3.7.64 \(\int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x))}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 (144+72 x+9 x^2+e^4 x^2 \log ^2(x))}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx\) [664]

3.7.64.1 Optimal result

Integrand size = 161, antiderivative size = 33 \begin {dmath*} \int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} \left (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x)\right )}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 \left (144+72 x+9 x^2+e^4 x^2 \log ^2(x)\right )}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx=\frac {5}{x \left (-x+e^{e^4+\frac {9 (4+x)^2}{x^2 \log ^2(x)}} x\right )} \end {dmath*}

output

5/(exp(9*(4+x)^2/x^2/ln(x)^2+exp(4))*x-x)/x
 

3.7.64.2 Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \begin {dmath*} \int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} \left (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x)\right )}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 \left (144+72 x+9 x^2+e^4 x^2 \log ^2(x)\right )}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx=\frac {5}{\left (-1+e^{e^4+\frac {9 (4+x)^2}{x^2 \log ^2(x)}}\right ) x^2} \end {dmath*}

input

Integrate[(10*x^2*Log[x]^3 + E^((144 + 72*x + 9*x^2 + E^4*x^2*Log[x]^2)/(x 
^2*Log[x]^2))*(1440 + 720*x + 90*x^2 + (1440 + 360*x)*Log[x] - 10*x^2*Log[ 
x]^3))/(x^5*Log[x]^3 - 2*E^((144 + 72*x + 9*x^2 + E^4*x^2*Log[x]^2)/(x^2*L 
og[x]^2))*x^5*Log[x]^3 + E^((2*(144 + 72*x + 9*x^2 + E^4*x^2*Log[x]^2))/(x 
^2*Log[x]^2))*x^5*Log[x]^3),x]
 

output

5/((-1 + E^(E^4 + (9*(4 + x)^2)/(x^2*Log[x]^2)))*x^2)
 

3.7.64.3 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.

\(\displaystyle \int \frac {\left (90 x^2-10 x^2 \log ^3(x)+720 x+(360 x+1440) \log (x)+1440\right ) \exp \left (\frac {9 x^2+e^4 x^2 \log ^2(x)+72 x+144}{x^2 \log ^2(x)}\right )+10 x^2 \log ^3(x)}{-2 x^5 \log ^3(x) \exp \left (\frac {9 x^2+e^4 x^2 \log ^2(x)+72 x+144}{x^2 \log ^2(x)}\right )+x^5 \log ^3(x) \exp \left (\frac {2 \left (9 x^2+e^4 x^2 \log ^2(x)+72 x+144\right )}{x^2 \log ^2(x)}\right )+x^5 \log ^3(x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (90 x^2-10 x^2 \log ^3(x)+720 x+(360 x+1440) \log (x)+1440\right ) \exp \left (\frac {9 x^2+e^4 x^2 \log ^2(x)+72 x+144}{x^2 \log ^2(x)}\right )+10 x^2 \log ^3(x)}{x^5 \left (1-e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 \log ^3(x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1440 e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{x^5 \log ^2(x) \left (e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}-1\right )^2}+\frac {1440 e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{x^5 \log ^3(x) \left (e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}-1\right )^2}+\frac {360 e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{x^4 \log ^2(x) \left (e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}-1\right )^2}+\frac {720 e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{x^4 \log ^3(x) \left (e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}-1\right )^2}-\frac {10 e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{x^3 \left (e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}-1\right )^2}+\frac {10}{x^3 \left (e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}-1\right )^2}+\frac {90 e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{x^3 \log ^3(x) \left (e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}-1\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 1440 \int \frac {e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{\left (-1+e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 x^5 \log ^2(x)}dx+1440 \int \frac {e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{\left (-1+e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 x^5 \log ^3(x)}dx+360 \int \frac {e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{\left (-1+e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 x^4 \log ^2(x)}dx+720 \int \frac {e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{\left (-1+e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 x^4 \log ^3(x)}dx+10 \int \frac {1}{\left (-1+e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 x^3}dx-10 \int \frac {e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{\left (-1+e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 x^3}dx+90 \int \frac {e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}}{\left (-1+e^{\frac {9 (x+4)^2}{x^2 \log ^2(x)}+e^4}\right )^2 x^3 \log ^3(x)}dx\)

input

Int[(10*x^2*Log[x]^3 + E^((144 + 72*x + 9*x^2 + E^4*x^2*Log[x]^2)/(x^2*Log 
[x]^2))*(1440 + 720*x + 90*x^2 + (1440 + 360*x)*Log[x] - 10*x^2*Log[x]^3)) 
/(x^5*Log[x]^3 - 2*E^((144 + 72*x + 9*x^2 + E^4*x^2*Log[x]^2)/(x^2*Log[x]^ 
2))*x^5*Log[x]^3 + E^((2*(144 + 72*x + 9*x^2 + E^4*x^2*Log[x]^2))/(x^2*Log 
[x]^2))*x^5*Log[x]^3),x]
 

output

$Aborted
 

3.7.64.3.1 Defintions of rubi rules used

rule 2009

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

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] 
]
 

