\(\int \frac {e^{-2 x+e^{-2 x} (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x (-2 x^3-2 x^2 \log (5)) \log (x)+e^{2 x} x \log ^2(x)+(e^x (-2 x^3-2 x^2 \log (5))+2 e^{2 x} x \log (x)) \log (-\frac {\log (x)}{-2+x})+e^{2 x} x \log ^2(-\frac {\log (x)}{-2+x}))} (e^x (4 x^2-2 x^3+(4 x-2 x^2) \log (5))+(-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+(-16 x^3+16 x^4-4 x^5) \log (5)+(-6 x^2+7 x^3-2 x^4) \log ^2(5)+e^x (4 x^2+4 x \log (5))) \log (x)+(-4 e^{2 x}+e^x (12 x^2-10 x^3+2 x^4+(8 x-8 x^2+2 x^3) \log (5))) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+(e^{2 x} (-4+2 x)+(-4 e^{2 x}+e^x (12 x^2-10 x^3+2 x^4+(8 x-8 x^2+2 x^3) \log (5))) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)) \log (-\frac {\log (x)}{-2+x})+e^{2 x} (-2+x) \log (x) \log ^2(-\frac {\log (x)}{-2+x}))}{(-2+x) \log (x)} \, dx\) [7076]

Optimal result

Integrand size = 414, antiderivative size = 32 \[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=e^{x \left (-e^{-x} x (x+\log (5))+\log (x)+\log \left (\frac {\log (x)}{2-x}\right )\right )^2} \]

[Out]

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

Rubi [F(-1)]

Timed out. \[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=\text {\$Aborted} \]

[In]

Int[(E^(-2*x + (x^5 + 2*x^4*Log[5] + x^3*Log[5]^2 + E^x*(-2*x^3 - 2*x^2*Log[5])*Log[x] + E^(2*x)*x*Log[x]^2 +
(E^x*(-2*x^3 - 2*x^2*Log[5]) + 2*E^(2*x)*x*Log[x])*Log[-(Log[x]/(-2 + x))] + E^(2*x)*x*Log[-(Log[x]/(-2 + x))]
^2)/E^(2*x))*(E^x*(4*x^2 - 2*x^3 + (4*x - 2*x^2)*Log[5]) + (-10*x^4 + 9*x^5 - 2*x^6 + E^(2*x)*(-4 + 2*x) + (-1
6*x^3 + 16*x^4 - 4*x^5)*Log[5] + (-6*x^2 + 7*x^3 - 2*x^4)*Log[5]^2 + E^x*(4*x^2 + 4*x*Log[5]))*Log[x] + (-4*E^
(2*x) + E^x*(12*x^2 - 10*x^3 + 2*x^4 + (8*x - 8*x^2 + 2*x^3)*Log[5]))*Log[x]^2 + E^(2*x)*(-2 + x)*Log[x]^3 + (
E^(2*x)*(-4 + 2*x) + (-4*E^(2*x) + E^x*(12*x^2 - 10*x^3 + 2*x^4 + (8*x - 8*x^2 + 2*x^3)*Log[5]))*Log[x] + E^(2
*x)*(-4 + 2*x)*Log[x]^2)*Log[-(Log[x]/(-2 + x))] + E^(2*x)*(-2 + x)*Log[x]*Log[-(Log[x]/(-2 + x))]^2))/((-2 +
x)*Log[x]),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [F]

\[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=\int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx \]

[In]

