3.33.51 \(\int \frac {-4+6 x-2 x^2+(2 x-x^2+x \log (x)) \log (24-12 x+12 \log (x))+(-4+4 x-x^2+(-2+x) \log (x)) \log (24-12 x+12 \log (x)) \log (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))})}{(-20 x^2+20 x^3-5 x^4+(-10 x^2+5 x^3) \log (x)) \log (24-12 x+12 \log (x))+(-40 x+40 x^2-10 x^3+(-20 x+10 x^2) \log (x)) \log (24-12 x+12 \log (x)) \log (\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))})+(-20+20 x-5 x^2+(-10+5 x) \log (x)) \log (24-12 x+12 \log (x)) \log ^2(\frac {1}{(-8+4 x) \log ^2(24-12 x+12 \log (x))})} \, dx\) [3251]

Optimal. Leaf size=31 \[ \frac {x}{5 \left (x+\log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )} \]

[Out]

x/(5*x+5*ln(1/4/ln(12*ln(x)-12*x+24)^2/(-2+x)))

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Rubi [A]
time = 1.68, antiderivative size = 36, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, integrand size = 237, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6820, 12, 6843, 32} \begin {gather*} -\frac {1}{5 \left (\frac {x}{\log \left (-\frac {1}{4 (2-x) \log ^2(12 (-x+\log (x)+2))}\right )}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 6*x - 2*x^2 + (2*x - x^2 + x*Log[x])*Log[24 - 12*x + 12*Log[x]] + (-4 + 4*x - x^2 + (-2 + x)*Log[x])
*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)])/((-20*x^2 + 20*x^3 - 5*x^4 + (-1
0*x^2 + 5*x^3)*Log[x])*Log[24 - 12*x + 12*Log[x]] + (-40*x + 40*x^2 - 10*x^3 + (-20*x + 10*x^2)*Log[x])*Log[24
- 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)] + (-20 + 20*x - 5*x^2 + (-10 + 5*x)*Log[
x])*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)]^2),x]

[Out]

-1/5*1/(1 + x/Log[-1/4*1/((2 - x)*Log[12*(2 - x + Log[x])]^2)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6843

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x] - q*v*D[w
, x])]}, Dist[c*p, Subst[Int[(b + a*x^p)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}
, x] && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (2-3 x+x^2\right )+(-2+x-\log (x)) \log (12 (2-x+\log (x))) \left (x+(-2+x) \log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )}{5 (2-x) (2-x+\log (x)) \log (12 (2-x+\log (x))) \left (x+\log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {2 \left (2-3 x+x^2\right )+(-2+x-\log (x)) \log (12 (2-x+\log (x))) \left (x+(-2+x) \log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )}{(2-x) (2-x+\log (x)) \log (12 (2-x+\log (x))) \left (x+\log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )^2} \, dx\\ &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\frac {x}{\log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )}\right )\\ &=-\frac {1}{5 \left (1+\frac {x}{\log \left (-\frac {1}{4 (2-x) \log ^2(12 (2-x+\log (x)))}\right )}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.11, size = 31, normalized size = 1.00 \begin {gather*} \frac {x}{5 \left (x+\log \left (\frac {1}{4 (-2+x) \log ^2(12 (2-x+\log (x)))}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 6*x - 2*x^2 + (2*x - x^2 + x*Log[x])*Log[24 - 12*x + 12*Log[x]] + (-4 + 4*x - x^2 + (-2 + x)*L
og[x])*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)])/((-20*x^2 + 20*x^3 - 5*x^4
+ (-10*x^2 + 5*x^3)*Log[x])*Log[24 - 12*x + 12*Log[x]] + (-40*x + 40*x^2 - 10*x^3 + (-20*x + 10*x^2)*Log[x])*
Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)] + (-20 + 20*x - 5*x^2 + (-10 + 5*x
)*Log[x])*Log[24 - 12*x + 12*Log[x]]*Log[1/((-8 + 4*x)*Log[24 - 12*x + 12*Log[x]]^2)]^2),x]

