\(\int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+(-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+(-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7) \log (3)) \log (x^2+2 x \log (3)+\log ^2(3))+(49152-73728 x+33792 x^2-4608 x^3+192 x^4+(36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+(36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5) \log (3)) \log (x^2+2 x \log (3)+\log ^2(3))) \log (\log (x^2+2 x \log (3)+\log ^2(3)))+(-24576+18432 x-1536 x^2+(-18432 x+16896 x^2-3456 x^3+192 x^4+(-18432+16896 x-3456 x^2+192 x^3) \log (3)) \log (x^2+2 x \log (3)+\log ^2(3))) \log ^2(\log (x^2+2 x \log (3)+\log ^2(3)))+(4096+(3072 x-512 x^2+(3072-512 x) \log (3)) \log (x^2+2 x \log (3)+\log ^2(3))) \log ^3(\log (x^2+2 x \log (3)+\log ^2(3)))}{(800 x+800 \log (3)) \log (x^2+2 x \log (3)+\log ^2(3))} \, dx\) [1533]

Optimal result

Integrand size = 385, antiderivative size = 29 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx=-5+\frac {1}{25} \left (-\frac {1}{4} (-4+x)^2+x+2 \log \left (\log \left ((x+\log (3))^2\right )\right )\right )^4 \] Output:

1/25*(2*ln(ln((ln(3)+x)^2))-1/4*(-4+x)^2+x)^4-5
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx=\frac {\left (16-12 x+x^2-8 \log \left (\log \left ((x+\log (3))^2\right )\right )\right )^4}{6400} \] Input:

Integrate[(-32768 + 73728*x - 61440*x^2 + 23040*x^3 - 3840*x^4 + 288*x^5 - 
 8*x^6 + (-24576*x + 59392*x^2 - 55296*x^3 + 24960*x^4 - 5760*x^5 + 696*x^ 
6 - 42*x^7 + x^8 + (-24576 + 59392*x - 55296*x^2 + 24960*x^3 - 5760*x^4 + 
696*x^5 - 42*x^6 + x^7)*Log[3])*Log[x^2 + 2*x*Log[3] + Log[3]^2] + (49152 
- 73728*x + 33792*x^2 - 4608*x^3 + 192*x^4 + (36864*x - 61440*x^2 + 34560* 
x^3 - 7680*x^4 + 720*x^5 - 24*x^6 + (36864 - 61440*x + 34560*x^2 - 7680*x^ 
3 + 720*x^4 - 24*x^5)*Log[3])*Log[x^2 + 2*x*Log[3] + Log[3]^2])*Log[Log[x^ 
2 + 2*x*Log[3] + Log[3]^2]] + (-24576 + 18432*x - 1536*x^2 + (-18432*x + 1 
6896*x^2 - 3456*x^3 + 192*x^4 + (-18432 + 16896*x - 3456*x^2 + 192*x^3)*Lo 
g[3])*Log[x^2 + 2*x*Log[3] + Log[3]^2])*Log[Log[x^2 + 2*x*Log[3] + Log[3]^ 
2]]^2 + (4096 + (3072*x - 512*x^2 + (3072 - 512*x)*Log[3])*Log[x^2 + 2*x*L 
og[3] + Log[3]^2])*Log[Log[x^2 + 2*x*Log[3] + Log[3]^2]]^3)/((800*x + 800* 
Log[3])*Log[x^2 + 2*x*Log[3] + Log[3]^2]),x]
 

Output:

(16 - 12*x + x^2 - 8*Log[Log[(x + Log[3])^2]])^4/6400
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {7239, 27, 25, 7237}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {-8 x^6+288 x^5-3840 x^4+23040 x^3-61440 x^2+\left (\left (-512 x^2+3072 x+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+4096\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-1536 x^2+\left (192 x^4-3456 x^3+16896 x^2+\left (192 x^3-3456 x^2+16896 x-18432\right ) \log (3)-18432 x\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+18432 x-24576\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (192 x^4-4608 x^3+33792 x^2+\left (-24 x^6+720 x^5-7680 x^4+34560 x^3-61440 x^2+\left (-24 x^5+720 x^4-7680 x^3+34560 x^2-61440 x+36864\right ) \log (3)+36864 x\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )-73728 x+49152\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (x^8-42 x^7+696 x^6-5760 x^5+24960 x^4-55296 x^3+59392 x^2+\left (x^7-42 x^6+696 x^5-5760 x^4+24960 x^3-55296 x^2+59392 x-24576\right ) \log (3)-24576 x\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+73728 x-32768}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left ((x-6) (x+\log (3)) \log \left ((x+\log (3))^2\right )-8\right ) \left (x^2-12 x-8 \log \left (\log \left ((x+\log (3))^2\right )\right )+16\right )^3}{800 (x+\log (3)) \log \left ((x+\log (3))^2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{800} \int -\frac {\left ((6-x) (x+\log (3)) \log \left ((x+\log (3))^2\right )+8\right ) \left (x^2-12 x-8 \log \left (\log \left ((x+\log (3))^2\right )\right )+16\right )^3}{(x+\log (3)) \log \left ((x+\log (3))^2\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{800} \int \frac {\left ((6-x) (x+\log (3)) \log \left ((x+\log (3))^2\right )+8\right ) \left (x^2-12 x-8 \log \left (\log \left ((x+\log (3))^2\right )\right )+16\right )^3}{(x+\log (3)) \log \left ((x+\log (3))^2\right )}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle \frac {\left (x^2-12 x-8 \log \left (\log \left ((x+\log (3))^2\right )\right )+16\right )^4}{6400}\)