Integrate[(E^(-2*x + (x^5 + 2*x^4*Log[5] + x^3*Log[5]^2 + E^x*(-2*x^3 - 2*x^2*Log[5])*Log[x] + E^(2*x)*x*Log[x
]^2 + (E^x*(-2*x^3 - 2*x^2*Log[5]) + 2*E^(2*x)*x*Log[x])*Log[-(Log[x]/(-2 + x))] + E^(2*x)*x*Log[-(Log[x]/(-2
+ x))]^2)/E^(2*x))*(E^x*(4*x^2 - 2*x^3 + (4*x - 2*x^2)*Log[5]) + (-10*x^4 + 9*x^5 - 2*x^6 + E^(2*x)*(-4 + 2*x)
+ (-16*x^3 + 16*x^4 - 4*x^5)*Log[5] + (-6*x^2 + 7*x^3 - 2*x^4)*Log[5]^2 + E^x*(4*x^2 + 4*x*Log[5]))*Log[x] +
(-4*E^(2*x) + E^x*(12*x^2 - 10*x^3 + 2*x^4 + (8*x - 8*x^2 + 2*x^3)*Log[5]))*Log[x]^2 + E^(2*x)*(-2 + x)*Log[x]
^3 + (E^(2*x)*(-4 + 2*x) + (-4*E^(2*x) + E^x*(12*x^2 - 10*x^3 + 2*x^4 + (8*x - 8*x^2 + 2*x^3)*Log[5]))*Log[x]
+ E^(2*x)*(-4 + 2*x)*Log[x]^2)*Log[-(Log[x]/(-2 + x))] + E^(2*x)*(-2 + x)*Log[x]*Log[-(Log[x]/(-2 + x))]^2))/(
(-2 + x)*Log[x]),x]

[Out]

Integrate[(E^(-2*x + (x^5 + 2*x^4*Log[5] + x^3*Log[5]^2 + E^x*(-2*x^3 - 2*x^2*Log[5])*Log[x] + E^(2*x)*x*Log[x
]^2 + (E^x*(-2*x^3 - 2*x^2*Log[5]) + 2*E^(2*x)*x*Log[x])*Log[-(Log[x]/(-2 + x))] + E^(2*x)*x*Log[-(Log[x]/(-2
+ x))]^2)/E^(2*x))*(E^x*(4*x^2 - 2*x^3 + (4*x - 2*x^2)*Log[5]) + (-10*x^4 + 9*x^5 - 2*x^6 + E^(2*x)*(-4 + 2*x)
+ (-16*x^3 + 16*x^4 - 4*x^5)*Log[5] + (-6*x^2 + 7*x^3 - 2*x^4)*Log[5]^2 + E^x*(4*x^2 + 4*x*Log[5]))*Log[x] +
(-4*E^(2*x) + E^x*(12*x^2 - 10*x^3 + 2*x^4 + (8*x - 8*x^2 + 2*x^3)*Log[5]))*Log[x]^2 + E^(2*x)*(-2 + x)*Log[x]
^3 + (E^(2*x)*(-4 + 2*x) + (-4*E^(2*x) + E^x*(12*x^2 - 10*x^3 + 2*x^4 + (8*x - 8*x^2 + 2*x^3)*Log[5]))*Log[x]
+ E^(2*x)*(-4 + 2*x)*Log[x]^2)*Log[-(Log[x]/(-2 + x))] + E^(2*x)*(-2 + x)*Log[x]*Log[-(Log[x]/(-2 + x))]^2))/(
(-2 + x)*Log[x]), x]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.20 (sec) , antiderivative size = 1224, normalized size of antiderivative = 38.25

\[\text {Expression too large to display}\]

[In]

int(((-2+x)*exp(x)^2*ln(x)*ln(-ln(x)/(-2+x))^2+((2*x-4)*exp(x)^2*ln(x)^2+(-4*exp(x)^2+((2*x^3-8*x^2+8*x)*ln(5)
+2*x^4-10*x^3+12*x^2)*exp(x))*ln(x)+(2*x-4)*exp(x)^2)*ln(-ln(x)/(-2+x))+(-2+x)*exp(x)^2*ln(x)^3+(-4*exp(x)^2+(
(2*x^3-8*x^2+8*x)*ln(5)+2*x^4-10*x^3+12*x^2)*exp(x))*ln(x)^2+((2*x-4)*exp(x)^2+(4*x*ln(5)+4*x^2)*exp(x)+(-2*x^
4+7*x^3-6*x^2)*ln(5)^2+(-4*x^5+16*x^4-16*x^3)*ln(5)-2*x^6+9*x^5-10*x^4)*ln(x)+((-2*x^2+4*x)*ln(5)-2*x^3+4*x^2)
*exp(x))*exp((x*exp(x)^2*ln(-ln(x)/(-2+x))^2+(2*x*exp(x)^2*ln(x)+(-2*x^2*ln(5)-2*x^3)*exp(x))*ln(-ln(x)/(-2+x)
)+x*exp(x)^2*ln(x)^2+(-2*x^2*ln(5)-2*x^3)*exp(x)*ln(x)+x^3*ln(5)^2+2*x^4*ln(5)+x^5)/exp(x)^2)/(-2+x)/exp(x)^2/
ln(x),x)

