3.12.63 \(\int \frac {-500 x^3+500 x^3 \log (x)+(500 x-500 x^2) \log ^2(x)+(-500 x+750 x^2) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} (-500 x \log ^2(x)+(500 x+128000 e^{256 x^2} x^3) \log ^3(x)+(250+e^{256 x^2} (-128000 x+128000 x^2)) \log ^5(x))}{\log ^5(x)} \, dx\) [1163]

3.12.63.1 Optimal result

Integrand size = 143, antiderivative size = 27 \begin {dmath*} \int \frac {-500 x^3+500 x^3 \log (x)+\left (500 x-500 x^2\right ) \log ^2(x)+\left (-500 x+750 x^2\right ) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} \left (-500 x \log ^2(x)+\left (500 x+128000 e^{256 x^2} x^3\right ) \log ^3(x)+\left (250+e^{256 x^2} \left (-128000 x+128000 x^2\right )\right ) \log ^5(x)\right )}{\log ^5(x)} \, dx=5 \left (5-5 \left (e^{e^{256 x^2}}+x+\frac {x^2}{\log ^2(x)}\right )\right )^2 \end {dmath*}

output

5*(5-5*exp(exp(256*x^2))-5*x-5*x^2/ln(x)^2)^2
 

3.12.63.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(27)=54\).

Time = 0.74 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \begin {dmath*} \int \frac {-500 x^3+500 x^3 \log (x)+\left (500 x-500 x^2\right ) \log ^2(x)+\left (-500 x+750 x^2\right ) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} \left (-500 x \log ^2(x)+\left (500 x+128000 e^{256 x^2} x^3\right ) \log ^3(x)+\left (250+e^{256 x^2} \left (-128000 x+128000 x^2\right )\right ) \log ^5(x)\right )}{\log ^5(x)} \, dx=125 \left (e^{2 e^{256 x^2}}+2 e^{e^{256 x^2}} (-1+x)+(-2+x) x+\frac {x^4}{\log ^4(x)}+\frac {2 x^2 \left (-1+e^{e^{256 x^2}}+x\right )}{\log ^2(x)}\right ) \end {dmath*}

input

Integrate[(-500*x^3 + 500*x^3*Log[x] + (500*x - 500*x^2)*Log[x]^2 + (-500* 
x + 750*x^2)*Log[x]^3 + 128000*E^(2*E^(256*x^2) + 256*x^2)*x*Log[x]^5 + (- 
250 + 250*x)*Log[x]^5 + E^E^(256*x^2)*(-500*x*Log[x]^2 + (500*x + 128000*E 
^(256*x^2)*x^3)*Log[x]^3 + (250 + E^(256*x^2)*(-128000*x + 128000*x^2))*Lo 
g[x]^5))/Log[x]^5,x]
 

output

125*(E^(2*E^(256*x^2)) + 2*E^E^(256*x^2)*(-1 + x) + (-2 + x)*x + x^4/Log[x 
]^4 + (2*x^2*(-1 + E^E^(256*x^2) + x))/Log[x]^2)
 

3.12.63.3 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.