[Out]

x/(5*(x + Log[1/(4*(-2 + x)*Log[12*(2 - x + Log[x])]^2)]))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 35.91, size = 732, normalized size = 23.61

method	result	size
risch	\(\frac {2 x}{5 \left (i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (x -2\right )}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (x -2\right )}\right ) \mathrm {csgn}\left (\frac {i}{x -2}\right )+i \pi \mathrm {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )\right )^{2} \mathrm {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )\right ) \mathrm {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i \ln \left (12 \ln \left (x \right )-12 x +24\right )^{2}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (x -2\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{x -2}\right )-i \pi \mathrm {csgn}\left (\frac {i}{\ln \left (12 \ln \left (x \right )-12 x +24\right )^{2} \left (x -2\right )}\right )^{3}-4 \ln \left (2\right )+2 x -2 \ln \left (x -2\right )-4 \ln \left (\ln \left (12 \ln \left (x \right )-12 x +24\right )\right )\right )}\)	\(283\)
default	\(\frac {2 \left (-x^{2} \ln \left (3\right ) \ln \left (x \right )+x^{3} \ln \left (3\right )-2 x^{2} \ln \left (2\right ) \ln \left (x \right )+2 x^{3} \ln \left (2\right )-x^{2} \ln \left (2+\ln \left (x \right )-x \right ) \ln \left (x \right )+x^{3} \ln \left (2+\ln \left (x \right )-x \right )+3 x \ln \left (3\right ) \ln \left (x \right )-5 x^{2} \ln \left (3\right )+6 x \ln \left (2\right ) \ln \left (x \right )-10 x^{2} \ln \left (2\right )+3 x \ln \left (x \right ) \ln \left (2+\ln \left (x \right )-x \right )-5 x^{2} \ln \left (2+\ln \left (x \right )-x \right )+6 x \ln \left (3\right )+12 x \ln \left (2\right )-2 x^{2}+6 x \ln \left (2+\ln \left (x \right )-x \right )+6 x -4\right ) x \left (2 \ln \left (3\right )+4 \ln \left (2\right )+2 \ln \left (2+\ln \left (x \right )-x \right )\right )}{5 \left (\ln \left (3\right )+2 \ln \left (2\right )+\ln \left (2+\ln \left (x \right )-x \right )\right ) \left (-8-2 x^{2} \ln \left (3\right ) \ln \left (x \right )+2 x^{3} \ln \left (2+\ln \left (x \right )-x \right )+12 x -4 x^{2}+12 x \ln \left (3\right )-10 x^{2} \ln \left (2+\ln \left (x \right )-x \right )+12 x \ln \left (2+\ln \left (x \right )-x \right )+24 x \ln \left (2\right )-20 x^{2} \ln \left (2\right )+4 x^{3} \ln \left (2\right )+6 x \ln \left (x \right ) \ln \left (2+\ln \left (x \right )-x \right )-4 x^{2} \ln \left (2\right ) \ln \left (x \right )+12 x \ln \left (2\right ) \ln \left (x \right )-2 x^{2} \ln \left (2+\ln \left (x \right )-x \right ) \ln \left (x \right )+6 x \ln \left (3\right ) \ln \left (x \right )+2 x^{3} \ln \left (3\right )-10 x^{2} \ln \left (3\right )\right ) \left (-i \pi \,\mathrm {csgn}\left (\frac {i}{x -2}\right ) \mathrm {csgn}\left (\frac {i}{\left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\left (x -2\right ) \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x -2}\right ) \mathrm {csgn}\left (\frac {i}{\left (x -2\right ) \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{\left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{\left (x -2\right ) \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )^{2}+i \pi \mathrm {csgn}\left (i \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )\right )^{2} \mathrm {csgn}\left (i \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )\right ) \mathrm {csgn}\left (i \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i}{\left (x -2\right ) \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i}{\left (x -2\right ) \left (2 i \ln \left (12\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )^{2}}\right )^{2}+2 i \pi +2 x -2 \ln \left (x -2\right )-4 \ln \left (2 i \left (\ln \left (3\right )+2 \ln \left (2\right )\right )+2 i \ln \left (2+\ln \left (x \right )-x \right )\right )\right )}\)	\(732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-2)*ln(x)-x^2+4*x-4)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12*ln(x)-12*x+24)^2)+(x*ln(x)-x^2+2*x)*ln(12
*ln(x)-12*x+24)-2*x^2+6*x-4)/(((5*x-10)*ln(x)-5*x^2+20*x-20)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12*ln(x)-12*
x+24)^2)^2+((10*x^2-20*x)*ln(x)-10*x^3+40*x^2-40*x)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12*ln(x)-12*x+24)^2)+
((5*x^3-10*x^2)*ln(x)-5*x^4+20*x^3-20*x^2)*ln(12*ln(x)-12*x+24)),x,method=_RETURNVERBOSE)