Input:

Int[(-32768 + 73728*x - 61440*x^2 + 23040*x^3 - 3840*x^4 + 288*x^5 - 8*x^6 
 + (-24576*x + 59392*x^2 - 55296*x^3 + 24960*x^4 - 5760*x^5 + 696*x^6 - 42 
*x^7 + x^8 + (-24576 + 59392*x - 55296*x^2 + 24960*x^3 - 5760*x^4 + 696*x^ 
5 - 42*x^6 + x^7)*Log[3])*Log[x^2 + 2*x*Log[3] + Log[3]^2] + (49152 - 7372 
8*x + 33792*x^2 - 4608*x^3 + 192*x^4 + (36864*x - 61440*x^2 + 34560*x^3 - 
7680*x^4 + 720*x^5 - 24*x^6 + (36864 - 61440*x + 34560*x^2 - 7680*x^3 + 72 
0*x^4 - 24*x^5)*Log[3])*Log[x^2 + 2*x*Log[3] + Log[3]^2])*Log[Log[x^2 + 2* 
x*Log[3] + Log[3]^2]] + (-24576 + 18432*x - 1536*x^2 + (-18432*x + 16896*x 
^2 - 3456*x^3 + 192*x^4 + (-18432 + 16896*x - 3456*x^2 + 192*x^3)*Log[3])* 
Log[x^2 + 2*x*Log[3] + Log[3]^2])*Log[Log[x^2 + 2*x*Log[3] + Log[3]^2]]^2 
+ (4096 + (3072*x - 512*x^2 + (3072 - 512*x)*Log[3])*Log[x^2 + 2*x*Log[3] 
+ Log[3]^2])*Log[Log[x^2 + 2*x*Log[3] + Log[3]^2]]^3)/((800*x + 800*Log[3] 
)*Log[x^2 + 2*x*Log[3] + Log[3]^2]),x]
 

Output:

(16 - 12*x + x^2 - 8*Log[Log[(x + Log[3])^2]])^4/6400
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(25)=50\).

Time = 2.23 (sec) , antiderivative size = 370, normalized size of antiderivative = 12.76

method	result	size
parallelrisch	\(-\frac {768 x}{25}-\frac {12 \ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right ) x^{4}}{5}+\frac {264 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{2} x^{2}}{25}+\frac {72 \ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right ) x^{3}}{5}-\frac {576 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{2} x}{25}-\frac {192 \ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right ) x^{2}}{5}+\frac {1152 \ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right ) x}{25}-\frac {512 \ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}{25}+\frac {x^{8}}{6400}+\frac {29 x^{6}}{200}-\frac {3 x^{7}}{400}+\frac {1536 \ln \left (3\right )}{25}-\frac {928 \ln \left (3\right )^{2}}{25}+\frac {928 x^{2}}{25}-\frac {576 x^{3}}{25}+\frac {39 x^{4}}{5}-\frac {36 x^{5}}{25}-\frac {128 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{3}}{25}+\frac {384 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{2}}{25}+\frac {16 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{4}}{25}+\frac {3 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{2} x^{4}}{50}+\frac {9 \ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right ) x^{5}}{50}-\frac {36 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{2} x^{3}}{25}+\frac {96 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{3} x}{25}-\frac {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right ) x^{6}}{200}-\frac {8 {\ln \left (\ln \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right )\right )}^{3} x^{2}}{25}\)	\(370\)