\(\displaystyle \int \frac {-500 x^3+500 x^3 \log (x)+128000 e^{256 x^2+2 e^{256 x^2}} x \log ^5(x)+\left (750 x^2-500 x\right ) \log ^3(x)+\left (500 x-500 x^2\right ) \log ^2(x)+e^{e^{256 x^2}} \left (\left (e^{256 x^2} \left (128000 x^2-128000 x\right )+250\right ) \log ^5(x)+\left (128000 e^{256 x^2} x^3+500 x\right ) \log ^3(x)-500 x \log ^2(x)\right )+(250 x-250) \log ^5(x)}{\log ^5(x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {250 \left (x^2+\left (e^{e^{256 x^2}}+x-1\right ) \log ^2(x)\right ) \left (\left (512 e^{256 x^2+e^{256 x^2}} x+1\right ) \log ^3(x)-2 x+2 x \log (x)\right )}{\log ^5(x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 250 \int -\frac {\left (x^2-\left (-x-e^{e^{256 x^2}}+1\right ) \log ^2(x)\right ) \left (-\left (\left (512 e^{256 x^2+e^{256 x^2}} x+1\right ) \log ^3(x)\right )-2 x \log (x)+2 x\right )}{\log ^5(x)}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -250 \int \frac {\left (x^2-\left (-x-e^{e^{256 x^2}}+1\right ) \log ^2(x)\right ) \left (-\left (\left (512 e^{256 x^2+e^{256 x^2}} x+1\right ) \log ^3(x)\right )-2 x \log (x)+2 x\right )}{\log ^5(x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -250 \int \left (-\frac {\left (\log ^3(x)+2 x \log (x)-2 x\right ) \left (x^2+\log ^2(x) x+e^{e^{256 x^2}} \log ^2(x)-\log ^2(x)\right )}{\log ^5(x)}-\frac {512 e^{256 x^2+e^{256 x^2}} x \left (x^2+\log ^2(x) x+e^{e^{256 x^2}} \log ^2(x)-\log ^2(x)\right )}{\log ^2(x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -250 \left (-\int e^{e^{256 x^2}}dx-512 \int e^{256 x^2+e^{256 x^2}} x^2dx+2 \int \frac {e^{e^{256 x^2}} x}{\log ^3(x)}dx-2 \int \frac {e^{e^{256 x^2}} x}{\log ^2(x)}dx-512 \int \frac {e^{256 x^2+e^{256 x^2}} x^3}{\log ^2(x)}dx-\frac {x^4}{2 \log ^4(x)}-\frac {x^3}{\log ^2(x)}-\frac {x^2}{2}+e^{e^{256 x^2}}-\frac {1}{2} e^{2 e^{256 x^2}}+\frac {x^2}{\log ^2(x)}+x\right )\)

input

Int[(-500*x^3 + 500*x^3*Log[x] + (500*x - 500*x^2)*Log[x]^2 + (-500*x + 75 
0*x^2)*Log[x]^3 + 128000*E^(2*E^(256*x^2) + 256*x^2)*x*Log[x]^5 + (-250 + 
250*x)*Log[x]^5 + E^E^(256*x^2)*(-500*x*Log[x]^2 + (500*x + 128000*E^(256* 
x^2)*x^3)*Log[x]^3 + (250 + E^(256*x^2)*(-128000*x + 128000*x^2))*Log[x]^5 
))/Log[x]^5,x]
 

output

$Aborted
 

3.12.63.3.1 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 2009

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

rule 7239

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

rule 7293

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

3.12.63.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(27)=54\).

Time = 1.90 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81

method	result	size
risch	\(125 x^{2}-250 x +\frac {125 x^{2} \left (2 x \ln \left (x \right )^{2}+x^{2}-2 \ln \left (x \right )^{2}\right )}{\ln \left (x \right )^{4}}+125 \,{\mathrm e}^{2 \,{\mathrm e}^{256 x^{2}}}+\frac {250 \left (x \ln \left (x \right )^{2}+x^{2}-\ln \left (x \right )^{2}\right ) {\mathrm e}^{{\mathrm e}^{256 x^{2}}}}{\ln \left (x \right )^{2}}\)	\(76\)
parallelrisch	\(-\frac {-125 x^{2} \ln \left (x \right )^{4}-250 \ln \left (x \right )^{4} {\mathrm e}^{{\mathrm e}^{256 x^{2}}} x -125 \,{\mathrm e}^{2 \,{\mathrm e}^{256 x^{2}}} \ln \left (x \right )^{4}-250 x^{3} \ln \left (x \right )^{2}-250 \ln \left (x \right )^{2} {\mathrm e}^{{\mathrm e}^{256 x^{2}}} x^{2}+250 x \ln \left (x \right )^{4}+250 \ln \left (x \right )^{4} {\mathrm e}^{{\mathrm e}^{256 x^{2}}}-125 x^{4}+250 x^{2} \ln \left (x \right )^{2}}{\ln \left (x \right )^{4}}\)	\(105\)