[Out]

2/5*(-x^2*ln(3)*ln(x)+x^3*ln(3)-2*x^2*ln(2)*ln(x)+2*x^3*ln(2)-x^2*ln(2+ln(x)-x)*ln(x)+x^3*ln(2+ln(x)-x)+3*x*ln
(3)*ln(x)-5*x^2*ln(3)+6*x*ln(2)*ln(x)-10*x^2*ln(2)+3*x*ln(x)*ln(2+ln(x)-x)-5*x^2*ln(2+ln(x)-x)+6*x*ln(3)+12*x*
ln(2)-2*x^2+6*x*ln(2+ln(x)-x)+6*x-4)*x*(2*ln(3)+4*ln(2)+2*ln(2+ln(x)-x))/(ln(3)+2*ln(2)+ln(2+ln(x)-x))/(-8-2*x
^2*ln(3)*ln(x)+2*x^3*ln(2+ln(x)-x)+12*x-4*x^2+12*x*ln(3)-10*x^2*ln(2+ln(x)-x)+12*x*ln(2+ln(x)-x)+24*x*ln(2)-20
*x^2*ln(2)+4*x^3*ln(2)+6*x*ln(x)*ln(2+ln(x)-x)-4*x^2*ln(2)*ln(x)+12*x*ln(2)*ln(x)-2*x^2*ln(2+ln(x)-x)*ln(x)+6*
x*ln(3)*ln(x)+2*x^3*ln(3)-10*x^2*ln(3))/(-I*Pi*csgn(I/(x-2))*csgn(I/(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)*csgn(I/(
x-2)/(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)+I*Pi*csgn(I/(x-2))*csgn(I/(x-2)/(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)^2+I*P
i*csgn(I/(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)*csgn(I/(x-2)/(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)^2+I*Pi*csgn(I*(2*I*l
n(12)+2*I*ln(2+ln(x)-x)))^2*csgn(I*(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)-2*I*Pi*csgn(I*(2*I*ln(12)+2*I*ln(2+ln(x)-
x)))*csgn(I*(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)^2+I*Pi*csgn(I*(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)^3+I*Pi*csgn(I/(x
-2)/(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)^3-2*I*Pi*csgn(I/(x-2)/(2*I*ln(12)+2*I*ln(2+ln(x)-x))^2)^2+2*I*Pi+2*x-2*l
n(x-2)-4*ln(2*I*(ln(3)+2*ln(2))+2*I*ln(2+ln(x)-x)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+x)*log(x)-x^2+4*x-4)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/log(12*log(x)-12*x+24)^2)+(x*log(x)-
x^2+2*x)*log(12*log(x)-12*x+24)-2*x^2+6*x-4)/(((5*x-10)*log(x)-5*x^2+20*x-20)*log(12*log(x)-12*x+24)*log(1/(4*
x-8)/log(12*log(x)-12*x+24)^2)^2+((10*x^2-20*x)*log(x)-10*x^3+40*x^2-40*x)*log(12*log(x)-12*x+24)*log(1/(4*x-8
)/log(12*log(x)-12*x+24)^2)+((5*x^3-10*x^2)*log(x)-5*x^4+20*x^3-20*x^2)*log(12*log(x)-12*x+24)),x, algorithm="
maxima")

[Out]

