3.14.18 \(\int \frac {(-208 x-60 x^2+108 x^3-36 x^4+4 x^5) \log (2 x)+(-60+164 x-57 x^2-15 x^3+9 x^4-x^5) \log (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4})}{(60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6) \log ^2(2 x)} \, dx\) [1318]

3.14.18.1 Optimal result

Integrand size = 152, antiderivative size = 25 \begin {dmath*} \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx=\frac {\log \left (\left (-4+\left (\frac {4}{3-x}+x\right )^2\right )^2\right )}{\log (2 x)} \end {dmath*}

output

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

3.14.18.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \begin {dmath*} \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx=\frac {\log \left (\frac {\left (-20+48 x-3 x^2-6 x^3+x^4\right )^2}{(-3+x)^4}\right )}{\log (2 x)} \end {dmath*}

input

Integrate[((-208*x - 60*x^2 + 108*x^3 - 36*x^4 + 4*x^5)*Log[2*x] + (-60 + 
164*x - 57*x^2 - 15*x^3 + 9*x^4 - x^5)*Log[(400 - 1920*x + 2424*x^2 - 48*x 
^3 - 607*x^4 + 132*x^5 + 30*x^6 - 12*x^7 + x^8)/(81 - 108*x + 54*x^2 - 12* 
x^3 + x^4)])/((60*x - 164*x^2 + 57*x^3 + 15*x^4 - 9*x^5 + x^6)*Log[2*x]^2) 
,x]
 

output

Log[(-20 + 48*x - 3*x^2 - 6*x^3 + x^4)^2/(-3 + x)^4]/Log[2*x]
 

3.14.18.3 Rubi [F]

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.

\(\displaystyle \int \frac {\left (4 x^5-36 x^4+108 x^3-60 x^2-208 x\right ) \log (2 x)+\left (-x^5+9 x^4-15 x^3-57 x^2+164 x-60\right ) \log \left (\frac {x^8-12 x^7+30 x^6+132 x^5-607 x^4-48 x^3+2424 x^2-1920 x+400}{x^4-12 x^3+54 x^2-108 x+81}\right )}{\left (x^6-9 x^5+15 x^4+57 x^3-164 x^2+60 x\right ) \log ^2(2 x)} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (4 x^5-36 x^4+108 x^3-60 x^2-208 x\right ) \log (2 x)+\left (-x^5+9 x^4-15 x^3-57 x^2+164 x-60\right ) \log \left (\frac {x^8-12 x^7+30 x^6+132 x^5-607 x^4-48 x^3+2424 x^2-1920 x+400}{x^4-12 x^3+54 x^2-108 x+81}\right )}{x \left (x^5-9 x^4+15 x^3+57 x^2-164 x+60\right ) \log ^2(2 x)}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {\left (4 x^5-36 x^4+108 x^3-60 x^2-208 x\right ) \log (2 x)+\left (-x^5+9 x^4-15 x^3-57 x^2+164 x-60\right ) \log \left (\frac {x^8-12 x^7+30 x^6+132 x^5-607 x^4-48 x^3+2424 x^2-1920 x+400}{x^4-12 x^3+54 x^2-108 x+81}\right )}{16 (x-3) x \log ^2(2 x)}+\frac {(x+2) \left (\left (4 x^5-36 x^4+108 x^3-60 x^2-208 x\right ) \log (2 x)+\left (-x^5+9 x^4-15 x^3-57 x^2+164 x-60\right ) \log \left (\frac {x^8-12 x^7+30 x^6+132 x^5-607 x^4-48 x^3+2424 x^2-1920 x+400}{x^4-12 x^3+54 x^2-108 x+81}\right )\right )}{64 x \left (x^2-5 x+2\right ) \log ^2(2 x)}+\frac {(-5 x-14) \left (\left (4 x^5-36 x^4+108 x^3-60 x^2-208 x\right ) \log (2 x)+\left (-x^5+9 x^4-15 x^3-57 x^2+164 x-60\right ) \log \left (\frac {x^8-12 x^7+30 x^6+132 x^5-607 x^4-48 x^3+2424 x^2-1920 x+400}{x^4-12 x^3+54 x^2-108 x+81}\right )\right )}{64 x \left (x^2-x-10\right ) \log ^2(2 x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \int \frac {(x-4) (x+1) \left (x^2-6 x+13\right )}{(x-3) \log (2 x)}dx+\frac {1}{16} \int \frac {(x-4) (x+1) (x+2) \left (x^2-6 x+13\right )}{\left (x^2-5 x+2\right ) \log (2 x)}dx-\frac {1}{16} \int \frac {(x-4) (x+1) (5 x+14) \left (x^2-6 x+13\right )}{\left (x^2-x-10\right ) \log (2 x)}dx-\int \frac {\log \left (\frac {\left (x^4-6 x^3-3 x^2+48 x-20\right )^2}{(x-3)^4}\right )}{x \log ^2(2 x)}dx\)

