3.6.70 \(\int \frac {-x+e^{-3-e^x+x} (2 x-x^2+e^x x^2) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx\) [570]

3.6.70.1 Optimal result

Integrand size = 117, antiderivative size = 26 \begin {dmath*} \int \frac {-x+e^{-3-e^x+x} \left (2 x-x^2+e^x x^2\right ) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx=1+\frac {x^2}{e^{-3-e^x+x}+\log (\log (2 x \log (4)))} \end {dmath*}

output

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

3.6.70.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \begin {dmath*} \int \frac {-x+e^{-3-e^x+x} \left (2 x-x^2+e^x x^2\right ) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx=\frac {e^{3+e^x} x^2}{e^x+e^{3+e^x} \log (\log (x \log (16)))} \end {dmath*}

input

Integrate[(-x + E^(-3 - E^x + x)*(2*x - x^2 + E^x*x^2)*Log[2*x*Log[4]] + 2 
*x*Log[2*x*Log[4]]*Log[Log[2*x*Log[4]]])/(E^(-6 - 2*E^x + 2*x)*Log[2*x*Log 
[4]] + 2*E^(-3 - E^x + x)*Log[2*x*Log[4]]*Log[Log[2*x*Log[4]]] + Log[2*x*L 
og[4]]*Log[Log[2*x*Log[4]]]^2),x]
 

output

(E^(3 + E^x)*x^2)/(E^x + E^(3 + E^x)*Log[Log[x*Log[16]]])
 

3.6.70.3 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.

\(\displaystyle \int \frac {e^{x-e^x-3} \left (e^x x^2-x^2+2 x\right ) \log (2 x \log (4))-x+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))+2 e^{x-e^x-3} \log (2 x \log (4)) \log (\log (2 x \log (4)))+e^{2 x-2 e^x-6} \log (2 x \log (4))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{2 \left (e^x+3\right )} \left (e^{x-e^x-3} \left (e^x x^2-x^2+2 x\right ) \log (2 x \log (4))-x+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))\right )}{\log (x \log (16)) \left (e^x+e^{e^x+3} \log (\log (x \log (16)))\right )^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \int e^{3+e^x} x^2dx+\int \frac {e^{3 \left (3+e^x\right )} x^2 \log ^2(\log (x \log (16)))}{\left (e^{3+e^x} \log (\log (x \log (16)))+e^x\right )^2}dx+\int \frac {e^{2 \left (3+e^x\right )} x^2 \log (\log (x \log (16)))}{\left (e^{3+e^x} \log (\log (x \log (16)))+e^x\right )^2}dx-\int \frac {e^{3+e^x} x^2}{e^{3+e^x} \log (\log (x \log (16)))+e^x}dx-2 \int \frac {e^{2 \left (3+e^x\right )} x^2 \log (\log (x \log (16)))}{e^{3+e^x} \log (\log (x \log (16)))+e^x}dx-\int \frac {e^{2 \left (3+e^x\right )} x}{\log (x \log (16)) \left (e^{3+e^x} \log (\log (x \log (16)))+e^x\right )^2}dx+2 \int \frac {e^{3+e^x} x}{e^{3+e^x} \log (\log (x \log (16)))+e^x}dx\)

input

Int[(-x + E^(-3 - E^x + x)*(2*x - x^2 + E^x*x^2)*Log[2*x*Log[4]] + 2*x*Log 
[2*x*Log[4]]*Log[Log[2*x*Log[4]]])/(E^(-6 - 2*E^x + 2*x)*Log[2*x*Log[4]] + 
 2*E^(-3 - E^x + x)*Log[2*x*Log[4]]*Log[Log[2*x*Log[4]]] + Log[2*x*Log[4]] 
*Log[Log[2*x*Log[4]]]^2),x]
 

output

$Aborted
 

3.6.70.3.1 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] 
]
 

3.6.70.4 Maple [A] (verified)

Time = 5.94 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {x^{2}}{{\mathrm e}^{-{\mathrm e}^{x}+x -3}+\ln \left (\ln \left (4 x \ln \left (2\right )\right )\right )}\)	\(23\)
parallelrisch	\(\frac {x^{2}}{{\mathrm e}^{-{\mathrm e}^{x}+x -3}+\ln \left (\ln \left (4 x \ln \left (2\right )\right )\right )}\)	\(23\)

input

int((2*x*ln(4*x*ln(2))*ln(ln(4*x*ln(2)))+(exp(x)*x^2-x^2+2*x)*ln(4*x*ln(2) 
)*exp(-exp(x)+x-3)-x)/(ln(4*x*ln(2))*ln(ln(4*x*ln(2)))^2+2*ln(4*x*ln(2))*e 
xp(-exp(x)+x-3)*ln(ln(4*x*ln(2)))+ln(4*x*ln(2))*exp(-exp(x)+x-3)^2),x,meth 
od=_RETURNVERBOSE)
 