-1/5*((-I*pi*(log(3) + 2*log(2)) - log(3)^2 - 4*log(3)*log(2) - 4*log(2)^2)*x^4 + (I*pi*(5*log(3) + 10*log(2)
+ 2) + 5*log(3)^2 + 4*(5*log(3) + 1)*log(2) + 20*log(2)^2 + 2*log(3))*x^3 - 6*(I*pi*(log(3) + 2*log(2) + 1) +
log(3)^2 + 2*(2*log(3) + 1)*log(2) + 4*log(2)^2 + log(3))*x^2 - 4*(-I*pi - log(3) - 2*log(2))*x - (x^4*(log(3)
+ 2*log(2)) - x^3*(5*log(3) + 10*log(2) + 2) + 6*x^2*(log(3) + 2*log(2) + 1) - (x^3*(log(3) + 2*log(2)) - 3*x
^2*(log(3) + 2*log(2)))*log(x) - 4*x)*log(x - log(x) - 2) + ((I*pi*(log(3) + 2*log(2)) + log(3)^2 + 4*log(3)*l
og(2) + 4*log(2)^2)*x^3 - 3*(I*pi*(log(3) + 2*log(2)) + log(3)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x^2)*log(x) +
((-I*pi - log(3) - 2*log(2))*x^4 - 5*(-I*pi - log(3) - 2*log(2))*x^3 - 6*(I*pi + log(3) + 2*log(2))*x^2 - (x^
4 - 5*x^3 + 6*x^2 - (x^3 - 3*x^2)*log(x))*log(x - log(x) - 2) + ((I*pi + log(3) + 2*log(2))*x^3 - 3*(I*pi + lo
g(3) + 2*log(2))*x^2)*log(x))*log(-x + log(x) + 2))/((I*pi*(log(3) + 2*log(2)) + log(3)^2 + 4*log(3)*log(2) +
4*log(2)^2)*x^4 - (4*(2*log(3) + 5)*log(2)^2 + 8*log(2)^3 + I*pi*(2*(log(3) + 5)*log(2) + 4*log(2)^2 + 5*log(3
)) + 5*log(3)^2 + 2*(log(3)^2 + 10*log(3) + 2)*log(2) + 2*log(3))*x^3 + 2*(4*(5*log(3) + 4)*log(2)^2 + 20*log(
2)^3 + I*pi*((5*log(3) + 6)*log(2) + 10*log(2)^2 + 3*log(3)) + 3*log(3)^2 + (5*log(3)^2 + 14*log(3) + 6)*log(2
) + 3*log(3))*x^2 - 4*(6*(2*log(3) + 1)*log(2)^2 + 12*log(2)^3 + 3*I*pi*(log(3)*log(2) + 2*log(2)^2) + (3*log(
3)^2 + 3*log(3) + 2)*log(2) + log(3))*x + 8*log(3)*log(2) + 16*log(2)^2 - 2*((I*pi*(log(3) + 2*log(2)) + log(3
)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x^3 + (-5*I*pi*(log(3) + 2*log(2)) - 5*log(3)^2 - 4*(5*log(3) + 1)*log(2)
- 20*log(2)^2 - 2*log(3))*x^2 + 6*(I*pi*(log(3) + 2*log(2)) + log(3)^2 + 2*(2*log(3) + 1)*log(2) + 4*log(2)^2
+ log(3))*x + (x^3*(log(3) + 2*log(2)) - 5*x^2*(log(3) + 2*log(2)) + 6*x*(log(3) + 2*log(2)) - (x^2*(log(3) +
2*log(2)) - 3*x*(log(3) + 2*log(2)))*log(x))*log(x - log(x) - 2) + ((-I*pi*(log(3) + 2*log(2)) - log(3)^2 - 4*
log(3)*log(2) - 4*log(2)^2)*x^2 + 3*(I*pi*(log(3) + 2*log(2)) + log(3)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x)*lo
g(x) + ((I*pi + log(3) + 2*log(2))*x^3 + (-5*I*pi - 5*log(3) - 10*log(2) - 2)*x^2 + 6*(I*pi + log(3) + 2*log(2
) + 1)*x + (x^3 - 5*x^2 - (x^2 - 3*x)*log(x) + 6*x)*log(x - log(x) - 2) + ((-I*pi - log(3) - 2*log(2))*x^2 + 3
*(I*pi + log(3) + 2*log(2))*x)*log(x) - 4)*log(-x + log(x) + 2) - 4*log(3) - 8*log(2))*log(I*pi + log(3) + 2*l
og(2) + log(x - log(x) - 2)) + (x^4*(log(3) + 2*log(2)) - (2*(log(3) + 5)*log(2) + 4*log(2)^2 + 5*log(3))*x^3
