\(\int \frac {e^{-2 x} ((-72+48 x-8 x^2+(18-12 x+2 x^2+18 x^4-12 x^5+2 x^6) \log (\frac {2+2 x^4}{x^4})) \log (x \log (\frac {2+2 x^4}{x^4}))+(-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7) \log (\frac {2+2 x^4}{x^4}) \log ^2(x \log (\frac {2+2 x^4}{x^4})))}{(x+x^5) \log (\frac {2+2 x^4}{x^4})} \, dx\) [6724]

Optimal result

Integrand size = 151, antiderivative size = 26 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=3+e^{-2 x} (-3+x)^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx \]

[In]

Int[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2*x^6)*Log[(2 + 2*x^4)/x^4])*Log[x*Log[(2 +
2*x^4)/x^4]] + (-24*x + 14*x^2 - 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*x^4)/x
^4]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]),x]

[Out]

-1/2*1/E^(2*x) - (11*ExpIntegralEi[-2*x])/2 + 36*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -2*x] - 18*EulerGam
ma*Log[x] - 18*(ExpIntegralE[1, 2*x] + ExpIntegralEi[-2*x])*Log[x] - 9*Log[2*x]^2 + (11*Log[x*Log[2 + 2/x^4]])
/(2*E^(2*x)) - (x*Log[x*Log[2 + 2/x^4]])/E^(2*x) + 18*ExpIntegralEi[-2*x]*Log[x*Log[2 + 2/x^4]] - (I/2)*E^(2*(
-1)^(1/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/x, x] + (I/2)*E^((1 + I)*Sqrt[2])*Defer[Int][ExpInteg
ralEi[-2*(-1)^(1/4) - 2*x]/x, x] - ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/x, x])/E^(2*(-1)^(1/4))
+ ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/x, x])/E^((1 + I)*Sqrt[2]) + (I/2)*E^(2*(-1)^(3/4))*Def
er[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/x, x] - ((I/2)*Defer[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/x, x])
/E^((1 - I)*Sqrt[2]) + ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(3/4) - 2*x]/x, x])/E^(2*(-1)^(3/4)) - (I/2)*E^(
(1 - I)*Sqrt[2])*Defer[Int][ExpIntegralEi[2*(-1)^(3/4) - 2*x]/x, x] + (11*Defer[Int][1/(E^(2*x)*((-1)^(1/4) -
x)*Log[2 + 2/x^4]), x])/2 - (-1)^(1/4)*Defer[Int][1/(E^(2*x)*((-1)^(1/4) - x)*Log[2 + 2/x^4]), x] + (11*Defer[
Int][1/(E^(2*x)*(-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x])/2 + (-1)^(3/4)*Defer[Int][1/(E^(2*x)*(-(-1)^(3/4) - x)*
Log[2 + 2/x^4]), x] + 22*Defer[Int][1/(E^(2*x)*x*Log[2 + 2/x^4]), x] - (11*Defer[Int][1/(E^(2*x)*((-1)^(1/4) +
x)*Log[2 + 2/x^4]), x])/2 - (-1)^(1/4)*Defer[Int][1/(E^(2*x)*((-1)^(1/4) + x)*Log[2 + 2/x^4]), x] - (11*Defer
[Int][1/(E^(2*x)*(-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x])/2 + (-1)^(3/4)*Defer[Int][1/(E^(2*x)*(-(-1)^(3/4) + x)
*Log[2 + 2/x^4]), x] + (I/2)*E^(2*(-1)^(1/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) - x)*
Log[2 + 2/x^4]), x] - (I/2)*E^((1 + I)*Sqrt[2])*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) - x
)*Log[2 + 2/x^4]), x] + (I/2)*E^(2*(-1)^(1/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4) - x
)*Log[2 + 2/x^4]), x] - (I/2)*E^((1 + I)*Sqrt[2])*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4)
- x)*Log[2 + 2/x^4]), x] + (2*I)*E^(2*(-1)^(1/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/(x*Log[2 + 2/x
^4]), x] - (2*I)*E^((1 + I)*Sqrt[2])*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/(x*Log[2 + 2/x^4]), x] - (I
/2)*E^(2*(-1)^(1/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x] + (I/
2)*E^((1 + I)*Sqrt[2])*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x] - (
I/2)*E^(2*(-1)^(1/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x] + (
I/2)*E^((1 + I)*Sqrt[2])*Defer[Int][ExpIntegralEi[-2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x]
+ ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) - x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(1/4))
- ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) - x)*Log[2 + 2/x^4]), x])/E^((1 + I)*Sqrt[2
]) + ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(1/
4)) - ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x])/E^((1 + I)*S
qrt[2]) + ((2*I)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/(x*Log[2 + 2/x^4]), x])/E^(2*(-1)^(1/4)) - ((2*I