output

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

3.6.70.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \begin {dmath*} \int \frac {-x+e^{-3-e^x+x} \left (2 x-x^2+e^x x^2\right ) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx=\frac {x^{2}}{e^{\left (x - e^{x} - 3\right )} + \log \left (\log \left (4 \, x \log \left (2\right )\right )\right )} \end {dmath*}

input

integrate((2*x*log(4*x*log(2))*log(log(4*x*log(2)))+(exp(x)*x^2-x^2+2*x)*l 
og(4*x*log(2))*exp(-exp(x)+x-3)-x)/(log(4*x*log(2))*log(log(4*x*log(2)))^2 
+2*log(4*x*log(2))*exp(-exp(x)+x-3)*log(log(4*x*log(2)))+log(4*x*log(2))*e 
xp(-exp(x)+x-3)^2),x, algorithm=\
 

output

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

3.6.70.6 Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \begin {dmath*} \int \frac {-x+e^{-3-e^x+x} \left (2 x-x^2+e^x x^2\right ) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx=\frac {x^{2}}{e^{x - e^{x} - 3} + \log {\left (\log {\left (4 x \log {\left (2 \right )} \right )} \right )}} \end {dmath*}

input

integrate((2*x*ln(4*x*ln(2))*ln(ln(4*x*ln(2)))+(exp(x)*x**2-x**2+2*x)*ln(4 
*x*ln(2))*exp(-exp(x)+x-3)-x)/(ln(4*x*ln(2))*ln(ln(4*x*ln(2)))**2+2*ln(4*x 
*ln(2))*exp(-exp(x)+x-3)*ln(ln(4*x*ln(2)))+ln(4*x*ln(2))*exp(-exp(x)+x-3)* 
*2),x)
 

output

x**2/(exp(x - exp(x) - 3) + log(log(4*x*log(2))))
 

3.6.70.7 Maxima [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \begin {dmath*} \int \frac {-x+e^{-3-e^x+x} \left (2 x-x^2+e^x x^2\right ) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx=\frac {x^{2} e^{\left (e^{x} + 3\right )}}{e^{\left (e^{x} + 3\right )} \log \left (2 \, \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (2\right )\right )\right ) + e^{x}} \end {dmath*}

input

integrate((2*x*log(4*x*log(2))*log(log(4*x*log(2)))+(exp(x)*x^2-x^2+2*x)*l 
og(4*x*log(2))*exp(-exp(x)+x-3)-x)/(log(4*x*log(2))*log(log(4*x*log(2)))^2 
+2*log(4*x*log(2))*exp(-exp(x)+x-3)*log(log(4*x*log(2)))+log(4*x*log(2))*e 
xp(-exp(x)+x-3)^2),x, algorithm=\
 

output

x^2*e^(e^x + 3)/(e^(e^x + 3)*log(2*log(2) + log(x) + log(log(2))) + e^x)
 

3.6.70.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3138 vs. \(2 (24) = 48\).

Time = 0.78 (sec) , antiderivative size = 3138, normalized size of antiderivative = 120.69 \begin {dmath*} \int \frac {-x+e^{-3-e^x+x} \left (2 x-x^2+e^x x^2\right ) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx=\text {Too large to display} \end {dmath*}

input

integrate((2*x*log(4*x*log(2))*log(log(4*x*log(2)))+(exp(x)*x^2-x^2+2*x)*l 
og(4*x*log(2))*exp(-exp(x)+x-3)-x)/(log(4*x*log(2))*log(log(4*x*log(2)))^2 
+2*log(4*x*log(2))*exp(-exp(x)+x-3)*log(log(4*x*log(2)))+log(4*x*log(2))*e 
xp(-exp(x)+x-3)^2),x, algorithm=\
 