+ 2*((5*log(3) + 6)*log(2) + 10*log(2)^2 + 3*log(3))*x^2 - 12*(log(3)*log(2) + 2*log(2)^2)*x - (x^3*(log(3) +
2*log(2)) - 5*x^2*(log(3) + 2*log(2)) + 6*x*(log(3) + 2*log(2)) - (x^2*(log(3) + 2*log(2)) - 3*x*(log(3) + 2*l
og(2)))*log(x))*log(x - 2) - (x^3*(log(3) + 2*log(2)) - (2*(log(3) + 3)*log(2) + 4*log(2)^2 + 3*log(3))*x^2 +
6*(log(3)*log(2) + 2*log(2)^2)*x)*log(x))*log(x - log(x) - 2) + ((-I*pi*(log(3) + 2*log(2)) - log(3)^2 - 4*log
(3)*log(2) - 4*log(2)^2)*x^3 + (5*I*pi*(log(3) + 2*log(2)) + 5*log(3)^2 + 4*(5*log(3) + 1)*log(2) + 20*log(2)^
2 + 2*log(3))*x^2 - 6*(I*pi*(log(3) + 2*log(2)) + log(3)^2 + 2*(2*log(3) + 1)*log(2) + 4*log(2)^2 + log(3))*x
+ ((I*pi*(log(3) + 2*log(2)) + log(3)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x^2 - 3*(I*pi*(log(3) + 2*log(2)) + lo
g(3)^2 + 4*log(3)*log(2) + 4*log(2)^2)*x)*log(x) + 4*log(3) + 8*log(2))*log(x - 2) + ((-I*pi*(log(3) + 2*log(2
)) - log(3)^2 - 4*log(3)*log(2) - 4*log(2)^2)*x^3 + (4*(2*log(3) + 3)*log(2)^2 + 8*log(2)^3 + I*pi*(2*(log(3)
+ 3)*log(2) + 4*log(2)^2 + 3*log(3)) + 3*log(3)^2 + 2*(log(3)^2 + 6*log(3))*log(2))*x^2 - 6*(log(3)^2*log(2) +
4*log(3)*log(2)^2 + 4*log(2)^3 + I*pi*(log(3)*log(2) + 2*log(2)^2))*x)*log(x) + ((I*pi + log(3) + 2*log(2))*x
^4 + (-I*pi*(2*log(2) + 5) - 2*(log(3) + 5)*log(2) - 4*log(2)^2 - 5*log(3) - 2)*x^3 - 2*(-I*pi*(5*log(2) + 3)
- (5*log(3) + 8)*log(2) - 10*log(2)^2 - 3*log(3) - 3)*x^2 - 4*(3*I*pi*log(2) + 3*(log(3) + 1)*log(2) + 6*log(2
)^2 + 1)*x + (x^4 - x^3*(2*log(2) + 5) + 2*x^2*(5*log(2) + 3) - 12*x*log(2) - (x^3 - 5*x^2 - (x^2 - 3*x)*log(x
) + 6*x)*log(x - 2) - (x^3 - x^2*(2*log(2) + 3) + 6*x*log(2))*log(x))*log(x - log(x) - 2) + ((-I*pi - log(3) -
2*log(2))*x^3 + (5*I*pi + 5*log(3) + 10*log(2) + 2)*x^2 - 6*(I*pi + log(3) + 2*log(2) + 1)*x + ((I*pi + log(3
) + 2*log(2))*x^2 - 3*(I*pi + log(3) + 2*log(2))*x)*log(x) + 4)*log(x - 2) + ((-I*pi - log(3) - 2*log(2))*x^3
+ (I*pi*(2*log(2) + 3) + 2*(log(3) + 3)*log(2) + 4*log(2)^2 + 3*log(3))*x^2 - 6*(I*pi*log(2) + log(3)*log(2) +
2*log(2)^2)*x)*log(x) + 8*log(2))*log(-x + log(x) + 2)) - 1/5*integrate(-2*((pi^2 - 2*I*pi*(log(3) + 2*log(2)
))*x^6 + (pi^2*(log(3) + 2*log(2) - 9) - I*pi*(log(3)^2 + 4*(log(3) - 8)*log(2) + 4*log(2)^2 - 16*log(3) - 2))
*x^5 - (pi^2*(4*log(3) + 8*log(2) - 31) - 2*I*pi*(2*log(3)^2 + 2*(4*log(3) - 23)*log(2) + 8*log(2)^2 - 23*log(
3) - 7))*x^4 + (pi^2*(log(3) + 2*log(2) - 51) - I*pi*(log(3)^2 + 4*(log(3) - 26)*log(2) + 4*log(2)^2 - 52*log(
3) - 38))*x^3 + 2*(2*pi^2*(3*log(3) + 6*log(2) ...

