3.23.87 \(\int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} (x^2)^{\frac {-125-25 x^2 \log (x)}{-2+x}} (500-250 x+(100 x^2-50 x^3) \log (x)+(125 x+50 x^2-25 x^3+(100 x^2-25 x^3) \log (x)) \log (2 x^2))}{4 x-4 x^2+x^3} \, dx\) [2287]

3.23.87.1 Optimal result
3.23.87.2 Mathematica [F]
3.23.87.3 Rubi [F]
3.23.87.4 Maple [A] (verified)
3.23.87.5 Fricas [A] (verification not implemented)
3.23.87.6 Sympy [A] (verification not implemented)
3.23.87.7 Maxima [B] (verification not implemented)
3.23.87.8 Giac [F]
3.23.87.9 Mupad [B] (verification not implemented)

3.23.87.1 Optimal result

Integrand size = 105, antiderivative size = 43 \[ \int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx=-4+2^{\frac {25 \left (5+x^2 \log (x)\right )}{2-x}} \left (x^2\right )^{\frac {25 \left (5+x^2 \log (x)\right )}{2-x}} \]

output
exp(25/(2-x)*(5+x^2*ln(x))*ln(2*x^2))-4
 
3.23.87.2 Mathematica [F]

\[ \int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx=\int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx \]

input
Integrate[(2^((-125 - 25*x^2*Log[x])/(-2 + x))*(x^2)^((-125 - 25*x^2*Log[x 
])/(-2 + x))*(500 - 250*x + (100*x^2 - 50*x^3)*Log[x] + (125*x + 50*x^2 - 
25*x^3 + (100*x^2 - 25*x^3)*Log[x])*Log[2*x^2]))/(4*x - 4*x^2 + x^3),x]
 
output
Integrate[(2^((-125 - 25*x^2*Log[x])/(-2 + x))*(x^2)^((-125 - 25*x^2*Log[x 
])/(-2 + x))*(500 - 250*x + (100*x^2 - 50*x^3)*Log[x] + (125*x + 50*x^2 - 
25*x^3 + (100*x^2 - 25*x^3)*Log[x])*Log[2*x^2]))/(4*x - 4*x^2 + x^3), x]
 
3.23.87.3 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.

\(\displaystyle \int \frac {2^{\frac {-25 x^2 \log (x)-125}{x-2}} \left (x^2\right )^{\frac {-25 x^2 \log (x)-125}{x-2}} \left (\left (100 x^2-50 x^3\right ) \log (x)+\left (-25 x^3+50 x^2+\left (100 x^2-25 x^3\right ) \log (x)+125 x\right ) \log \left (2 x^2\right )-250 x+500\right )}{x^3-4 x^2+4 x} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {2^{\frac {-25 x^2 \log (x)-125}{x-2}} \left (x^2\right )^{\frac {-25 x^2 \log (x)-125}{x-2}} \left (\left (100 x^2-50 x^3\right ) \log (x)+\left (-25 x^3+50 x^2+\left (100 x^2-25 x^3\right ) \log (x)+125 x\right ) \log \left (2 x^2\right )-250 x+500\right )}{x \left (x^2-4 x+4\right )}dx\)

\(\Big \downarrow \) 7277

\(\displaystyle 4 \int \frac {25\ 2^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}-2} \left (x^2\right )^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}} \left (-10 x+2 \left (2 x^2-x^3\right ) \log (x)+\left (-x^3+2 x^2+5 x+\left (4 x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+20\right )}{(2-x)^2 x}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 100 \int \frac {2^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}-2} \left (x^2\right )^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}} \left (-10 x+2 \left (2 x^2-x^3\right ) \log (x)+\left (-x^3+2 x^2+5 x+\left (4 x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+20\right )}{(2-x)^2 x}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle 100 \int \frac {2^{\frac {-25 \log (x) x^2-2 x-121}{x-2}} \left (x^2\right )^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}} \left (-10 x+2 \left (2 x^2-x^3\right ) \log (x)+\left (-x^3+2 x^2+5 x+\left (4 x^2-x^3\right ) \log (x)\right ) \log \left (2 x^2\right )+20\right )}{(2-x)^2 x}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 100 \int \left (-\frac {2^{\frac {-25 \log (x) x^2-2 x-121}{x-2}+1} \left (\log (x) x^2+5\right ) \left (x^2\right )^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}}}{(x-2) x}-\frac {2^{\frac {-25 \log (x) x^2-2 x-121}{x-2}} \left (\log (x) x^2+x^2-4 \log (x) x-2 x-5\right ) \log \left (2 x^2\right ) \left (x^2\right )^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}}}{(x-2)^2}\right )dx\)