output

(4*x^4*e^(6*x - 2*e^x + 6)*log(2)^2*log(2*log(2) + log(x) + log(log(2))) - 
 8*x^4*e^(5*x - 2*e^x + 6)*log(2)^2*log(2*log(2) + log(x) + log(log(2))) + 
 4*x^4*e^(4*x - 2*e^x + 6)*log(2)^2*log(2*log(2) + log(x) + log(log(2))) + 
 4*x^4*e^(6*x - 2*e^x + 6)*log(2)*log(x)*log(2*log(2) + log(x) + log(log(2 
))) - 8*x^4*e^(5*x - 2*e^x + 6)*log(2)*log(x)*log(2*log(2) + log(x) + log( 
log(2))) + 4*x^4*e^(4*x - 2*e^x + 6)*log(2)*log(x)*log(2*log(2) + log(x) + 
 log(log(2))) + x^4*e^(6*x - 2*e^x + 6)*log(x)^2*log(2*log(2) + log(x) + l 
og(log(2))) - 2*x^4*e^(5*x - 2*e^x + 6)*log(x)^2*log(2*log(2) + log(x) + l 
og(log(2))) + x^4*e^(4*x - 2*e^x + 6)*log(x)^2*log(2*log(2) + log(x) + log 
(log(2))) + 4*x^4*e^(6*x - 2*e^x + 6)*log(2)*log(2*log(2) + log(x) + log(l 
og(2)))*log(log(2)) - 8*x^4*e^(5*x - 2*e^x + 6)*log(2)*log(2*log(2) + log( 
x) + log(log(2)))*log(log(2)) + 4*x^4*e^(4*x - 2*e^x + 6)*log(2)*log(2*log 
(2) + log(x) + log(log(2)))*log(log(2)) + 2*x^4*e^(6*x - 2*e^x + 6)*log(x) 
*log(2*log(2) + log(x) + log(log(2)))*log(log(2)) - 4*x^4*e^(5*x - 2*e^x + 
 6)*log(x)*log(2*log(2) + log(x) + log(log(2)))*log(log(2)) + 2*x^4*e^(4*x 
 - 2*e^x + 6)*log(x)*log(2*log(2) + log(x) + log(log(2)))*log(log(2)) + x^ 
4*e^(6*x - 2*e^x + 6)*log(2*log(2) + log(x) + log(log(2)))*log(log(2))^2 - 
 2*x^4*e^(5*x - 2*e^x + 6)*log(2*log(2) + log(x) + log(log(2)))*log(log(2) 
)^2 + x^4*e^(4*x - 2*e^x + 6)*log(2*log(2) + log(x) + log(log(2)))*log(log 
(2))^2 + 4*x^4*e^(7*x - 3*e^x + 3)*log(2)^2 - 8*x^4*e^(6*x - 3*e^x + 3)...
 

3.6.70.9 Mupad [F(-1)]

Timed out. \begin {dmath*} \int \frac {-x+e^{-3-e^x+x} \left (2 x-x^2+e^x x^2\right ) \log (2 x \log (4))+2 x \log (2 x \log (4)) \log (\log (2 x \log (4)))}{e^{-6-2 e^x+2 x} \log (2 x \log (4))+2 e^{-3-e^x+x} \log (2 x \log (4)) \log (\log (2 x \log (4)))+\log (2 x \log (4)) \log ^2(\log (2 x \log (4)))} \, dx=\int \frac {{\mathrm {e}}^{x-{\mathrm {e}}^x-3}\,\ln \left (4\,x\,\ln \left (2\right )\right )\,\left (2\,x+x^2\,{\mathrm {e}}^x-x^2\right )-x+2\,x\,\ln \left (\ln \left (4\,x\,\ln \left (2\right )\right )\right )\,\ln \left (4\,x\,\ln \left (2\right )\right )}{\ln \left (4\,x\,\ln \left (2\right )\right )\,{\ln \left (\ln \left (4\,x\,\ln \left (2\right )\right )\right )}^2+2\,{\mathrm {e}}^{x-{\mathrm {e}}^x-3}\,\ln \left (4\,x\,\ln \left (2\right )\right )\,\ln \left (\ln \left (4\,x\,\ln \left (2\right )\right )\right )+{\mathrm {e}}^{2\,x-2\,{\mathrm {e}}^x-6}\,\ln \left (4\,x\,\ln \left (2\right )\right )} \,d x \end {dmath*}

input

int((exp(x - exp(x) - 3)*log(4*x*log(2))*(2*x + x^2*exp(x) - x^2) - x + 2* 
x*log(log(4*x*log(2)))*log(4*x*log(2)))/(log(log(4*x*log(2)))^2*log(4*x*lo 
g(2)) + exp(2*x - 2*exp(x) - 6)*log(4*x*log(2)) + 2*log(log(4*x*log(2)))*e 
xp(x - exp(x) - 3)*log(4*x*log(2))),x)
 

output

int((exp(x - exp(x) - 3)*log(4*x*log(2))*(2*x + x^2*exp(x) - x^2) - x + 2* 
x*log(log(4*x*log(2)))*log(4*x*log(2)))/(log(log(4*x*log(2)))^2*log(4*x*lo 
g(2)) + exp(2*x - 2*exp(x) - 6)*log(4*x*log(2)) + 2*log(log(4*x*log(2)))*e 
xp(x - exp(x) - 3)*log(4*x*log(2))), x)