input

int((128000*x*exp(256*x^2)*ln(x)^5*exp(exp(256*x^2))^2+(((128000*x^2-12800 
0*x)*exp(256*x^2)+250)*ln(x)^5+(128000*x^3*exp(256*x^2)+500*x)*ln(x)^3-500 
*x*ln(x)^2)*exp(exp(256*x^2))+(250*x-250)*ln(x)^5+(750*x^2-500*x)*ln(x)^3+ 
(-500*x^2+500*x)*ln(x)^2+500*x^3*ln(x)-500*x^3)/ln(x)^5,x,method=_RETURNVE 
RBOSE)
 

output

125*x^2-250*x+125*x^2*(2*x*ln(x)^2+x^2-2*ln(x)^2)/ln(x)^4+125*exp(2*exp(25 
6*x^2))+250*(x*ln(x)^2+x^2-ln(x)^2)/ln(x)^2*exp(exp(256*x^2))
 

3.12.63.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (22) = 44\).

Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.85 \begin {dmath*} \int \frac {-500 x^3+500 x^3 \log (x)+\left (500 x-500 x^2\right ) \log ^2(x)+\left (-500 x+750 x^2\right ) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} \left (-500 x \log ^2(x)+\left (500 x+128000 e^{256 x^2} x^3\right ) \log ^3(x)+\left (250+e^{256 x^2} \left (-128000 x+128000 x^2\right )\right ) \log ^5(x)\right )}{\log ^5(x)} \, dx=\frac {125 \, {\left ({\left (x^{2} - 2 \, x\right )} \log \left (x\right )^{4} + e^{\left (2 \, e^{\left (256 \, x^{2}\right )}\right )} \log \left (x\right )^{4} + x^{4} + 2 \, {\left (x^{3} - x^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left ({\left (x - 1\right )} \log \left (x\right )^{4} + x^{2} \log \left (x\right )^{2}\right )} e^{\left (e^{\left (256 \, x^{2}\right )}\right )}\right )}}{\log \left (x\right )^{4}} \end {dmath*}

input

integrate((128000*x*exp(256*x^2)*log(x)^5*exp(exp(256*x^2))^2+(((128000*x^ 
2-128000*x)*exp(256*x^2)+250)*log(x)^5+(128000*x^3*exp(256*x^2)+500*x)*log 
(x)^3-500*x*log(x)^2)*exp(exp(256*x^2))+(250*x-250)*log(x)^5+(750*x^2-500* 
x)*log(x)^3+(-500*x^2+500*x)*log(x)^2+500*x^3*log(x)-500*x^3)/log(x)^5,x, 
algorithm=\
 

output

125*((x^2 - 2*x)*log(x)^4 + e^(2*e^(256*x^2))*log(x)^4 + x^4 + 2*(x^3 - x^ 
2)*log(x)^2 + 2*((x - 1)*log(x)^4 + x^2*log(x)^2)*e^(e^(256*x^2)))/log(x)^ 
4
 

3.12.63.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).

Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.07 \begin {dmath*} \int \frac {-500 x^3+500 x^3 \log (x)+\left (500 x-500 x^2\right ) \log ^2(x)+\left (-500 x+750 x^2\right ) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} \left (-500 x \log ^2(x)+\left (500 x+128000 e^{256 x^2} x^3\right ) \log ^3(x)+\left (250+e^{256 x^2} \left (-128000 x+128000 x^2\right )\right ) \log ^5(x)\right )}{\log ^5(x)} \, dx=125 x^{2} - 250 x + \frac {125 x^{4} + \left (250 x^{3} - 250 x^{2}\right ) \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{4}} + \frac {\left (250 x^{2} + 250 x \log {\left (x \right )}^{2} - 250 \log {\left (x \right )}^{2}\right ) e^{e^{256 x^{2}}} + 125 e^{2 e^{256 x^{2}}} \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{2}} \end {dmath*}

