*x^3)/(x + 1) - (6*x + 1
5*x^2 + 9*x^3)/(x + 1) + log(x)*((6*x + 3*x^2)/(x + 1) - (6*x + 6*x^2)/(x + 1))) - (12*x^2 - (6*x^2*log(x + lo
g(x))*log(16*log(x + log(x))^2)*(x + log(x)))/(x + 1))/log(16*log(x + log(x))^2)^2 - ((6*x^2*log(x + log(x))*(
x + log(x)))/(x + 1) - (3*x^2*log(x + log(x))*log(16*log(x + log(x))^2)*(x + log(x))*(2*x + log(x + log(x)) +
2*x^2*log(x + log(x)) + 2*log(x + log(x))*log(x) + x^2 + 4*x*log(x + log(x)) + x*log(x + log(x))*log(x) + 1))/
(x + 1)^3)/log(16*log(x + log(x))^2) - log(x + log(x))^2*((84*x + 312*x^2 + 432*x^3 + 264*x^4 + 60*x^5)/(3*x +
3*x^2 + x^3 + 1) - ((213*x)/2 + (639*x^2)/2 + (831*x^3)/2 + 243*x^4 + 54*x^5 + 21/2)/(3*x + 3*x^2 + x^3 + 1)
- (18*x + 6*x^2 + 50/3)/(3*x + 3*x^2 + x^3 + 1) - log(x)^2*((9*x + 3*x^2 + 3)/(3*x + 3*x^2 + x^3 + 1) - 3) + l
og(x)*((3*(16*x + 13*x^2 + (11*x^3)/3 + 22/3))/(3*x + 3*x^2 + x^3 + 1) - (18*x + 6*x^2 + 14)/(3*x + 3*x^2 + x^
3 + 1) + (36*x + 108*x^2 + 108*x^3 + 36*x^4)/(3*x + 3*x^2 + x^3 + 1) - (120*x + 192*x^2 + 119*x^3 + 27*x^4 + 2
6)/(3*x + 3*x^2 + x^3 + 1) + 18) + (18*(8*x + (13*x^2)/2 + (11*x^3)/6 + 11/3))/(3*x + 3*x^2 + x^3 + 1) + (3*((
209*x)/6 + (155*x^2)/6 + (121*x^3)/18 + 109/6))/(3*x + 3*x^2 + x^3 + 1) - (36*x + 108*x^2 + 108*x^3 + 36*x^4)/
(3*x + 3*x^2 + x^3 + 1) + (108*x + 324*x^2 + 324*x^3 + 108*x^4)/(3*x + 3*x^2 + x^3 + 1) - (613*x + 730*x^2 + (
1181*x^3)/3 + 81*x^4 + 613/3)/(3*x + 3*x^2 + x^3 + 1) + 111)