input

Int[((-208*x - 60*x^2 + 108*x^3 - 36*x^4 + 4*x^5)*Log[2*x] + (-60 + 164*x 
- 57*x^2 - 15*x^3 + 9*x^4 - x^5)*Log[(400 - 1920*x + 2424*x^2 - 48*x^3 - 6 
07*x^4 + 132*x^5 + 30*x^6 - 12*x^7 + x^8)/(81 - 108*x + 54*x^2 - 12*x^3 + 
x^4)])/((60*x - 164*x^2 + 57*x^3 + 15*x^4 - 9*x^5 + x^6)*Log[2*x]^2),x]
 

output

$Aborted
 

3.14.18.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

3.14.18.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(25)=50\).

Time = 14.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {x^{8}-12 x^{7}+30 x^{6}+132 x^{5}-607 x^{4}-48 x^{3}+2424 x^{2}-1920 x +400}{x^{4}-12 x^{3}+54 x^{2}-108 x +81}\right )}{\ln \left (2 x \right )}\)	\(68\)
risch	\(\frac {2 \ln \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )}{\ln \left (2 x \right )}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{4}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{3}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{2}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{3}\right )+i \pi {\operatorname {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )\right )}^{2} \operatorname {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right )-i \pi \operatorname {csgn}\left (i \left (-3+x \right )^{3}\right )^{3}-i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{3}\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{4}\right )+i \pi {\operatorname {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right )}^{3}-i \pi \operatorname {csgn}\left (i \left (-3+x \right )^{4}\right )^{3}+i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{3}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (-3+x \right )^{2}\right )^{3}-i \pi \,\operatorname {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (-3+x \right )^{4}}\right )}^{2}-2 i \pi \,\operatorname {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )\right ) {\operatorname {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right )}^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-3+x \right )^{4}}\right ) \operatorname {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (-3+x \right )^{4}}\right )+i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )^{3}\right ) \operatorname {csgn}\left (i \left (-3+x \right )^{4}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (-3+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-3+x \right )^{2}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (-3+x \right )^{4}}\right )}^{3}-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-3+x \right )^{4}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (-3+x \right )^{4}}\right )}^{2}+8 \ln \left (-3+x \right )}{2 \ln \left (2 x \right )}\)	\(621\)

input

int(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*ln((x^8-12*x^7+30*x^6+132*x^5-607 
*x^4-48*x^3+2424*x^2-1920*x+400)/(x^4-12*x^3+54*x^2-108*x+81))+(4*x^5-36*x 
^4+108*x^3-60*x^2-208*x)*ln(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x^2+60*x)/l 
n(2*x)^2,x,method=_RETURNVERBOSE)
 

output

ln((x^8-12*x^7+30*x^6+132*x^5-607*x^4-48*x^3+2424*x^2-1920*x+400)/(x^4-12* 
x^3+54*x^2-108*x+81))/ln(2*x)
 

3.14.18.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (23) = 46\).

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \begin {dmath*} \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx=\frac {\log \left (\frac {x^{8} - 12 \, x^{7} + 30 \, x^{6} + 132 \, x^{5} - 607 \, x^{4} - 48 \, x^{3} + 2424 \, x^{2} - 1920 \, x + 400}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81}\right )}{\log \left (2 \, x\right )} \end {dmath*}

input

integrate(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*log((x^8-12*x^7+30*x^6+132* 
x^5-607*x^4-48*x^3+2424*x^2-1920*x+400)/(x^4-12*x^3+54*x^2-108*x+81))+(4*x 
^5-36*x^4+108*x^3-60*x^2-208*x)*log(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x^2 
+60*x)/log(2*x)^2,x, algorithm=\
 

output

log((x^8 - 12*x^7 + 30*x^6 + 132*x^5 - 607*x^4 - 48*x^3 + 2424*x^2 - 1920* 
x + 400)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81))/log(2*x)
 