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Fricas [A]
time = 0.45, size = 27, normalized size = 0.87 \begin {gather*} \frac {x}{5 \, {\left (x + \log \left (\frac {1}{4 \, {\left (x - 2\right )} \log \left (-12 \, x + 12 \, \log \left (x\right ) + 24\right )^{2}}\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+x)*log(x)-x^2+4*x-4)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/log(12*log(x)-12*x+24)^2)+(x*log(x)-
x^2+2*x)*log(12*log(x)-12*x+24)-2*x^2+6*x-4)/(((5*x-10)*log(x)-5*x^2+20*x-20)*log(12*log(x)-12*x+24)*log(1/(4*
x-8)/log(12*log(x)-12*x+24)^2)^2+((10*x^2-20*x)*log(x)-10*x^3+40*x^2-40*x)*log(12*log(x)-12*x+24)*log(1/(4*x-8
)/log(12*log(x)-12*x+24)^2)+((5*x^3-10*x^2)*log(x)-5*x^4+20*x^3-20*x^2)*log(12*log(x)-12*x+24)),x, algorithm="
fricas")

[Out]

1/5*x/(x + log(1/4/((x - 2)*log(-12*x + 12*log(x) + 24)^2)))

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Sympy [A]
time = 1.05, size = 27, normalized size = 0.87 \begin {gather*} \frac {x}{5 x + 5 \log {\left (\frac {1}{\left (4 x - 8\right ) \log {\left (- 12 x + 12 \log {\left (x \right )} + 24 \right )}^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+x)*ln(x)-x**2+4*x-4)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12*ln(x)-12*x+24)**2)+(x*ln(x)-x**2+
2*x)*ln(12*ln(x)-12*x+24)-2*x**2+6*x-4)/(((5*x-10)*ln(x)-5*x**2+20*x-20)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(
12*ln(x)-12*x+24)**2)**2+((10*x**2-20*x)*ln(x)-10*x**3+40*x**2-40*x)*ln(12*ln(x)-12*x+24)*ln(1/(4*x-8)/ln(12*l
n(x)-12*x+24)**2)+((5*x**3-10*x**2)*ln(x)-5*x**4+20*x**3-20*x**2)*ln(12*ln(x)-12*x+24)),x)

[Out]

x/(5*x + 5*log(1/((4*x - 8)*log(-12*x + 12*log(x) + 24)**2)))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1588 vs. \(2 (27) = 54\).
time = 150.19, size = 1588, normalized size = 51.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+x)*log(x)-x^2+4*x-4)*log(12*log(x)-12*x+24)*log(1/(4*x-8)/log(12*log(x)-12*x+24)^2)+(x*log(x)-
x^2+2*x)*log(12*log(x)-12*x+24)-2*x^2+6*x-4)/(((5*x-10)*log(x)-5*x^2+20*x-20)*log(12*log(x)-12*x+24)*log(1/(4*
x-8)/log(12*log(x)-12*x+24)^2)^2+((10*x^2-20*x)*log(x)-10*x^3+40*x^2-40*x)*log(12*log(x)-12*x+24)*log(1/(4*x-8
)/log(12*log(x)-12*x+24)^2)+((5*x^3-10*x^2)*log(x)-5*x^4+20*x^3-20*x^2)*log(12*log(x)-12*x+24)),x, algorithm="
giac")