input

integrate((128000*x*exp(256*x**2)*ln(x)**5*exp(exp(256*x**2))**2+(((128000 
*x**2-128000*x)*exp(256*x**2)+250)*ln(x)**5+(128000*x**3*exp(256*x**2)+500 
*x)*ln(x)**3-500*x*ln(x)**2)*exp(exp(256*x**2))+(250*x-250)*ln(x)**5+(750* 
x**2-500*x)*ln(x)**3+(-500*x**2+500*x)*ln(x)**2+500*x**3*ln(x)-500*x**3)/l 
n(x)**5,x)
 

output

125*x**2 - 250*x + (125*x**4 + (250*x**3 - 250*x**2)*log(x)**2)/log(x)**4 
+ ((250*x**2 + 250*x*log(x)**2 - 250*log(x)**2)*exp(exp(256*x**2)) + 125*e 
xp(2*exp(256*x**2))*log(x)**2)/log(x)**2
 

3.12.63.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (22) = 44\).

Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.74 \begin {dmath*} \int \frac {-500 x^3+500 x^3 \log (x)+\left (500 x-500 x^2\right ) \log ^2(x)+\left (-500 x+750 x^2\right ) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} \left (-500 x \log ^2(x)+\left (500 x+128000 e^{256 x^2} x^3\right ) \log ^3(x)+\left (250+e^{256 x^2} \left (-128000 x+128000 x^2\right )\right ) \log ^5(x)\right )}{\log ^5(x)} \, dx=125 \, x^{2} - 250 \, x + \frac {125 \, {\left (e^{\left (2 \, e^{\left (256 \, x^{2}\right )}\right )} \log \left (x\right )^{4} + x^{4} + 2 \, {\left (x^{3} - x^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left ({\left (x - 1\right )} \log \left (x\right )^{4} + x^{2} \log \left (x\right )^{2}\right )} e^{\left (e^{\left (256 \, x^{2}\right )}\right )}\right )}}{\log \left (x\right )^{4}} \end {dmath*}

input

integrate((128000*x*exp(256*x^2)*log(x)^5*exp(exp(256*x^2))^2+(((128000*x^ 
2-128000*x)*exp(256*x^2)+250)*log(x)^5+(128000*x^3*exp(256*x^2)+500*x)*log 
(x)^3-500*x*log(x)^2)*exp(exp(256*x^2))+(250*x-250)*log(x)^5+(750*x^2-500* 
x)*log(x)^3+(-500*x^2+500*x)*log(x)^2+500*x^3*log(x)-500*x^3)/log(x)^5,x, 
algorithm=\
 

output

125*x^2 - 250*x + 125*(e^(2*e^(256*x^2))*log(x)^4 + x^4 + 2*(x^3 - x^2)*lo 
g(x)^2 + 2*((x - 1)*log(x)^4 + x^2*log(x)^2)*e^(e^(256*x^2)))/log(x)^4
 

3.12.63.8 Giac [F]

\begin {dmath*} \int \frac {-500 x^3+500 x^3 \log (x)+\left (500 x-500 x^2\right ) \log ^2(x)+\left (-500 x+750 x^2\right ) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} \left (-500 x \log ^2(x)+\left (500 x+128000 e^{256 x^2} x^3\right ) \log ^3(x)+\left (250+e^{256 x^2} \left (-128000 x+128000 x^2\right )\right ) \log ^5(x)\right )}{\log ^5(x)} \, dx=\int { \frac {250 \, {\left (512 \, x e^{\left (256 \, x^{2} + 2 \, e^{\left (256 \, x^{2}\right )}\right )} \log \left (x\right )^{5} + {\left (x - 1\right )} \log \left (x\right )^{5} + 2 \, x^{3} \log \left (x\right ) + {\left (3 \, x^{2} - 2 \, x\right )} \log \left (x\right )^{3} - 2 \, x^{3} - 2 \, {\left (x^{2} - x\right )} \log \left (x\right )^{2} + {\left ({\left (512 \, {\left (x^{2} - x\right )} e^{\left (256 \, x^{2}\right )} + 1\right )} \log \left (x\right )^{5} + 2 \, {\left (256 \, x^{3} e^{\left (256 \, x^{2}\right )} + x\right )} \log \left (x\right )^{3} - 2 \, x \log \left (x\right )^{2}\right )} e^{\left (e^{\left (256 \, x^{2}\right )}\right )}\right )}}{\log \left (x\right )^{5}} \,d x } \end {dmath*}