[Out]

(-2+x)^(-I*x*csgn(I/(-2+x))*Pi)*x^(-2*x^2*ln(5)*exp(-x))*ln(x)^(-2*x^2*ln(5)*exp(-x))*ln(x)^(I*x*csgn(I/(-2+x)
)*Pi)*5^(I*x^2*csgn(I/(-2+x))*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))*Pi*exp(-x))*x^(I*x*csgn(I*ln(x)/(-2+x))*Pi)*l
n(x)^(-2*x^3*exp(-x))*5^(2*I*x^2*Pi*exp(-x))*(-2+x)^(-2*I*x*Pi)*5^(-I*x^2*csgn(I*ln(x))*Pi*exp(-x))*ln(x)^(2*x
*ln(x))*x^(I*x*csgn(I*ln(x))*Pi)*ln(x)^(I*x*csgn(I*ln(x))*Pi)*ln(x)^(-2*I*x*Pi)*(-2+x)^(-2*x*ln(x))*(-2+x)^(2*
x^3*exp(-x))*(-2+x)^(2*I*x*Pi)*25^(exp(-2*x)*x^4)*ln(x)^(2*I*x*Pi)*5^(-I*x^2*csgn(I/(-2+x))*Pi*exp(-x))*(-2+x)
^(-I*x*csgn(I*ln(x)/(-2+x))*Pi)*ln(x)^(-2*x*ln(-2+x))*5^(-I*x^2*csgn(I*ln(x)/(-2+x))*Pi*exp(-x))*5^(-2*I*x^2*P
i*exp(-x))*(-2+x)^(-I*x*csgn(I*ln(x))*Pi)*ln(x)^(I*x*csgn(I*ln(x)/(-2+x))*Pi)*x^(-2*I*x*Pi)*(-2+x)^(2*x^2*ln(5
)*exp(-x))*x^(2*I*x*Pi)*ln(x)^(-I*x*csgn(I/(-2+x))*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))*Pi)*(-2+x)^(I*x*csgn(I/(
-2+x))*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))*Pi)*x^(I*x*csgn(I/(-2+x))*Pi)*x^(-2*x^3*exp(-x))*x^(-I*x*csgn(I/(-2+
x))*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))*Pi)*exp(-1/4*x*(-4*exp(-2*x)*x^4+4*Pi^2-4*ln(-2+x)^2-4*ln(ln(x))^2-4*ln
(x)^2-4*ln(5)^2*exp(-2*x)*x^2+4*csgn(I*ln(x)/(-2+x))^4*Pi^2+8*I*x^2*Pi*exp(-x)+4*csgn(I*ln(x))*csgn(I*ln(x)/(-
2+x))^2*Pi^2+4*csgn(I/(-2+x))*csgn(I*ln(x)/(-2+x))^2*Pi^2-2*csgn(I/(-2+x))*csgn(I*ln(x))^2*csgn(I*ln(x)/(-2+x)
)^3*Pi^2-2*csgn(I*ln(x))*csgn(I/(-2+x))^2*csgn(I*ln(x)/(-2+x))^3*Pi^2+csgn(I/(-2+x))^2*csgn(I*ln(x))^2*csgn(I*
ln(x)/(-2+x))^2*Pi^2+4*csgn(I/(-2+x))*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))^3*Pi^2-4*csgn(I/(-2+x))*csgn(I*ln(x))
*csgn(I*ln(x)/(-2+x))*Pi^2+2*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))^5*Pi^2+2*csgn(I/(-2+x))*csgn(I*ln(x)/(-2+x))^5
*Pi^2+csgn(I*ln(x))^2*csgn(I*ln(x)/(-2+x))^4*Pi^2+4*I*x^2*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))^2*Pi*exp(-x)+4*I*
x^2*csgn(I/(-2+x))*csgn(I*ln(x)/(-2+x))^2*Pi*exp(-x)+csgn(I*ln(x)/(-2+x))^6*Pi^2-4*csgn(I*ln(x)/(-2+x))^5*Pi^2
+4*csgn(I*ln(x)/(-2+x))^3*Pi^2-8*csgn(I*ln(x)/(-2+x))^2*Pi^2-4*I*x^2*csgn(I/(-2+x))*csgn(I*ln(x))*csgn(I*ln(x)
/(-2+x))*Pi*exp(-x)+csgn(I/(-2+x))^2*csgn(I*ln(x)/(-2+x))^4*Pi^2-4*csgn(I*ln(x))*csgn(I*ln(x)/(-2+x))^4*Pi^2-4
*csgn(I/(-2+x))*csgn(I*ln(x)/(-2+x))^4*Pi^2+4*I*x^2*csgn(I*ln(x)/(-2+x))^3*Pi*exp(-x)-8*I*x^2*csgn(I*ln(x)/(-2
+x))^2*Pi*exp(-x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (32) = 64\).

Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.59 \[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=e^{\left ({\left (x^{5} + 2 \, x^{4} \log \left (5\right ) + x^{3} \log \left (5\right )^{2} + x e^{\left (2 \, x\right )} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} \log \left (-\frac {\log \left (x\right )}{x - 2}\right )^{2} - 2 \, {\left (x^{3} + x^{2} \log \left (5\right )\right )} e^{x} \log \left (x\right ) - 2 \, x e^{\left (2 \, x\right )} + 2 \, {\left (x e^{\left (2 \, x\right )} \log \left (x\right ) - {\left (x^{3} + x^{2} \log \left (5\right )\right )} e^{x}\right )} \log \left (-\frac {\log \left (x\right )}{x - 2}\right )\right )} e^{\left (-2 \, x\right )} + 2 \, x\right )} \]

[In]

integrate(((-2+x)*exp(x)^2*log(x)*log(-log(x)/(-2+x))^2+((2*x-4)*exp(x)^2*log(x)^2+(-4*exp(x)^2+((2*x^3-8*x^2+
8*x)*log(5)+2*x^4-10*x^3+12*x^2)*exp(x))*log(x)+(2*x-4)*exp(x)^2)*log(-log(x)/(-2+x))+(-2+x)*exp(x)^2*log(x)^3
+(-4*exp(x)^2+((2*x^3-8*x^2+8*x)*log(5)+2*x^4-10*x^3+12*x^2)*exp(x))*log(x)^2+((2*x-4)*exp(x)^2+(4*x*log(5)+4*
x^2)*exp(x)+(-2*x^4+7*x^3-6*x^2)*log(5)^2+(-4*x^5+16*x^4-16*x^3)*log(5)-2*x^6+9*x^5-10*x^4)*log(x)+((-2*x^2+4*
x)*log(5)-2*x^3+4*x^2)*exp(x))*exp((x*exp(x)^2*log(-log(x)/(-2+x))^2+(2*x*exp(x)^2*log(x)+(-2*x^2*log(5)-2*x^3
)*exp(x))*log(-log(x)/(-2+x))+x*exp(x)^2*log(x)^2+(-2*x^2*log(5)-2*x^3)*exp(x)*log(x)+x^3*log(5)^2+2*x^4*log(5
)+x^5)/exp(x)^2)/(-2+x)/exp(x)^2/log(x),x, algorithm="fricas")

[Out]

e^((x^5 + 2*x^4*log(5) + x^3*log(5)^2 + x*e^(2*x)*log(x)^2 + x*e^(2*x)*log(-log(x)/(x - 2))^2 - 2*(x^3 + x^2*l
og(5))*e^x*log(x) - 2*x*e^(2*x) + 2*(x*e^(2*x)*log(x) - (x^3 + x^2*log(5))*e^x)*log(-log(x)/(x - 2)))*e^(-2*x)
+ 2*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (26) = 52\).