)*Defer[Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/(x*Log[2 + 2/x^4]), x])/E^((1 + I)*Sqrt[2]) - ((I/2)*Defer[Int]
[ExpIntegralEi[2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(1/4)) + ((I/2)*Defer[Int]
[ExpIntegralEi[2*(-1)^(1/4) - 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x])/E^((1 + I)*Sqrt[2]) - ((I/2)*Defer[I
nt][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(1/4)) + ((I/2)*Defer[
Int][ExpIntegralEi[2*(-1)^(1/4) - 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x])/E^((1 + I)*Sqrt[2]) - (I/2)*E^(
2*(-1)^(3/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/(((-1)^(1/4) - x)*Log[2 + 2/x^4]), x] + ((I/2)*Def
er[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/(((-1)^(1/4) - x)*Log[2 + 2/x^4]), x])/E^((1 - I)*Sqrt[2]) - (I/2)*
E^(2*(-1)^(3/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/((-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x] + ((I/2)
*Defer[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/((-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x])/E^((1 - I)*Sqrt[2]) - (
2*I)*E^(2*(-1)^(3/4))*Defer[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/(x*Log[2 + 2/x^4]), x] + ((2*I)*Defer[Int]
[ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/(x*Log[2 + 2/x^4]), x])/E^((1 - I)*Sqrt[2]) + (I/2)*E^(2*(-1)^(3/4))*Defer
[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x] - ((I/2)*Defer[Int][ExpIntegral
Ei[-2*(-1)^(3/4) - 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x])/E^((1 - I)*Sqrt[2]) + (I/2)*E^(2*(-1)^(3/4))*De
fer[Int][ExpIntegralEi[-2*(-1)^(3/4) - 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x] - ((I/2)*Defer[Int][ExpInte
gralEi[-2*(-1)^(3/4) - 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x])/E^((1 - I)*Sqrt[2]) - ((I/2)*Defer[Int][Ex
pIntegralEi[2*(-1)^(3/4) - 2*x]/(((-1)^(1/4) - x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(3/4)) + (I/2)*E^((1 - I)*Sqr
t[2])*Defer[Int][ExpIntegralEi[2*(-1)^(3/4) - 2*x]/(((-1)^(1/4) - x)*Log[2 + 2/x^4]), x] - ((I/2)*Defer[Int][E
xpIntegralEi[2*(-1)^(3/4) - 2*x]/((-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(3/4)) + (I/2)*E^((1 - I)*S
qrt[2])*Defer[Int][ExpIntegralEi[2*(-1)^(3/4) - 2*x]/((-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x] - ((2*I)*Defer[Int
][ExpIntegralEi[2*(-1)^(3/4) - 2*x]/(x*Log[2 + 2/x^4]), x])/E^(2*(-1)^(3/4)) + (2*I)*E^((1 - I)*Sqrt[2])*Defer
[Int][ExpIntegralEi[2*(-1)^(3/4) - 2*x]/(x*Log[2 + 2/x^4]), x] + ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(3/4)
- 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(3/4)) - (I/2)*E^((1 - I)*Sqrt[2])*Defer[Int][ExpInteg
ralEi[2*(-1)^(3/4) - 2*x]/(((-1)^(1/4) + x)*Log[2 + 2/x^4]), x] + ((I/2)*Defer[Int][ExpIntegralEi[2*(-1)^(3/4)
- 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x])/E^(2*(-1)^(3/4)) - (I/2)*E^((1 - I)*Sqrt[2])*Defer[Int][ExpInt
egralEi[2*(-1)^(3/4) - 2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x] + 18*Defer[Int][ExpIntegralEi[-2*x]/(((-1)^
(1/4) - x)*Log[2 + 2/x^4]), x] + 18*Defer[Int][ExpIntegralEi[-2*x]/((-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x] + 72
*Defer[Int][ExpIntegralEi[-2*x]/(x*Log[2 + 2/x^4]), x] - 18*Defer[Int][ExpIntegralEi[-2*x]/(((-1)^(1/4) + x)*L
og[2 + 2/x^4]), x] - 18*Defer[Int][ExpIntegralEi[-2*x]/((-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x] - (18 + 2*I)*Def
er[Int][Log[x*Log[2 + 2/x^4]]/(E^(2*x)*((-1)^(1/4) - x)*Log[2 + 2/x^4]), x] + 12*(-1)^(1/4)*Defer[Int][Log[x*L
og[2 + 2/x^4]]/(E^(2*x)*((-1)^(1/4) - x)*Log[2 + 2/x^4]), x] - (18 - 2*I)*Defer[Int][Log[x*Log[2 + 2/x^4]]/(E^
(2*x)*(-(-1)^(3/4) - x)*Log[2 + 2/x^4]), x] - 12*(-1)^(3/4)*Defer[Int][Log[x*Log[2 + 2/x^4]]/(E^(2*x)*(-(-1)^(
3/4) - x)*Log[2 + 2/x^4]), x] - 72*Defer[Int][Log[x*Log[2 + 2/x^4]]/(E^(2*x)*x*Log[2 + 2/x^4]), x] + (18 + 2*I
)*Defer[Int][Log[x*Log[2 + 2/x^4]]/(E^(2*x)*((-1)^(1/4) + x)*Log[2 + 2/x^4]), x] + 12*(-1)^(1/4)*Defer[Int][Lo