3.7.64.4 Maple [A] (verified)

Time = 5.50 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18

method	result	size
risch	\(\frac {5}{x^{2} \left ({\mathrm e}^{\frac {x^{2} {\mathrm e}^{4} \ln \left (x \right )^{2}+9 x^{2}+72 x +144}{x^{2} \ln \left (x \right )^{2}}}-1\right )}\)	\(39\)
parallelrisch	\(\frac {5}{x^{2} \left ({\mathrm e}^{\frac {x^{2} {\mathrm e}^{4} \ln \left (x \right )^{2}+9 x^{2}+72 x +144}{x^{2} \ln \left (x \right )^{2}}}-1\right )}\)	\(39\)

input

int(((-10*x^2*ln(x)^3+(360*x+1440)*ln(x)+90*x^2+720*x+1440)*exp((x^2*exp(4 
)*ln(x)^2+9*x^2+72*x+144)/x^2/ln(x)^2)+10*x^2*ln(x)^3)/(x^5*ln(x)^3*exp((x 
^2*exp(4)*ln(x)^2+9*x^2+72*x+144)/x^2/ln(x)^2)^2-2*x^5*ln(x)^3*exp((x^2*ex 
p(4)*ln(x)^2+9*x^2+72*x+144)/x^2/ln(x)^2)+x^5*ln(x)^3),x,method=_RETURNVER 
BOSE)
 

output

5/x^2/(exp((x^2*exp(4)*ln(x)^2+9*x^2+72*x+144)/x^2/ln(x)^2)-1)
 

3.7.64.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \begin {dmath*} \int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} \left (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x)\right )}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 \left (144+72 x+9 x^2+e^4 x^2 \log ^2(x)\right )}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx=\frac {5}{x^{2} e^{\left (\frac {x^{2} e^{4} \log \left (x\right )^{2} + 9 \, x^{2} + 72 \, x + 144}{x^{2} \log \left (x\right )^{2}}\right )} - x^{2}} \end {dmath*}

input

integrate(((-10*x^2*log(x)^3+(360*x+1440)*log(x)+90*x^2+720*x+1440)*exp((x 
^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)+10*x^2*log(x)^3)/(x^5*log 
(x)^3*exp((x^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)^2-2*x^5*log(x 
)^3*exp((x^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)+x^5*log(x)^3),x 
, algorithm=\
 

output

5/(x^2*e^((x^2*e^4*log(x)^2 + 9*x^2 + 72*x + 144)/(x^2*log(x)^2)) - x^2)
 

3.7.64.6 Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \begin {dmath*} \int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} \left (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x)\right )}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 \left (144+72 x+9 x^2+e^4 x^2 \log ^2(x)\right )}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx=\frac {5}{x^{2} e^{\frac {x^{2} e^{4} \log {\left (x \right )}^{2} + 9 x^{2} + 72 x + 144}{x^{2} \log {\left (x \right )}^{2}}} - x^{2}} \end {dmath*}

input

integrate(((-10*x**2*ln(x)**3+(360*x+1440)*ln(x)+90*x**2+720*x+1440)*exp(( 
x**2*exp(4)*ln(x)**2+9*x**2+72*x+144)/x**2/ln(x)**2)+10*x**2*ln(x)**3)/(x* 
*5*ln(x)**3*exp((x**2*exp(4)*ln(x)**2+9*x**2+72*x+144)/x**2/ln(x)**2)**2-2 
*x**5*ln(x)**3*exp((x**2*exp(4)*ln(x)**2+9*x**2+72*x+144)/x**2/ln(x)**2)+x 
**5*ln(x)**3),x)
 

output

5/(x**2*exp((x**2*exp(4)*log(x)**2 + 9*x**2 + 72*x + 144)/(x**2*log(x)**2) 
) - x**2)
 