Time = 65.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.72 \[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=e^{\left (x^{5} + 2 x^{4} \log {\left (5 \right )} + x^{3} \log {\left (5 \right )}^{2} + x e^{2 x} \log {\left (x \right )}^{2} + x e^{2 x} \log {\left (- \frac {\log {\left (x \right )}}{x - 2} \right )}^{2} + \left (- 2 x^{3} - 2 x^{2} \log {\left (5 \right )}\right ) e^{x} \log {\left (x \right )} + \left (2 x e^{2 x} \log {\left (x \right )} + \left (- 2 x^{3} - 2 x^{2} \log {\left (5 \right )}\right ) e^{x}\right ) \log {\left (- \frac {\log {\left (x \right )}}{x - 2} \right )}\right ) e^{- 2 x}} \]

[In]

integrate(((-2+x)*exp(x)**2*ln(x)*ln(-ln(x)/(-2+x))**2+((2*x-4)*exp(x)**2*ln(x)**2+(-4*exp(x)**2+((2*x**3-8*x*
*2+8*x)*ln(5)+2*x**4-10*x**3+12*x**2)*exp(x))*ln(x)+(2*x-4)*exp(x)**2)*ln(-ln(x)/(-2+x))+(-2+x)*exp(x)**2*ln(x
)**3+(-4*exp(x)**2+((2*x**3-8*x**2+8*x)*ln(5)+2*x**4-10*x**3+12*x**2)*exp(x))*ln(x)**2+((2*x-4)*exp(x)**2+(4*x
*ln(5)+4*x**2)*exp(x)+(-2*x**4+7*x**3-6*x**2)*ln(5)**2+(-4*x**5+16*x**4-16*x**3)*ln(5)-2*x**6+9*x**5-10*x**4)*
ln(x)+((-2*x**2+4*x)*ln(5)-2*x**3+4*x**2)*exp(x))*exp((x*exp(x)**2*ln(-ln(x)/(-2+x))**2+(2*x*exp(x)**2*ln(x)+(
-2*x**2*ln(5)-2*x**3)*exp(x))*ln(-ln(x)/(-2+x))+x*exp(x)**2*ln(x)**2+(-2*x**2*ln(5)-2*x**3)*exp(x)*ln(x)+x**3*
ln(5)**2+2*x**4*ln(5)+x**5)/exp(x)**2)/(-2+x)/exp(x)**2/ln(x),x)

[Out]

exp((x**5 + 2*x**4*log(5) + x**3*log(5)**2 + x*exp(2*x)*log(x)**2 + x*exp(2*x)*log(-log(x)/(x - 2))**2 + (-2*x
**3 - 2*x**2*log(5))*exp(x)*log(x) + (2*x*exp(2*x)*log(x) + (-2*x**3 - 2*x**2*log(5))*exp(x))*log(-log(x)/(x -
2)))*exp(-2*x))