[Out]

1/5*(2*x^4*log(2)*log(-12*x + 12*log(x) + 24) - 2*x^3*log(2)*log(x)*log(-12*x + 12*log(x) + 24) + x^4*log(-3*x
+ 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - x^3*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24)
- 10*x^3*log(2)*log(-12*x + 12*log(x) + 24) + 6*x^2*log(2)*log(x)*log(-12*x + 12*log(x) + 24) - 5*x^3*log(-3*
x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + 3*x^2*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) +
24) - 2*x^3*log(-12*x + 12*log(x) + 24) + 12*x^2*log(2)*log(-12*x + 12*log(x) + 24) + 6*x^2*log(-3*x + 3*log(x
) + 6)*log(-12*x + 12*log(x) + 24) + 6*x^2*log(-12*x + 12*log(x) + 24) - 4*x*log(-12*x + 12*log(x) + 24))/(2*x
^4*log(2)*log(-12*x + 12*log(x) + 24) - 4*x^3*log(2)^2*log(-12*x + 12*log(x) + 24) - 2*x^3*log(2)*log(x*log(-1
2*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(-12*x + 12*log(x) + 24) - 2*x^3*log(2)*log(x)*l
og(-12*x + 12*log(x) + 24) + 4*x^2*log(2)^2*log(x)*log(-12*x + 12*log(x) + 24) + 2*x^2*log(2)*log(x*log(-12*x
+ 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(x)*log(-12*x + 12*log(x) + 24) + x^4*log(-3*x + 3*l
og(x) + 6)*log(-12*x + 12*log(x) + 24) - 2*x^3*log(2)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - x
^3*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(-3*x + 3*log(x) + 6)*log(-12*x +
12*log(x) + 24) - x^3*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + 2*x^2*log(2)*log(x)*log(-
3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + x^2*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*l
og(x) + 24)^2)*log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - 10*x^3*log(2)*log(-12*x + 12*log(
x) + 24) + 20*x^2*log(2)^2*log(-12*x + 12*log(x) + 24) + 10*x^2*log(2)*log(x*log(-12*x + 12*log(x) + 24)^2 - 2
*log(-12*x + 12*log(x) + 24)^2)*log(-12*x + 12*log(x) + 24) + 6*x^2*log(2)*log(x)*log(-12*x + 12*log(x) + 24)
- 12*x*log(2)^2*log(x)*log(-12*x + 12*log(x) + 24) - 6*x*log(2)*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-1
2*x + 12*log(x) + 24)^2)*log(x)*log(-12*x + 12*log(x) + 24) - 5*x^3*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*lo
g(x) + 24) + 10*x^2*log(2)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + 5*x^2*log(x*log(-12*x + 12*l
og(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) + 3*x^2*
log(x)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - 6*x*log(2)*log(x)*log(-3*x + 3*log(x) + 6)*log(-
12*x + 12*log(x) + 24) - 3*x*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(x)*log
(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - 4*x^3*log(2) + 8*x^2*log(2)^2 + 4*x^2*log(2)*log(x*log(-12
*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2) - 2*x^3*log(-3*x + 3*log(x) + 6) + 4*x^2*log(2)*log(
-3*x + 3*log(x) + 6) + 2*x^2*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(-3*x +
3*log(x) + 6) + 12*x^2*log(2)*log(-12*x + 12*log(x) + 24) - 24*x*log(2)^2*log(-12*x + 12*log(x) + 24) - 12*x*
log(2)*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(-12*x + 12*log(x) + 24) + 6*