3.7.64.7 Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \begin {dmath*} \int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} \left (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x)\right )}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 \left (144+72 x+9 x^2+e^4 x^2 \log ^2(x)\right )}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx=\frac {5}{x^{2} e^{\left (\frac {9}{\log \left (x\right )^{2}} + \frac {72}{x \log \left (x\right )^{2}} + \frac {144}{x^{2} \log \left (x\right )^{2}} + e^{4}\right )} - x^{2}} \end {dmath*}

input

integrate(((-10*x^2*log(x)^3+(360*x+1440)*log(x)+90*x^2+720*x+1440)*exp((x 
^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)+10*x^2*log(x)^3)/(x^5*log 
(x)^3*exp((x^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)^2-2*x^5*log(x 
)^3*exp((x^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)+x^5*log(x)^3),x 
, algorithm=\
 

output

5/(x^2*e^(9/log(x)^2 + 72/(x*log(x)^2) + 144/(x^2*log(x)^2) + e^4) - x^2)
 

3.7.64.8 Giac [F]

\begin {dmath*} \int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} \left (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x)\right )}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 \left (144+72 x+9 x^2+e^4 x^2 \log ^2(x)\right )}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx=\int { \frac {10 \, {\left (x^{2} \log \left (x\right )^{3} - {\left (x^{2} \log \left (x\right )^{3} - 9 \, x^{2} - 36 \, {\left (x + 4\right )} \log \left (x\right ) - 72 \, x - 144\right )} e^{\left (\frac {x^{2} e^{4} \log \left (x\right )^{2} + 9 \, x^{2} + 72 \, x + 144}{x^{2} \log \left (x\right )^{2}}\right )}\right )}}{x^{5} e^{\left (\frac {2 \, {\left (x^{2} e^{4} \log \left (x\right )^{2} + 9 \, x^{2} + 72 \, x + 144\right )}}{x^{2} \log \left (x\right )^{2}}\right )} \log \left (x\right )^{3} - 2 \, x^{5} e^{\left (\frac {x^{2} e^{4} \log \left (x\right )^{2} + 9 \, x^{2} + 72 \, x + 144}{x^{2} \log \left (x\right )^{2}}\right )} \log \left (x\right )^{3} + x^{5} \log \left (x\right )^{3}} \,d x } \end {dmath*}

input

integrate(((-10*x^2*log(x)^3+(360*x+1440)*log(x)+90*x^2+720*x+1440)*exp((x 
^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)+10*x^2*log(x)^3)/(x^5*log 
(x)^3*exp((x^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)^2-2*x^5*log(x 
)^3*exp((x^2*exp(4)*log(x)^2+9*x^2+72*x+144)/x^2/log(x)^2)+x^5*log(x)^3),x 
, algorithm=\
 

output

undef
 

3.7.64.9 Mupad [B] (verification not implemented)

Time = 14.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.09 \begin {dmath*} \int \frac {10 x^2 \log ^3(x)+e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} \left (1440+720 x+90 x^2+(1440+360 x) \log (x)-10 x^2 \log ^3(x)\right )}{x^5 \log ^3(x)-2 e^{\frac {144+72 x+9 x^2+e^4 x^2 \log ^2(x)}{x^2 \log ^2(x)}} x^5 \log ^3(x)+e^{\frac {2 \left (144+72 x+9 x^2+e^4 x^2 \log ^2(x)\right )}{x^2 \log ^2(x)}} x^5 \log ^3(x)} \, dx=\frac {80\,\ln \left (x\right )+x\,\left (20\,\ln \left (x\right )+40\right )+5\,x^2+80}{x^2\,\left ({\mathrm {e}}^{{\mathrm {e}}^4+\frac {9}{{\ln \left (x\right )}^2}+\frac {72}{x\,{\ln \left (x\right )}^2}+\frac {144}{x^2\,{\ln \left (x\right )}^2}}-1\right )\,\left (x+4\right )\,\left (x+4\,\ln \left (x\right )+4\right )} \end {dmath*}

input

int((10*x^2*log(x)^3 + exp((72*x + 9*x^2 + x^2*exp(4)*log(x)^2 + 144)/(x^2 
*log(x)^2))*(720*x + log(x)*(360*x + 1440) - 10*x^2*log(x)^3 + 90*x^2 + 14 
40))/(x^5*log(x)^3 - 2*x^5*exp((72*x + 9*x^2 + x^2*exp(4)*log(x)^2 + 144)/ 
(x^2*log(x)^2))*log(x)^3 + x^5*exp((2*(72*x + 9*x^2 + x^2*exp(4)*log(x)^2 
+ 144))/(x^2*log(x)^2))*log(x)^3),x)
 

output

(80*log(x) + x*(20*log(x) + 40) + 5*x^2 + 80)/(x^2*(exp(exp(4) + 9/log(x)^ 
2 + 72/(x*log(x)^2) + 144/(x^2*log(x)^2)) - 1)*(x + 4)*(x + 4*log(x) + 4))