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(((-2+x)*exp(x)^2*log(x)*log(-log(x)/(-2+x))^2+((2*x-4)*exp(x)^2*log(x)^2+(-4*exp(x)^2+((2*x^3-8*x^2+
8*x)*log(5)+2*x^4-10*x^3+12*x^2)*exp(x))*log(x)+(2*x-4)*exp(x)^2)*log(-log(x)/(-2+x))+(-2+x)*exp(x)^2*log(x)^3
+(-4*exp(x)^2+((2*x^3-8*x^2+8*x)*log(5)+2*x^4-10*x^3+12*x^2)*exp(x))*log(x)^2+((2*x-4)*exp(x)^2+(4*x*log(5)+4*
x^2)*exp(x)+(-2*x^4+7*x^3-6*x^2)*log(5)^2+(-4*x^5+16*x^4-16*x^3)*log(5)-2*x^6+9*x^5-10*x^4)*log(x)+((-2*x^2+4*
x)*log(5)-2*x^3+4*x^2)*exp(x))*exp((x*exp(x)^2*log(-log(x)/(-2+x))^2+(2*x*exp(x)^2*log(x)+(-2*x^2*log(5)-2*x^3
)*exp(x))*log(-log(x)/(-2+x))+x*exp(x)^2*log(x)^2+(-2*x^2*log(5)-2*x^3)*exp(x)*log(x)+x^3*log(5)^2+2*x^4*log(5
)+x^5)/exp(x)^2)/(-2+x)/exp(x)^2/log(x),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=\int { \frac {{\left ({\left (x - 2\right )} e^{\left (2 \, x\right )} \log \left (x\right )^{3} + {\left (x - 2\right )} e^{\left (2 \, x\right )} \log \left (x\right ) \log \left (-\frac {\log \left (x\right )}{x - 2}\right )^{2} + 2 \, {\left ({\left (x^{4} - 5 \, x^{3} + 6 \, x^{2} + {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (5\right )\right )} e^{x} - 2 \, e^{\left (2 \, x\right )}\right )} \log \left (x\right )^{2} - 2 \, {\left (x^{3} - 2 \, x^{2} + {\left (x^{2} - 2 \, x\right )} \log \left (5\right )\right )} e^{x} - {\left (2 \, x^{6} - 9 \, x^{5} + 10 \, x^{4} + {\left (2 \, x^{4} - 7 \, x^{3} + 6 \, x^{2}\right )} \log \left (5\right )^{2} - 2 \, {\left (x - 2\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{2} + x \log \left (5\right )\right )} e^{x} + 4 \, {\left (x^{5} - 4 \, x^{4} + 4 \, x^{3}\right )} \log \left (5\right )\right )} \log \left (x\right ) + 2 \, {\left ({\left (x - 2\right )} e^{\left (2 \, x\right )} \log \left (x\right )^{2} + {\left (x - 2\right )} e^{\left (2 \, x\right )} + {\left ({\left (x^{4} - 5 \, x^{3} + 6 \, x^{2} + {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (5\right )\right )} e^{x} - 2 \, e^{\left (2 \, x\right )}\right )} \log \left (x\right )\right )} \log \left (-\frac {\log \left (x\right )}{x - 2}\right )\right )} e^{\left ({\left (x^{5} + 2 \, x^{4} \log \left (5\right ) + x^{3} \log \left (5\right )^{2} + x e^{\left (2 \, x\right )} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} \log \left (-\frac {\log \left (x\right )}{x - 2}\right )^{2} - 2 \, {\left (x^{3} + x^{2} \log \left (5\right )\right )} e^{x} \log \left (x\right ) + 2 \, {\left (x e^{\left (2 \, x\right )} \log \left (x\right ) - {\left (x^{3} + x^{2} \log \left (5\right )\right )} e^{x}\right )} \log \left (-\frac {\log \left (x\right )}{x - 2}\right )\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )}}{{\left (x - 2\right )} \log \left (x\right )} \,d x } \]

[In]

integrate(((-2+x)*exp(x)^2*log(x)*log(-log(x)/(-2+x))^2+((2*x-4)*exp(x)^2*log(x)^2+(-4*exp(x)^2+((2*x^3-8*x^2+
8*x)*log(5)+2*x^4-10*x^3+12*x^2)*exp(x))*log(x)+(2*x-4)*exp(x)^2)*log(-log(x)/(-2+x))+(-2+x)*exp(x)^2*log(x)^3
+(-4*exp(x)^2+((2*x^3-8*x^2+8*x)*log(5)+2*x^4-10*x^3+12*x^2)*exp(x))*log(x)^2+((2*x-4)*exp(x)^2+(4*x*log(5)+4*
x^2)*exp(x)+(-2*x^4+7*x^3-6*x^2)*log(5)^2+(-4*x^5+16*x^4-16*x^3)*log(5)-2*x^6+9*x^5-10*x^4)*log(x)+((-2*x^2+4*
x)*log(5)-2*x^3+4*x^2)*exp(x))*exp((x*exp(x)^2*log(-log(x)/(-2+x))^2+(2*x*exp(x)^2*log(x)+(-2*x^2*log(5)-2*x^3
)*exp(x))*log(-log(x)/(-2+x))+x*exp(x)^2*log(x)^2+(-2*x^2*log(5)-2*x^3)*exp(x)*log(x)+x^3*log(5)^2+2*x^4*log(5
)+x^5)/exp(x)^2)/(-2+x)/exp(x)^2/log(x),x, algorithm="giac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 16.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.47 \[ \int \frac {e^{-2 x+e^{-2 x} \left (x^5+2 x^4 \log (5)+x^3 \log ^2(5)+e^x \left (-2 x^3-2 x^2 \log (5)\right ) \log (x)+e^{2 x} x \log ^2(x)+\left (e^x \left (-2 x^3-2 x^2 \log (5)\right )+2 e^{2 x} x \log (x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} x \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )} \left (e^x \left (4 x^2-2 x^3+\left (4 x-2 x^2\right ) \log (5)\right )+\left (-10 x^4+9 x^5-2 x^6+e^{2 x} (-4+2 x)+\left (-16 x^3+16 x^4-4 x^5\right ) \log (5)+\left (-6 x^2+7 x^3-2 x^4\right ) \log ^2(5)+e^x \left (4 x^2+4 x \log (5)\right )\right ) \log (x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log ^2(x)+e^{2 x} (-2+x) \log ^3(x)+\left (e^{2 x} (-4+2 x)+\left (-4 e^{2 x}+e^x \left (12 x^2-10 x^3+2 x^4+\left (8 x-8 x^2+2 x^3\right ) \log (5)\right )\right ) \log (x)+e^{2 x} (-4+2 x) \log ^2(x)\right ) \log \left (-\frac {\log (x)}{-2+x}\right )+e^{2 x} (-2+x) \log (x) \log ^2\left (-\frac {\log (x)}{-2+x}\right )\right )}{(-2+x) \log (x)} \, dx=\frac {5^{2\,x^4\,{\mathrm {e}}^{-2\,x}}\,x^{2\,x\,\ln \left (-\frac {\ln \left (x\right )}{x-2}\right )}\,{\mathrm {e}}^{x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{x\,{\ln \left (-\frac {\ln \left (x\right )}{x-2}\right )}^2}\,{\mathrm {e}}^{x^3\,{\mathrm {e}}^{-2\,x}\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{x^5\,{\mathrm {e}}^{-2\,x}}}{x^{2\,x^3\,{\mathrm {e}}^{-x}}\,x^{2\,x^2\,{\mathrm {e}}^{-x}\,\ln \left (5\right )}\,{\left (-\frac {\ln \left (x\right )}{x-2}\right )}^{2\,x^3\,{\mathrm {e}}^{-x}}\,{\left (-\frac {\ln \left (x\right )}{x-2}\right )}^{2\,x^2\,{\mathrm {e}}^{-x}\,\ln \left (5\right )}} \]