Input:

int(((((-512*x+3072)*ln(3)-512*x^2+3072*x)*ln(ln(3)^2+2*x*ln(3)+x^2)+4096) 
*ln(ln(ln(3)^2+2*x*ln(3)+x^2))^3+(((192*x^3-3456*x^2+16896*x-18432)*ln(3)+ 
192*x^4-3456*x^3+16896*x^2-18432*x)*ln(ln(3)^2+2*x*ln(3)+x^2)-1536*x^2+184 
32*x-24576)*ln(ln(ln(3)^2+2*x*ln(3)+x^2))^2+(((-24*x^5+720*x^4-7680*x^3+34 
560*x^2-61440*x+36864)*ln(3)-24*x^6+720*x^5-7680*x^4+34560*x^3-61440*x^2+3 
6864*x)*ln(ln(3)^2+2*x*ln(3)+x^2)+192*x^4-4608*x^3+33792*x^2-73728*x+49152 
)*ln(ln(ln(3)^2+2*x*ln(3)+x^2))+((x^7-42*x^6+696*x^5-5760*x^4+24960*x^3-55 
296*x^2+59392*x-24576)*ln(3)+x^8-42*x^7+696*x^6-5760*x^5+24960*x^4-55296*x 
^3+59392*x^2-24576*x)*ln(ln(3)^2+2*x*ln(3)+x^2)-8*x^6+288*x^5-3840*x^4+230 
40*x^3-61440*x^2+73728*x-32768)/(800*ln(3)+800*x)/ln(ln(3)^2+2*x*ln(3)+x^2 
),x,method=_RETURNVERBOSE)
 

Output:

-768/25*x-12/5*ln(ln(ln(3)^2+2*x*ln(3)+x^2))*x^4+264/25*ln(ln(ln(3)^2+2*x* 
ln(3)+x^2))^2*x^2+72/5*ln(ln(ln(3)^2+2*x*ln(3)+x^2))*x^3-576/25*ln(ln(ln(3 
)^2+2*x*ln(3)+x^2))^2*x-192/5*ln(ln(ln(3)^2+2*x*ln(3)+x^2))*x^2+1152/25*ln 
(ln(ln(3)^2+2*x*ln(3)+x^2))*x-512/25*ln(ln(ln(3)^2+2*x*ln(3)+x^2))+1/6400* 
x^8+29/200*x^6-3/400*x^7+1536/25*ln(3)-928/25*ln(3)^2+928/25*x^2-576/25*x^ 
3+39/5*x^4-36/25*x^5-128/25*ln(ln(ln(3)^2+2*x*ln(3)+x^2))^3+384/25*ln(ln(l 
n(3)^2+2*x*ln(3)+x^2))^2+16/25*ln(ln(ln(3)^2+2*x*ln(3)+x^2))^4+3/50*ln(ln( 
ln(3)^2+2*x*ln(3)+x^2))^2*x^4+9/50*ln(ln(ln(3)^2+2*x*ln(3)+x^2))*x^5-36/25 
*ln(ln(ln(3)^2+2*x*ln(3)+x^2))^2*x^3+96/25*ln(ln(ln(3)^2+2*x*ln(3)+x^2))^3 
*x-1/200*ln(ln(ln(3)^2+2*x*ln(3)+x^2))*x^6-8/25*ln(ln(ln(3)^2+2*x*ln(3)+x^ 
2))^3*x^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (25) = 50\).

Time = 0.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.76 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx=\frac {1}{6400} \, x^{8} - \frac {3}{400} \, x^{7} + \frac {29}{200} \, x^{6} - \frac {36}{25} \, x^{5} + \frac {39}{5} \, x^{4} - \frac {8}{25} \, {\left (x^{2} - 12 \, x + 16\right )} \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right )^{3} + \frac {16}{25} \, \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right )^{4} - \frac {576}{25} \, x^{3} + \frac {3}{50} \, {\left (x^{4} - 24 \, x^{3} + 176 \, x^{2} - 384 \, x + 256\right )} \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right )^{2} + \frac {928}{25} \, x^{2} - \frac {1}{200} \, {\left (x^{6} - 36 \, x^{5} + 480 \, x^{4} - 2880 \, x^{3} + 7680 \, x^{2} - 9216 \, x + 4096\right )} \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right ) - \frac {768}{25} \, x \] Input:

integrate(((((-512*x+3072)*log(3)-512*x^2+3072*x)*log(log(3)^2+2*x*log(3)+ 
x^2)+4096)*log(log(log(3)^2+2*x*log(3)+x^2))^3+(((192*x^3-3456*x^2+16896*x 
-18432)*log(3)+192*x^4-3456*x^3+16896*x^2-18432*x)*log(log(3)^2+2*x*log(3) 
+x^2)-1536*x^2+18432*x-24576)*log(log(log(3)^2+2*x*log(3)+x^2))^2+(((-24*x 
^5+720*x^4-7680*x^3+34560*x^2-61440*x+36864)*log(3)-24*x^6+720*x^5-7680*x^ 
4+34560*x^3-61440*x^2+36864*x)*log(log(3)^2+2*x*log(3)+x^2)+192*x^4-4608*x 
^3+33792*x^2-73728*x+49152)*log(log(log(3)^2+2*x*log(3)+x^2))+((x^7-42*x^6 
+696*x^5-5760*x^4+24960*x^3-55296*x^2+59392*x-24576)*log(3)+x^8-42*x^7+696 
*x^6-5760*x^5+24960*x^4-55296*x^3+59392*x^2-24576*x)*log(log(3)^2+2*x*log( 
3)+x^2)-8*x^6+288*x^5-3840*x^4+23040*x^3-61440*x^2+73728*x-32768)/(800*log 
(3)+800*x)/log(log(3)^2+2*x*log(3)+x^2),x, algorithm="fricas")
 