input

integrate((128000*x*exp(256*x^2)*log(x)^5*exp(exp(256*x^2))^2+(((128000*x^ 
2-128000*x)*exp(256*x^2)+250)*log(x)^5+(128000*x^3*exp(256*x^2)+500*x)*log 
(x)^3-500*x*log(x)^2)*exp(exp(256*x^2))+(250*x-250)*log(x)^5+(750*x^2-500* 
x)*log(x)^3+(-500*x^2+500*x)*log(x)^2+500*x^3*log(x)-500*x^3)/log(x)^5,x, 
algorithm=\
 

output

integrate(250*(512*x*e^(256*x^2 + 2*e^(256*x^2))*log(x)^5 + (x - 1)*log(x) 
^5 + 2*x^3*log(x) + (3*x^2 - 2*x)*log(x)^3 - 2*x^3 - 2*(x^2 - x)*log(x)^2 
+ ((512*(x^2 - x)*e^(256*x^2) + 1)*log(x)^5 + 2*(256*x^3*e^(256*x^2) + x)* 
log(x)^3 - 2*x*log(x)^2)*e^(e^(256*x^2)))/log(x)^5, x)
 

3.12.63.9 Mupad [B] (verification not implemented)

Time = 14.79 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04 \begin {dmath*} \int \frac {-500 x^3+500 x^3 \log (x)+\left (500 x-500 x^2\right ) \log ^2(x)+\left (-500 x+750 x^2\right ) \log ^3(x)+128000 e^{2 e^{256 x^2}+256 x^2} x \log ^5(x)+(-250+250 x) \log ^5(x)+e^{e^{256 x^2}} \left (-500 x \log ^2(x)+\left (500 x+128000 e^{256 x^2} x^3\right ) \log ^3(x)+\left (250+e^{256 x^2} \left (-128000 x+128000 x^2\right )\right ) \log ^5(x)\right )}{\log ^5(x)} \, dx=125\,{\mathrm {e}}^{2\,{\mathrm {e}}^{256\,x^2}}-250\,{\mathrm {e}}^{{\mathrm {e}}^{256\,x^2}}-250\,x-\frac {250\,x^2}{{\ln \left (x\right )}^2}+\frac {250\,x^3}{{\ln \left (x\right )}^2}+\frac {125\,x^4}{{\ln \left (x\right )}^4}+125\,x^2+250\,x\,{\mathrm {e}}^{{\mathrm {e}}^{256\,x^2}}+\frac {250\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{256\,x^2}}}{{\ln \left (x\right )}^2} \end {dmath*}

input

int((log(x)^2*(500*x - 500*x^2) - log(x)^3*(500*x - 750*x^2) + 500*x^3*log 
(x) - 500*x^3 + log(x)^5*(250*x - 250) - exp(exp(256*x^2))*(500*x*log(x)^2 
 + log(x)^5*(exp(256*x^2)*(128000*x - 128000*x^2) - 250) - log(x)^3*(500*x 
 + 128000*x^3*exp(256*x^2))) + 128000*x*exp(2*exp(256*x^2))*exp(256*x^2)*l 
og(x)^5)/log(x)^5,x)
 

output

125*exp(2*exp(256*x^2)) - 250*exp(exp(256*x^2)) - 250*x - (250*x^2)/log(x) 
^2 + (250*x^3)/log(x)^2 + (125*x^4)/log(x)^4 + 125*x^2 + 250*x*exp(exp(256 
*x^2)) + (250*x^2*exp(exp(256*x^2)))/log(x)^2