[In]

int((exp(exp(-2*x)*(x^3*log(5)^2 - log(-log(x)/(x - 2))*(exp(x)*(2*x^2*log(5) + 2*x^3) - 2*x*exp(2*x)*log(x))
+ 2*x^4*log(5) + x^5 + x*exp(2*x)*log(x)^2 + x*exp(2*x)*log(-log(x)/(x - 2))^2 - exp(x)*log(x)*(2*x^2*log(5) +
2*x^3)))*exp(-2*x)*(exp(x)*(log(5)*(4*x - 2*x^2) + 4*x^2 - 2*x^3) - log(x)*(log(5)^2*(6*x^2 - 7*x^3 + 2*x^4)
- exp(x)*(4*x*log(5) + 4*x^2) + log(5)*(16*x^3 - 16*x^4 + 4*x^5) - exp(2*x)*(2*x - 4) + 10*x^4 - 9*x^5 + 2*x^6
) - log(x)^2*(4*exp(2*x) - exp(x)*(log(5)*(8*x - 8*x^2 + 2*x^3) + 12*x^2 - 10*x^3 + 2*x^4)) + log(-log(x)/(x -
2))*(exp(2*x)*(2*x - 4) - log(x)*(4*exp(2*x) - exp(x)*(log(5)*(8*x - 8*x^2 + 2*x^3) + 12*x^2 - 10*x^3 + 2*x^4
)) + exp(2*x)*log(x)^2*(2*x - 4)) + exp(2*x)*log(x)^3*(x - 2) + exp(2*x)*log(x)*log(-log(x)/(x - 2))^2*(x - 2)
))/(log(x)*(x - 2)),x)

[Out]

(5^(2*x^4*exp(-2*x))*x^(2*x*log(-log(x)/(x - 2)))*exp(x*log(x)^2)*exp(x*log(-log(x)/(x - 2))^2)*exp(x^3*exp(-2
*x)*log(5)^2)*exp(x^5*exp(-2*x)))/(x^(2*x^3*exp(-x))*x^(2*x^2*exp(-x)*log(5))*(-log(x)/(x - 2))^(2*x^3*exp(-x)
)*(-log(x)/(x - 2))^(2*x^2*exp(-x)*log(5)))