Output:

1/6400*x^8 - 3/400*x^7 + 29/200*x^6 - 36/25*x^5 + 39/5*x^4 - 8/25*(x^2 - 1 
2*x + 16)*log(log(x^2 + 2*x*log(3) + log(3)^2))^3 + 16/25*log(log(x^2 + 2* 
x*log(3) + log(3)^2))^4 - 576/25*x^3 + 3/50*(x^4 - 24*x^3 + 176*x^2 - 384* 
x + 256)*log(log(x^2 + 2*x*log(3) + log(3)^2))^2 + 928/25*x^2 - 1/200*(x^6 
 - 36*x^5 + 480*x^4 - 2880*x^3 + 7680*x^2 - 9216*x + 4096)*log(log(x^2 + 2 
*x*log(3) + log(3)^2)) - 768/25*x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (24) = 48\).

Time = 0.52 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.14 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx=\frac {x^{8}}{6400} - \frac {3 x^{7}}{400} + \frac {29 x^{6}}{200} - \frac {36 x^{5}}{25} + \frac {39 x^{4}}{5} - \frac {576 x^{3}}{25} + \frac {928 x^{2}}{25} - \frac {768 x}{25} + \left (- \frac {8 x^{2}}{25} + \frac {96 x}{25} - \frac {128}{25}\right ) \log {\left (\log {\left (x^{2} + 2 x \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )} \right )}^{3} + \left (\frac {3 x^{4}}{50} - \frac {36 x^{3}}{25} + \frac {264 x^{2}}{25} - \frac {576 x}{25} + \frac {384}{25}\right ) \log {\left (\log {\left (x^{2} + 2 x \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )} \right )}^{2} + \left (- \frac {x^{6}}{200} + \frac {9 x^{5}}{50} - \frac {12 x^{4}}{5} + \frac {72 x^{3}}{5} - \frac {192 x^{2}}{5} + \frac {1152 x}{25}\right ) \log {\left (\log {\left (x^{2} + 2 x \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )} \right )} + \frac {16 \log {\left (\log {\left (x^{2} + 2 x \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )} \right )}^{4}}{25} - \frac {512 \log {\left (\log {\left (x^{2} + 2 x \log {\left (3 \right )} + \log {\left (3 \right )}^{2} \right )} \right )}}{25} \] Input:

integrate(((((-512*x+3072)*ln(3)-512*x**2+3072*x)*ln(ln(3)**2+2*x*ln(3)+x* 
*2)+4096)*ln(ln(ln(3)**2+2*x*ln(3)+x**2))**3+(((192*x**3-3456*x**2+16896*x 
-18432)*ln(3)+192*x**4-3456*x**3+16896*x**2-18432*x)*ln(ln(3)**2+2*x*ln(3) 
+x**2)-1536*x**2+18432*x-24576)*ln(ln(ln(3)**2+2*x*ln(3)+x**2))**2+(((-24* 
x**5+720*x**4-7680*x**3+34560*x**2-61440*x+36864)*ln(3)-24*x**6+720*x**5-7 
680*x**4+34560*x**3-61440*x**2+36864*x)*ln(ln(3)**2+2*x*ln(3)+x**2)+192*x* 
*4-4608*x**3+33792*x**2-73728*x+49152)*ln(ln(ln(3)**2+2*x*ln(3)+x**2))+((x 
**7-42*x**6+696*x**5-5760*x**4+24960*x**3-55296*x**2+59392*x-24576)*ln(3)+ 
x**8-42*x**7+696*x**6-5760*x**5+24960*x**4-55296*x**3+59392*x**2-24576*x)* 
ln(ln(3)**2+2*x*ln(3)+x**2)-8*x**6+288*x**5-3840*x**4+23040*x**3-61440*x** 
2+73728*x-32768)/(800*ln(3)+800*x)/ln(ln(3)**2+2*x*ln(3)+x**2),x)
 