3.14.18.6 Sympy [F(-2)]

Exception generated. \begin {dmath*} \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx=\text {Exception raised: TypeError} \end {dmath*}

input

integrate(((-x**5+9*x**4-15*x**3-57*x**2+164*x-60)*ln((x**8-12*x**7+30*x** 
6+132*x**5-607*x**4-48*x**3+2424*x**2-1920*x+400)/(x**4-12*x**3+54*x**2-10 
8*x+81))+(4*x**5-36*x**4+108*x**3-60*x**2-208*x)*ln(2*x))/(x**6-9*x**5+15* 
x**4+57*x**3-164*x**2+60*x)/ln(2*x)**2,x)
 

output

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

3.14.18.7 Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \begin {dmath*} \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx=\frac {2 \, {\left (\log \left (x^{2} - x - 10\right ) + \log \left (x^{2} - 5 \, x + 2\right ) - 2 \, \log \left (x - 3\right )\right )}}{\log \left (2\right ) + \log \left (x\right )} \end {dmath*}

input

integrate(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*log((x^8-12*x^7+30*x^6+132* 
x^5-607*x^4-48*x^3+2424*x^2-1920*x+400)/(x^4-12*x^3+54*x^2-108*x+81))+(4*x 
^5-36*x^4+108*x^3-60*x^2-208*x)*log(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x^2 
+60*x)/log(2*x)^2,x, algorithm=\
 

output

2*(log(x^2 - x - 10) + log(x^2 - 5*x + 2) - 2*log(x - 3))/(log(2) + log(x) 
)
 

3.14.18.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (23) = 46\).

Time = 0.50 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96 \begin {dmath*} \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx=\frac {\log \left (x^{8} - 12 \, x^{7} + 30 \, x^{6} + 132 \, x^{5} - 607 \, x^{4} - 48 \, x^{3} + 2424 \, x^{2} - 1920 \, x + 400\right )}{\log \left (2 \, x\right )} - \frac {\log \left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}{\log \left (2 \, x\right )} \end {dmath*}

input

integrate(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*log((x^8-12*x^7+30*x^6+132* 
x^5-607*x^4-48*x^3+2424*x^2-1920*x+400)/(x^4-12*x^3+54*x^2-108*x+81))+(4*x 
^5-36*x^4+108*x^3-60*x^2-208*x)*log(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x^2 
+60*x)/log(2*x)^2,x, algorithm=\
 

output

log(x^8 - 12*x^7 + 30*x^6 + 132*x^5 - 607*x^4 - 48*x^3 + 2424*x^2 - 1920*x 
 + 400)/log(2*x) - log(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)/log(2*x)
 

3.14.18.9 Mupad [B] (verification not implemented)

Time = 18.43 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \begin {dmath*} \int \frac {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx=\frac {\ln \left (\frac {x^8-12\,x^7+30\,x^6+132\,x^5-607\,x^4-48\,x^3+2424\,x^2-1920\,x+400}{x^4-12\,x^3+54\,x^2-108\,x+81}\right )}{\ln \left (2\,x\right )} \end {dmath*}

input

int(-(log((2424*x^2 - 1920*x - 48*x^3 - 607*x^4 + 132*x^5 + 30*x^6 - 12*x^ 
7 + x^8 + 400)/(54*x^2 - 108*x - 12*x^3 + x^4 + 81))*(57*x^2 - 164*x + 15* 
x^3 - 9*x^4 + x^5 + 60) + log(2*x)*(208*x + 60*x^2 - 108*x^3 + 36*x^4 - 4* 
x^5))/(log(2*x)^2*(60*x - 164*x^2 + 57*x^3 + 15*x^4 - 9*x^5 + x^6)),x)
 

output

log((2424*x^2 - 1920*x - 48*x^3 - 607*x^4 + 132*x^5 + 30*x^6 - 12*x^7 + x^ 
8 + 400)/(54*x^2 - 108*x - 12*x^3 + x^4 + 81))/log(2*x)