\(\Big \downarrow \) 7292

\(\displaystyle 100 \int \left (-\frac {2^{\frac {-25 \log (x) x^2-x-123}{x-2}} \left (\log (x) x^2+5\right ) \left (x^2\right )^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}}}{(x-2) x}-\frac {2^{\frac {-25 \log (x) x^2-2 x-121}{x-2}} \left (\log (x) x^2+x^2-4 \log (x) x-2 x-5\right ) \log \left (2 x^2\right ) \left (x^2\right )^{\frac {25 \left (\log (x) x^2+5\right )}{2-x}}}{(x-2)^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 100 \int \left (\frac {2^{\frac {-25 \log (x) x^2-x-123}{x-2}} \left (\log (x) x^2+5\right ) \left (x^2\right )^{-\frac {25 \left (\log (x) x^2+5\right )}{x-2}}}{(2-x) x}+\frac {2^{\frac {-25 \log (x) x^2-2 x-121}{x-2}} \left (-\log (x) x^2-x^2+4 \log (x) x+2 x+5\right ) \log \left (2 x^2\right ) \left (x^2\right )^{-\frac {25 \left (\log (x) x^2+5\right )}{x-2}}}{(2-x)^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle 100 \int \left (\frac {2^{\frac {-25 \log (x) x^2-x-123}{x-2}} \left (\log (x) x^2+5\right ) \left (x^2\right )^{-\frac {25 \left (\log (x) x^2+5\right )}{x-2}}}{(2-x) x}+\frac {2^{\frac {-25 \log (x) x^2-2 x-121}{x-2}} \left (-\log (x) x^2-x^2+4 \log (x) x+2 x+5\right ) \log \left (2 x^2\right ) \left (x^2\right )^{-\frac {25 \left (\log (x) x^2+5\right )}{x-2}}}{(2-x)^2}\right )dx\)

input
Int[(2^((-125 - 25*x^2*Log[x])/(-2 + x))*(x^2)^((-125 - 25*x^2*Log[x])/(-2 
 + x))*(500 - 250*x + (100*x^2 - 50*x^3)*Log[x] + (125*x + 50*x^2 - 25*x^3 
 + (100*x^2 - 25*x^3)*Log[x])*Log[2*x^2]))/(4*x - 4*x^2 + x^3),x]
 
output
$Aborted
 

3.23.87.3.1 Defintions of rubi rules used

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 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 7277
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> 
 Simp[1/(4^p*c^p)   Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} 
, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] &&  !AlgebraicFu 
nctionQ[u, x]
 

rule 7292
Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7299
Int[u_, x_] :> CannotIntegrate[u, x]
 
3.23.87.4 Maple [A] (verified)

Time = 3.76 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.53

method result size
parallelrisch \({\mathrm e}^{\frac {\left (-25 x^{2} \ln \left (x \right )-125\right ) \ln \left (2 x^{2}\right )}{-2+x}}\) \(23\)
risch \({\mathrm e}^{-\frac {25 \left (5+x^{2} \ln \left (x \right )\right ) \left (-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+4 \ln \left (x \right )+2 \ln \left (2\right )\right )}{2 \left (-2+x \right )}}\) \(75\)

input
int((((-25*x^3+100*x^2)*ln(x)-25*x^3+50*x^2+125*x)*ln(2*x^2)+(-50*x^3+100* 
x^2)*ln(x)-250*x+500)*exp((-25*x^2*ln(x)-125)*ln(2*x^2)/(-2+x))/(x^3-4*x^2 
+4*x),x,method=_RETURNVERBOSE)
 
output
exp((-25*x^2*ln(x)-125)*ln(2*x^2)/(-2+x))
 
3.23.87.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx=e^{\left (-\frac {25 \, {\left (2 \, x^{2} \log \left (x\right )^{2} + {\left (x^{2} \log \left (2\right ) + 10\right )} \log \left (x\right ) + 5 \, \log \left (2\right )\right )}}{x - 2}\right )} \]

input
integrate((((-25*x^3+100*x^2)*log(x)-25*x^3+50*x^2+125*x)*log(2*x^2)+(-50* 
x^3+100*x^2)*log(x)-250*x+500)*exp((-25*x^2*log(x)-125)*log(2*x^2)/(-2+x)) 
/(x^3-4*x^2+4*x),x, algorithm=\
 
output
e^(-25*(2*x^2*log(x)^2 + (x^2*log(2) + 10)*log(x) + 5*log(2))/(x - 2))
 
