\(\int \frac {-2 x \log (x)+(3 x+x \log ^2(x)) \log (3+\log ^2(x))+(6+2 \log ^2(x)) \log ^2(3+\log ^2(x))+((-3 x-x \log ^2(x)) \log (3+\log ^2(x))+(-15+3 \log (5)+3 \log (\frac {3}{4 x^2})+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log ^2(x)) \log ^2(3+\log ^2(x))) \log (\frac {-x+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log (3+\log ^2(x))}{\log (3+\log ^2(x))})}{((-3 x-x \log ^2(x)) \log (3+\log ^2(x))+(-15+3 \log (5)+3 \log (\frac {3}{4 x^2})+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log ^2(x)) \log ^2(3+\log ^2(x))) \log ^2(\frac {-x+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log (3+\log ^2(x))}{\log (3+\log ^2(x))})} \, dx\) [2536]

Optimal result

Integrand size = 244, antiderivative size = 29 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \]

[Out]

x/ln(ln(5)+ln(3/4/x^2)-5-x/ln(ln(x)^2+3))

Rubi [F]

\[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx \]

[In]

Int[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log[x]^2)*Log[3 + Log[x]^2]^2 + ((-3*x - x*Lo
g[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log[
3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]])/(((-3*x - x*L
og[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log
[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]]^2),x]

[Out]

-3*Defer[Int][x/((3 + Log[x]^2)*(x + 5*Log[3 + Log[x]^2] - Log[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 + Log[15/
(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] - Defer[Int][(x*Log[x]^2)/((3 + Log[x]^2)*(x + 5*Log[3 + Log[x]^2] - Lo
g[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] - 2*Defer[Int][(x*Log[
x])/((3 + Log[x]^2)*Log[3 + Log[x]^2]*(-x - 5*Log[3 + Log[x]^2] + Log[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 +
Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] + 6*Defer[Int][Log[3 + Log[x]^2]/((3 + Log[x]^2)*(-x - 5*Log[3 +
Log[x]^2] + Log[15/(4*x^2)]*Log[3 + Log[x]^2])*Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] + 2*Def
er[Int][(Log[x]^2*Log[3 + Log[x]^2])/((3 + Log[x]^2)*(-x - 5*Log[3 + Log[x]^2] + Log[15/(4*x^2)]*Log[3 + Log[x
]^2])*Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]^2), x] + Defer[Int][Log[-5 + Log[15/(4*x^2)] - x/Log[3 +
Log[x]^2]]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \log (x)-x \left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right )-2 \left (3+\log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (x-\left (-5+\log \left (\frac {15}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )\right ) \log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (x-\left (-5+\log \left (\frac {15}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx \\ & = \int \left (\frac {-2 x \log (x)+3 x \log \left (3+\log ^2(x)\right )+x \log ^2(x) \log \left (3+\log ^2(x)\right )+6 \log ^2\left (3+\log ^2(x)\right )+2 \log ^2(x) \log ^2\left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}\right ) \, dx \\ & = \int \frac {-2 x \log (x)+3 x \log \left (3+\log ^2(x)\right )+x \log ^2(x) \log \left (3+\log ^2(x)\right )+6 \log ^2\left (3+\log ^2(x)\right )+2 \log ^2(x) \log ^2\left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx \\ & = \int \left (-\frac {3 x}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}-\frac {x \log ^2(x)}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}-\frac {2 x \log (x)}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {6 \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}+\frac {2 \log ^2(x) \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )}\right ) \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx \\ & = -\left (2 \int \frac {x \log (x)}{\left (3+\log ^2(x)\right ) \log \left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx\right )+2 \int \frac {\log ^2(x) \log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx-3 \int \frac {x}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+6 \int \frac {\log \left (3+\log ^2(x)\right )}{\left (3+\log ^2(x)\right ) \left (-x-5 \log \left (3+\log ^2(x)\right )+\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx-\int \frac {x \log ^2(x)}{\left (3+\log ^2(x)\right ) \left (x+5 \log \left (3+\log ^2(x)\right )-\log \left (\frac {15}{4 x^2}\right ) \log \left (3+\log ^2(x)\right )\right ) \log ^2\left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx+\int \frac {1}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \]

[In]

Integrate[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log[x]^2)*Log[3 + Log[x]^2]^2 + ((-3*x
- x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2
)*Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]])/(((-3*x
- x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^
2)*Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/Log[3 + Log[x]^2]]^2),x]

[Out]

x/Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^2]]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.06 (sec) , antiderivative size = 1746, normalized size of antiderivative = 60.21

\[\text {Expression too large to display}\]

[In]

int(((((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15)*ln(ln(x)^2+3)^2+(-x*ln(x)^2-3*x)*ln(ln(x)^2+3))
*ln(((ln(3/4/x^2)+ln(5)-5)*ln(ln(x)^2+3)-x)/ln(ln(x)^2+3))+(2*ln(x)^2+6)*ln(ln(x)^2+3)^2+(x*ln(x)^2+3*x)*ln(ln
(x)^2+3)-2*x*ln(x))/(((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15)*ln(ln(x)^2+3)^2+(-x*ln(x)^2-3*x)
*ln(ln(x)^2+3))/ln(((ln(3/4/x^2)+ln(5)-5)*ln(ln(x)^2+3)-x)/ln(ln(x)^2+3))^2,x)

[Out]