Output:

x**8/6400 - 3*x**7/400 + 29*x**6/200 - 36*x**5/25 + 39*x**4/5 - 576*x**3/2 
5 + 928*x**2/25 - 768*x/25 + (-8*x**2/25 + 96*x/25 - 128/25)*log(log(x**2 
+ 2*x*log(3) + log(3)**2))**3 + (3*x**4/50 - 36*x**3/25 + 264*x**2/25 - 57 
6*x/25 + 384/25)*log(log(x**2 + 2*x*log(3) + log(3)**2))**2 + (-x**6/200 + 
 9*x**5/50 - 12*x**4/5 + 72*x**3/5 - 192*x**2/5 + 1152*x/25)*log(log(x**2 
+ 2*x*log(3) + log(3)**2)) + 16*log(log(x**2 + 2*x*log(3) + log(3)**2))**4 
/25 - 512*log(log(x**2 + 2*x*log(3) + log(3)**2))/25
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (25) = 50\).

Time = 0.19 (sec) , antiderivative size = 264, normalized size of antiderivative = 9.10 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx=\frac {1}{6400} \, x^{8} - \frac {3}{400} \, x^{7} - \frac {1}{200} \, x^{6} {\left (\log \left (2\right ) - 29\right )} + \frac {9}{50} \, x^{5} {\left (\log \left (2\right ) - 8\right )} + \frac {3}{50} \, {\left (\log \left (2\right )^{2} - 40 \, \log \left (2\right ) + 130\right )} x^{4} - \frac {36}{25} \, {\left (\log \left (2\right )^{2} - 10 \, \log \left (2\right ) + 16\right )} x^{3} - \frac {8}{25} \, {\left (x^{2} - 12 \, x - 8 \, \log \left (2\right ) + 16\right )} \log \left (\log \left (x + \log \left (3\right )\right )\right )^{3} + \frac {16}{25} \, \log \left (\log \left (x + \log \left (3\right )\right )\right )^{4} - \frac {8}{25} \, {\left (\log \left (2\right )^{3} - 33 \, \log \left (2\right )^{2} + 120 \, \log \left (2\right ) - 116\right )} x^{2} + \frac {3}{50} \, {\left (x^{4} - 24 \, x^{3} - 16 \, x^{2} {\left (\log \left (2\right ) - 11\right )} + 192 \, x {\left (\log \left (2\right ) - 2\right )} + 64 \, \log \left (2\right )^{2} - 256 \, \log \left (2\right ) + 256\right )} \log \left (\log \left (x + \log \left (3\right )\right )\right )^{2} + \frac {96}{25} \, {\left (\log \left (2\right )^{3} - 6 \, \log \left (2\right )^{2} + 12 \, \log \left (2\right ) - 8\right )} x - \frac {1}{200} \, {\left (x^{6} - 36 \, x^{5} - 24 \, x^{4} {\left (\log \left (2\right ) - 20\right )} + 576 \, x^{3} {\left (\log \left (2\right ) - 5\right )} + 192 \, {\left (\log \left (2\right )^{2} - 22 \, \log \left (2\right ) + 40\right )} x^{2} - 512 \, \log \left (2\right )^{3} - 2304 \, {\left (\log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 4\right )} x + 3072 \, \log \left (2\right )^{2} - 6144 \, \log \left (2\right )\right )} \log \left (\log \left (x + \log \left (3\right )\right )\right ) - \frac {512}{25} \, \log \left (\log \left (x + \log \left (3\right )\right )\right ) \] Input:

integrate(((((-512*x+3072)*log(3)-512*x^2+3072*x)*log(log(3)^2+2*x*log(3)+ 
x^2)+4096)*log(log(log(3)^2+2*x*log(3)+x^2))^3+(((192*x^3-3456*x^2+16896*x 
-18432)*log(3)+192*x^4-3456*x^3+16896*x^2-18432*x)*log(log(3)^2+2*x*log(3) 
+x^2)-1536*x^2+18432*x-24576)*log(log(log(3)^2+2*x*log(3)+x^2))^2+(((-24*x 
^5+720*x^4-7680*x^3+34560*x^2-61440*x+36864)*log(3)-24*x^6+720*x^5-7680*x^ 
4+34560*x^3-61440*x^2+36864*x)*log(log(3)^2+2*x*log(3)+x^2)+192*x^4-4608*x 
^3+33792*x^2-73728*x+49152)*log(log(log(3)^2+2*x*log(3)+x^2))+((x^7-42*x^6 
+696*x^5-5760*x^4+24960*x^3-55296*x^2+59392*x-24576)*log(3)+x^8-42*x^7+696 
*x^6-5760*x^5+24960*x^4-55296*x^3+59392*x^2-24576*x)*log(log(3)^2+2*x*log( 
3)+x^2)-8*x^6+288*x^5-3840*x^4+23040*x^3-61440*x^2+73728*x-32768)/(800*log 
(3)+800*x)/log(log(3)^2+2*x*log(3)+x^2),x, algorithm="maxima")
 