3.23.87.6 Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.56 \[ \int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx=e^{\frac {\left (- 25 x^{2} \log {\left (x \right )} - 125\right ) \left (2 \log {\left (x \right )} + \log {\left (2 \right )}\right )}{x - 2}} \]

input
integrate((((-25*x**3+100*x**2)*ln(x)-25*x**3+50*x**2+125*x)*ln(2*x**2)+(- 
50*x**3+100*x**2)*ln(x)-250*x+500)*exp((-25*x**2*ln(x)-125)*ln(2*x**2)/(-2 
+x))/(x**3-4*x**2+4*x),x)
 
output
exp((-25*x**2*log(x) - 125)*(2*log(x) + log(2))/(x - 2))
 
3.23.87.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (25) = 50\).

Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.58 \[ \int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx=e^{\left (-25 \, x \log \left (2\right ) \log \left (x\right ) - 50 \, x \log \left (x\right )^{2} - 50 \, \log \left (2\right ) \log \left (x\right ) - 100 \, \log \left (x\right )^{2} - \frac {100 \, \log \left (2\right ) \log \left (x\right )}{x - 2} - \frac {200 \, \log \left (x\right )^{2}}{x - 2} - \frac {125 \, \log \left (2\right )}{x - 2} - \frac {250 \, \log \left (x\right )}{x - 2}\right )} \]

input
integrate((((-25*x^3+100*x^2)*log(x)-25*x^3+50*x^2+125*x)*log(2*x^2)+(-50* 
x^3+100*x^2)*log(x)-250*x+500)*exp((-25*x^2*log(x)-125)*log(2*x^2)/(-2+x)) 
/(x^3-4*x^2+4*x),x, algorithm=\
 
output
e^(-25*x*log(2)*log(x) - 50*x*log(x)^2 - 50*log(2)*log(x) - 100*log(x)^2 - 
 100*log(2)*log(x)/(x - 2) - 200*log(x)^2/(x - 2) - 125*log(2)/(x - 2) - 2 
50*log(x)/(x - 2))
 
3.23.87.8 Giac [F]

\[ \int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx=\int { -\frac {25 \, {\left ({\left (x^{3} - 2 \, x^{2} + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x\right ) - 5 \, x\right )} \log \left (2 \, x^{2}\right ) + 2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right ) + 10 \, x - 20\right )}}{{\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \left (2 \, x^{2}\right )^{\frac {25 \, {\left (x^{2} \log \left (x\right ) + 5\right )}}{x - 2}}} \,d x } \]

input
integrate((((-25*x^3+100*x^2)*log(x)-25*x^3+50*x^2+125*x)*log(2*x^2)+(-50* 
x^3+100*x^2)*log(x)-250*x+500)*exp((-25*x^2*log(x)-125)*log(2*x^2)/(-2+x)) 
/(x^3-4*x^2+4*x),x, algorithm=\
 
output
integrate(-25*((x^3 - 2*x^2 + (x^3 - 4*x^2)*log(x) - 5*x)*log(2*x^2) + 2*( 
x^3 - 2*x^2)*log(x) + 10*x - 20)/((x^3 - 4*x^2 + 4*x)*(2*x^2)^(25*(x^2*log 
(x) + 5)/(x - 2))), x)
 
3.23.87.9 Mupad [B] (verification not implemented)

Time = 14.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \frac {2^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (x^2\right )^{\frac {-125-25 x^2 \log (x)}{-2+x}} \left (500-250 x+\left (100 x^2-50 x^3\right ) \log (x)+\left (125 x+50 x^2-25 x^3+\left (100 x^2-25 x^3\right ) \log (x)\right ) \log \left (2 x^2\right )\right )}{4 x-4 x^2+x^3} \, dx=\frac {{\left (\frac {1}{42535295865117307932921825928971026432\,x^{250}}\right )}^{\frac {1}{x-2}}}{x^{\frac {25\,\left (x^2\,\ln \left (x^2\right )+x^2\,\ln \left (2\right )\right )}{x-2}}} \]

input
int((exp(-(log(2*x^2)*(25*x^2*log(x) + 125))/(x - 2))*(log(x)*(100*x^2 - 5 
0*x^3) - 250*x + log(2*x^2)*(125*x + log(x)*(100*x^2 - 25*x^3) + 50*x^2 - 
25*x^3) + 500))/(4*x - 4*x^2 + x^3),x)
 
output
(1/(42535295865117307932921825928971026432*x^250))^(1/(x - 2))/x^((25*(x^2 
*log(x^2) + x^2*log(2)))/(x - 2))