2*I*x/(-Pi*csgn(I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-2*I*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2+I*ln(l
n(x)^2+3)*Pi*csgn(I*x^2)^3-4*ln(ln(x)^2+3)*ln(x)+2*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*ln(3/4)-10*ln(ln(x)^2+3
)-2*x)*csgn(I/ln(ln(x)^2+3))*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn
(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*
I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))+Pi*csgn(I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-2*I*ln(ln(x
)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2+I*ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3-4*ln(ln(x)^2+3)*ln(x)+2*ln(ln(x)^2+3)*ln(5)
+2*ln(ln(x)^2+3)*ln(3/4)-10*ln(ln(x)^2+3)-2*x)*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*P
i*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(
x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))^2-Pi*csgn(I/ln(ln(x)^2+3))*csgn(I/ln(ln(x)^2
+3)*(-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^
2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5))
)^2+Pi*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3
)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+
2*I*ln(ln(x)^2+3)*ln(5)))^3-Pi*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*cs
gn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-
4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))*csgn((-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2
*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/
4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5))/ln(ln(x)^2+3))^2+Pi*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(l
n(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi
*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))*csgn((-2*I*x-10*I*l
n(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)
*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5))/ln(ln(x)^2+3))+Pi
*csgn((-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*
x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5
))/ln(ln(x)^2+3))^3-Pi*csgn((-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3
)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+
2*I*ln(ln(x)^2+3)*ln(5))/ln(ln(x)^2+3))^2+Pi-2*I*ln(2)-2*I*ln(ln(ln(x)^2+3))+2*I*ln(-2*I*x-10*I*ln(ln(x)^2+3)-
ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^
2)^3+2*I*ln(ln(x)^2+3)*(ln(3)-2*ln(2))-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (27) = 54\).

Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (\frac {{\left (\log \left (5\right ) + \log \left (\frac {3}{4 \, x^{2}}\right ) - 5\right )} \log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right ) - x}{\log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right )}\right )} \]

[In]

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*
log(log(x)^2+3))*log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+
3)^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-1
5)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(lo
g(x)^2+3))^2,x, algorithm="fricas")

[Out]

x/log(((log(5) + log(3/4/x^2) - 5)*log(1/4*log(3/4)^2 - 1/2*log(3/4)*log(3/4/x^2) + 1/4*log(3/4/x^2)^2 + 3) -
x)/log(1/4*log(3/4)^2 - 1/2*log(3/4)*log(3/4/x^2) + 1/4*log(3/4/x^2)^2 + 3))

Sympy [A] (verification not implemented)

Time = 107.45 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log {\left (\frac {- x + \left (- 2 \log {\left (x \right )} - 5 + \log {\left (\frac {3}{4} \right )} + \log {\left (5 \right )}\right ) \log {\left (\log {\left (x \right )}^{2} + 3 \right )}}{\log {\left (\log {\left (x \right )}^{2} + 3 \right )}} \right )}} \]

[In]

integrate(((((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3*ln(3/4/x**2)+3*ln(5)-15)*ln(ln(x)**2+3)**2+(-x*ln(x)**2-3*x)*ln
(ln(x)**2+3))*ln(((ln(3/4/x**2)+ln(5)-5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))+(2*ln(x)**2+6)*ln(ln(x)**2+3)**2+(x
*ln(x)**2+3*x)*ln(ln(x)**2+3)-2*x*ln(x))/(((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3*ln(3/4/x**2)+3*ln(5)-15)*ln(ln(x)
**2+3)**2+(-x*ln(x)**2-3*x)*ln(ln(x)**2+3))/ln(((ln(3/4/x**2)+ln(5)-5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))**2,x)

[Out]

x/log((-x + (-2*log(x) - 5 + log(3/4) + log(5))*log(log(x)**2 + 3))/log(log(x)**2 + 3))

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left ({\left (\log \left (5\right ) + \log \left (3\right ) - 2 \, \log \left (2\right ) - 5\right )} \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \]

[In]

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*
log(log(x)^2+3))*log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+
3)^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-1
5)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(lo
g(x)^2+3))^2,x, algorithm="maxima")

[Out]

x/(log((log(5) + log(3) - 2*log(2) - 5)*log(log(x)^2 + 3) - 2*log(log(x)^2 + 3)*log(x) - x) - log(log(log(x)^2
+ 3)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (27) = 54\).

Time = 2.71 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (\log \left (5\right ) \log \left (\log \left (x\right )^{2} + 3\right ) + \log \left (3\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (2\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x - 5 \, \log \left (\log \left (x\right )^{2} + 3\right )\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \]

[In]

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*
log(log(x)^2+3))*log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+
3)^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-1
5)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(lo
g(x)^2+3))^2,x, algorithm="giac")

[Out]

x/(log(log(5)*log(log(x)^2 + 3) + log(3)*log(log(x)^2 + 3) - 2*log(2)*log(log(x)^2 + 3) - 2*log(log(x)^2 + 3)*
log(x) - x - 5*log(log(x)^2 + 3)) - log(log(log(x)^2 + 3)))

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\int \frac {\ln \left (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )-2\,x\,\ln \left (x\right )+\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )+{\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (2\,{\ln \left (x\right )}^2+6\right )}{{\ln \left (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )}^2\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )} \,d x \]

[In]

int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))*(log(log(x)^2 + 3)^2*(3*log
(5) + 3*log(3/(4*x^2)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))
- 2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2*log(x)^2 + 6))/(log(-(x - log(log
(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2
)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))),x)

[Out]

int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))*(log(log(x)^2 + 3)^2*(3*log
(5) + 3*log(3/(4*x^2)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))
- 2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2*log(x)^2 + 6))/(log(-(x - log(log
(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2
)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2))), x)