Output:

1/6400*x^8 - 3/400*x^7 - 1/200*x^6*(log(2) - 29) + 9/50*x^5*(log(2) - 8) + 
 3/50*(log(2)^2 - 40*log(2) + 130)*x^4 - 36/25*(log(2)^2 - 10*log(2) + 16) 
*x^3 - 8/25*(x^2 - 12*x - 8*log(2) + 16)*log(log(x + log(3)))^3 + 16/25*lo 
g(log(x + log(3)))^4 - 8/25*(log(2)^3 - 33*log(2)^2 + 120*log(2) - 116)*x^ 
2 + 3/50*(x^4 - 24*x^3 - 16*x^2*(log(2) - 11) + 192*x*(log(2) - 2) + 64*lo 
g(2)^2 - 256*log(2) + 256)*log(log(x + log(3)))^2 + 96/25*(log(2)^3 - 6*lo 
g(2)^2 + 12*log(2) - 8)*x - 1/200*(x^6 - 36*x^5 - 24*x^4*(log(2) - 20) + 5 
76*x^3*(log(2) - 5) + 192*(log(2)^2 - 22*log(2) + 40)*x^2 - 512*log(2)^3 - 
 2304*(log(2)^2 - 4*log(2) + 4)*x + 3072*log(2)^2 - 6144*log(2))*log(log(x 
 + log(3))) - 512/25*log(log(x + log(3)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (25) = 50\).

Time = 1.66 (sec) , antiderivative size = 183, normalized size of antiderivative = 6.31 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx=\frac {1}{6400} \, x^{8} - \frac {3}{400} \, x^{7} + \frac {29}{200} \, x^{6} - \frac {36}{25} \, x^{5} + \frac {39}{5} \, x^{4} - \frac {8}{25} \, {\left (x^{2} - 12 \, x + 16\right )} \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right )^{3} + \frac {16}{25} \, \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right )^{4} - \frac {576}{25} \, x^{3} + \frac {3}{50} \, {\left (x^{4} - 24 \, x^{3} + 176 \, x^{2} - 384 \, x + 256\right )} \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right )^{2} + \frac {928}{25} \, x^{2} - \frac {1}{200} \, {\left (x^{6} - 36 \, x^{5} + 480 \, x^{4} - 2880 \, x^{3} + 7680 \, x^{2} - 9216 \, x\right )} \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right ) - \frac {768}{25} \, x - \frac {512}{25} \, \log \left (\log \left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )\right ) \] Input:

integrate(((((-512*x+3072)*log(3)-512*x^2+3072*x)*log(log(3)^2+2*x*log(3)+ 
x^2)+4096)*log(log(log(3)^2+2*x*log(3)+x^2))^3+(((192*x^3-3456*x^2+16896*x 
-18432)*log(3)+192*x^4-3456*x^3+16896*x^2-18432*x)*log(log(3)^2+2*x*log(3) 
+x^2)-1536*x^2+18432*x-24576)*log(log(log(3)^2+2*x*log(3)+x^2))^2+(((-24*x 
^5+720*x^4-7680*x^3+34560*x^2-61440*x+36864)*log(3)-24*x^6+720*x^5-7680*x^ 
4+34560*x^3-61440*x^2+36864*x)*log(log(3)^2+2*x*log(3)+x^2)+192*x^4-4608*x 
^3+33792*x^2-73728*x+49152)*log(log(log(3)^2+2*x*log(3)+x^2))+((x^7-42*x^6 
+696*x^5-5760*x^4+24960*x^3-55296*x^2+59392*x-24576)*log(3)+x^8-42*x^7+696 
*x^6-5760*x^5+24960*x^4-55296*x^3+59392*x^2-24576*x)*log(log(3)^2+2*x*log( 
3)+x^2)-8*x^6+288*x^5-3840*x^4+23040*x^3-61440*x^2+73728*x-32768)/(800*log 
(3)+800*x)/log(log(3)^2+2*x*log(3)+x^2),x, algorithm="giac")
 

Output:

1/6400*x^8 - 3/400*x^7 + 29/200*x^6 - 36/25*x^5 + 39/5*x^4 - 8/25*(x^2 - 1 
2*x + 16)*log(log(x^2 + 2*x*log(3) + log(3)^2))^3 + 16/25*log(log(x^2 + 2* 
x*log(3) + log(3)^2))^4 - 576/25*x^3 + 3/50*(x^4 - 24*x^3 + 176*x^2 - 384* 
x + 256)*log(log(x^2 + 2*x*log(3) + log(3)^2))^2 + 928/25*x^2 - 1/200*(x^6 
 - 36*x^5 + 480*x^4 - 2880*x^3 + 7680*x^2 - 9216*x)*log(log(x^2 + 2*x*log( 
3) + log(3)^2)) - 768/25*x - 512/25*log(log(x^2 + 2*x*log(3) + log(3)^2))
 

Mupad [B] (verification not implemented)

Time = 4.41 (sec) , antiderivative size = 187, normalized size of antiderivative = 6.45 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx=\frac {16\,{\ln \left (\ln \left (x^2+2\,\ln \left (3\right )\,x+{\ln \left (3\right )}^2\right )\right )}^4}{25}-\frac {512\,\ln \left (\ln \left (x^2+2\,\ln \left (3\right )\,x+{\ln \left (3\right )}^2\right )\right )}{25}-{\ln \left (\ln \left (x^2+2\,\ln \left (3\right )\,x+{\ln \left (3\right )}^2\right )\right )}^3\,\left (\frac {8\,x^2}{25}-\frac {96\,x}{25}+\frac {128}{25}\right )-\frac {768\,x}{25}+{\ln \left (\ln \left (x^2+2\,\ln \left (3\right )\,x+{\ln \left (3\right )}^2\right )\right )}^2\,\left (\frac {3\,x^4}{50}-\frac {36\,x^3}{25}+\frac {264\,x^2}{25}-\frac {576\,x}{25}+\frac {384}{25}\right )+\ln \left (\ln \left (x^2+2\,\ln \left (3\right )\,x+{\ln \left (3\right )}^2\right )\right )\,\left (-\frac {x^6}{200}+\frac {9\,x^5}{50}-\frac {12\,x^4}{5}+\frac {72\,x^3}{5}-\frac {192\,x^2}{5}+\frac {1152\,x}{25}\right )+\frac {928\,x^2}{25}-\frac {576\,x^3}{25}+\frac {39\,x^4}{5}-\frac {36\,x^5}{25}+\frac {29\,x^6}{200}-\frac {3\,x^7}{400}+\frac {x^8}{6400} \] Input:

int(-(log(log(2*x*log(3) + log(3)^2 + x^2))*(73728*x + log(2*x*log(3) + lo 
g(3)^2 + x^2)*(61440*x^2 - 36864*x - 34560*x^3 + 7680*x^4 - 720*x^5 + 24*x 
^6 + log(3)*(61440*x - 34560*x^2 + 7680*x^3 - 720*x^4 + 24*x^5 - 36864)) - 
 33792*x^2 + 4608*x^3 - 192*x^4 - 49152) - log(2*x*log(3) + log(3)^2 + x^2 
)*(59392*x^2 - 24576*x - 55296*x^3 + 24960*x^4 - 5760*x^5 + 696*x^6 - 42*x 
^7 + x^8 + log(3)*(59392*x - 55296*x^2 + 24960*x^3 - 5760*x^4 + 696*x^5 - 
42*x^6 + x^7 - 24576)) - 73728*x - log(log(2*x*log(3) + log(3)^2 + x^2))^2 
*(18432*x + log(2*x*log(3) + log(3)^2 + x^2)*(log(3)*(16896*x - 3456*x^2 + 
 192*x^3 - 18432) - 18432*x + 16896*x^2 - 3456*x^3 + 192*x^4) - 1536*x^2 - 
 24576) + log(log(2*x*log(3) + log(3)^2 + x^2))^3*(log(2*x*log(3) + log(3) 
^2 + x^2)*(log(3)*(512*x - 3072) - 3072*x + 512*x^2) - 4096) + 61440*x^2 - 
 23040*x^3 + 3840*x^4 - 288*x^5 + 8*x^6 + 32768)/(log(2*x*log(3) + log(3)^ 
2 + x^2)*(800*x + 800*log(3))),x)
 

Output:

(16*log(log(2*x*log(3) + log(3)^2 + x^2))^4)/25 - (512*log(log(2*x*log(3) 
+ log(3)^2 + x^2)))/25 - log(log(2*x*log(3) + log(3)^2 + x^2))^3*((8*x^2)/ 
25 - (96*x)/25 + 128/25) - (768*x)/25 + log(log(2*x*log(3) + log(3)^2 + x^ 
2))^2*((264*x^2)/25 - (576*x)/25 - (36*x^3)/25 + (3*x^4)/50 + 384/25) + lo 
g(log(2*x*log(3) + log(3)^2 + x^2))*((1152*x)/25 - (192*x^2)/5 + (72*x^3)/ 
5 - (12*x^4)/5 + (9*x^5)/50 - x^6/200) + (928*x^2)/25 - (576*x^3)/25 + (39 
*x^4)/5 - (36*x^5)/25 + (29*x^6)/200 - (3*x^7)/400 + x^8/6400
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 359, normalized size of antiderivative = 12.38 \[ \int \frac {-32768+73728 x-61440 x^2+23040 x^3-3840 x^4+288 x^5-8 x^6+\left (-24576 x+59392 x^2-55296 x^3+24960 x^4-5760 x^5+696 x^6-42 x^7+x^8+\left (-24576+59392 x-55296 x^2+24960 x^3-5760 x^4+696 x^5-42 x^6+x^7\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )+\left (49152-73728 x+33792 x^2-4608 x^3+192 x^4+\left (36864 x-61440 x^2+34560 x^3-7680 x^4+720 x^5-24 x^6+\left (36864-61440 x+34560 x^2-7680 x^3+720 x^4-24 x^5\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log \left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (-24576+18432 x-1536 x^2+\left (-18432 x+16896 x^2-3456 x^3+192 x^4+\left (-18432+16896 x-3456 x^2+192 x^3\right ) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^2\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )+\left (4096+\left (3072 x-512 x^2+(3072-512 x) \log (3)\right ) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right ) \log ^3\left (\log \left (x^2+2 x \log (3)+\log ^2(3)\right )\right )}{(800 x+800 \log (3)) \log \left (x^2+2 x \log (3)+\log ^2(3)\right )} \, dx =\text {Too large to display} \] Input:

int(((((-512*x+3072)*log(3)-512*x^2+3072*x)*log(log(3)^2+2*x*log(3)+x^2)+4 
096)*log(log(log(3)^2+2*x*log(3)+x^2))^3+(((192*x^3-3456*x^2+16896*x-18432 
)*log(3)+192*x^4-3456*x^3+16896*x^2-18432*x)*log(log(3)^2+2*x*log(3)+x^2)- 
1536*x^2+18432*x-24576)*log(log(log(3)^2+2*x*log(3)+x^2))^2+(((-24*x^5+720 
*x^4-7680*x^3+34560*x^2-61440*x+36864)*log(3)-24*x^6+720*x^5-7680*x^4+3456 
0*x^3-61440*x^2+36864*x)*log(log(3)^2+2*x*log(3)+x^2)+192*x^4-4608*x^3+337 
92*x^2-73728*x+49152)*log(log(log(3)^2+2*x*log(3)+x^2))+((x^7-42*x^6+696*x 
^5-5760*x^4+24960*x^3-55296*x^2+59392*x-24576)*log(3)+x^8-42*x^7+696*x^6-5 
760*x^5+24960*x^4-55296*x^3+59392*x^2-24576*x)*log(log(3)^2+2*x*log(3)+x^2 
)-8*x^6+288*x^5-3840*x^4+23040*x^3-61440*x^2+73728*x-32768)/(800*log(3)+80 
0*x)/log(log(3)^2+2*x*log(3)+x^2),x)
 

Output:

(4096*log(log(log(3)**2 + 2*log(3)*x + x**2))**4 - 2048*log(log(log(3)**2 
+ 2*log(3)*x + x**2))**3*x**2 + 24576*log(log(log(3)**2 + 2*log(3)*x + x** 
2))**3*x - 32768*log(log(log(3)**2 + 2*log(3)*x + x**2))**3 + 384*log(log( 
log(3)**2 + 2*log(3)*x + x**2))**2*x**4 - 9216*log(log(log(3)**2 + 2*log(3 
)*x + x**2))**2*x**3 + 67584*log(log(log(3)**2 + 2*log(3)*x + x**2))**2*x* 
*2 - 147456*log(log(log(3)**2 + 2*log(3)*x + x**2))**2*x + 98304*log(log(l 
og(3)**2 + 2*log(3)*x + x**2))**2 - 32*log(log(log(3)**2 + 2*log(3)*x + x* 
*2))*x**6 + 1152*log(log(log(3)**2 + 2*log(3)*x + x**2))*x**5 - 15360*log( 
log(log(3)**2 + 2*log(3)*x + x**2))*x**4 + 92160*log(log(log(3)**2 + 2*log 
(3)*x + x**2))*x**3 - 245760*log(log(log(3)**2 + 2*log(3)*x + x**2))*x**2 
+ 294912*log(log(log(3)**2 + 2*log(3)*x + x**2))*x - 131072*log(log(log(3) 
**2 + 2*log(3)*x + x**2)) + x**8 - 48*x**7 + 928*x**6 - 9216*x**5 + 49920* 
x**4 - 147456*x**3 + 237568*x**2 - 196608*x)/6400