x^2*log(-3*x + 3*log(x) + 6)*log(-12*x + 12*log(x) + 24) - 12*x*log(2)*log(-3*x + 3*log(x) + 6)*log(-12*x + 12
*log(x) + 24) - 6*x*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(-3*x + 3*log(x)
+ 6)*log(-12*x + 12*log(x) + 24) + 12*x^2*log(2) - 24*x*log(2)^2 - 12*x*log(2)*log(x*log(-12*x + 12*log(x) +
24)^2 - 2*log(-12*x + 12*log(x) + 24)^2) + 6*x^2*log(-3*x + 3*log(x) + 6) - 12*x*log(2)*log(-3*x + 3*log(x) +
6) - 6*x*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2)*log(-3*x + 3*log(x) + 6) - 8*x
*log(2) + 16*log(2)^2 + 8*log(2)*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-12*x + 12*log(x) + 24)^2) - 4*x*
log(-3*x + 3*log(x) + 6) + 8*log(2)*log(-3*x + 3*log(x) + 6) + 4*log(x*log(-12*x + 12*log(x) + 24)^2 - 2*log(-
12*x + 12*log(x) + 24)^2)*log(-3*x + 3*log(x) + 6))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {6\,x+\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (2\,x+x\,\ln \left (x\right )-x^2\right )-2\,x^2+\ln \left (\frac {1}{{\ln \left (12\,\ln \left (x\right )-12\,x+24\right )}^2\,\left (4\,x-8\right )}\right )\,\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (4\,x+\ln \left (x\right )\,\left (x-2\right )-x^2-4\right )-4}{-\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (20\,x+\ln \left (x\right )\,\left (5\,x-10\right )-5\,x^2-20\right )\,{\ln \left (\frac {1}{{\ln \left (12\,\ln \left (x\right )-12\,x+24\right )}^2\,\left (4\,x-8\right )}\right )}^2+\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (40\,x+\ln \left (x\right )\,\left (20\,x-10\,x^2\right )-40\,x^2+10\,x^3\right )\,\ln \left (\frac {1}{{\ln \left (12\,\ln \left (x\right )-12\,x+24\right )}^2\,\left (4\,x-8\right )}\right )+\ln \left (12\,\ln \left (x\right )-12\,x+24\right )\,\left (\ln \left (x\right )\,\left (10\,x^2-5\,x^3\right )+20\,x^2-20\,x^3+5\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + log(12*log(x) - 12*x + 24)*(2*x + x*log(x) - x^2) - 2*x^2 + log(1/(log(12*log(x) - 12*x + 24)^2*(4
*x - 8)))*log(12*log(x) - 12*x + 24)*(4*x + log(x)*(x - 2) - x^2 - 4) - 4)/(log(12*log(x) - 12*x + 24)*(log(x)
*(10*x^2 - 5*x^3) + 20*x^2 - 20*x^3 + 5*x^4) - log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))^2*log(12*log(x)
- 12*x + 24)*(20*x + log(x)*(5*x - 10) - 5*x^2 - 20) + log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))*log(12
*log(x) - 12*x + 24)*(40*x + log(x)*(20*x - 10*x^2) - 40*x^2 + 10*x^3)),x)

[Out]

-int((6*x + log(12*log(x) - 12*x + 24)*(2*x + x*log(x) - x^2) - 2*x^2 + log(1/(log(12*log(x) - 12*x + 24)^2*(4
*x - 8)))*log(12*log(x) - 12*x + 24)*(4*x + log(x)*(x - 2) - x^2 - 4) - 4)/(log(12*log(x) - 12*x + 24)*(log(x)
*(10*x^2 - 5*x^3) + 20*x^2 - 20*x^3 + 5*x^4) - log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))^2*log(12*log(x)
- 12*x + 24)*(20*x + log(x)*(5*x - 10) - 5*x^2 - 20) + log(1/(log(12*log(x) - 12*x + 24)^2*(4*x - 8)))*log(12
*log(x) - 12*x + 24)*(40*x + log(x)*(20*x - 10*x^2) - 40*x^2 + 10*x^3)), x)

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