g[x*Log[2 + 2/x^4]]/(E^(2*x)*((-1)^(1/4) + x)*Log[2 + 2/x^4]), x] + (18 - 2*I)*Defer[Int][Log[x*Log[2 + 2/x^4]
]/(E^(2*x)*(-(-1)^(3/4) + x)*Log[2 + 2/x^4]), x] - 12*(-1)^(3/4)*Defer[Int][Log[x*Log[2 + 2/x^4]]/(E^(2*x)*(-(
-1)^(3/4) + x)*Log[2 + 2/x^4]), x] - 24*Defer[Int][Log[x*Log[2 + 2/x^4]]^2/E^(2*x), x] + 14*Defer[Int][(x*Log[
x*Log[2 + 2/x^4]]^2)/E^(2*x), x] - 2*Defer[Int][(x^2*Log[x*Log[2 + 2/x^4]]^2)/E^(2*x), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{x \left (1+x^4\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx \\ & = \int \frac {2 e^{-2 x} (3-x) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right ) \left (4 (-3+x)+\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right ) \left (3-x+(-4+x) x \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )\right )\right )}{x \left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx \\ & = 2 \int \frac {e^{-2 x} (3-x) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right ) \left (4 (-3+x)+\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right ) \left (3-x+(-4+x) x \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )\right )\right )}{x \left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx \\ & = 2 \int \left (\frac {e^{-2 x} (-3+x)^2 \left (-4+\log \left (2+\frac {2}{x^4}\right )+x^4 \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )}-e^{-2 x} (-4+x) (-3+x) \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right )\right ) \, dx \\ & = 2 \int \frac {e^{-2 x} (-3+x)^2 \left (-4+\log \left (2+\frac {2}{x^4}\right )+x^4 \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx-2 \int e^{-2 x} (-4+x) (-3+x) \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx \\ & = 2 \int \left (\frac {9 e^{-2 x} \left (-4+\log \left (2+\frac {2}{x^4}\right )+x^4 \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \log \left (2+\frac {2}{x^4}\right )}-\frac {e^{-2 x} \left (6-x+9 x^3\right ) \left (-4+\log \left (2+\frac {2}{x^4}\right )+x^4 \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )}\right ) \, dx-2 \int \left (12 e^{-2 x} \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right )-7 e^{-2 x} x \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right )+e^{-2 x} x^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right )\right ) \, dx \\ & = -\left (2 \int \frac {e^{-2 x} \left (6-x+9 x^3\right ) \left (-4+\log \left (2+\frac {2}{x^4}\right )+x^4 \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx\right )-2 \int e^{-2 x} x^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx+14 \int e^{-2 x} x \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx+18 \int \frac {e^{-2 x} \left (-4+\log \left (2+\frac {2}{x^4}\right )+x^4 \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \log \left (2+\frac {2}{x^4}\right )} \, dx-24 \int e^{-2 x} \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx \\ & = -\left (2 \int \frac {e^{-2 x} \left (6-x+9 x^3\right ) \left (-4+\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx\right )-2 \int e^{-2 x} x^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx+14 \int e^{-2 x} x \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx+18 \int \frac {e^{-2 x} \left (-4+\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )\right ) \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \log \left (2+\frac {2}{x^4}\right )} \, dx-24 \int e^{-2 x} \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx \\ & = -\left (2 \int e^{-2 x} x^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx\right )-2 \int \left (\frac {6 e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4}-\frac {e^{-2 x} x \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4}+\frac {9 e^{-2 x} x^3 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4}+\frac {6 e^{-2 x} x^4 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4}-\frac {e^{-2 x} x^5 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4}+\frac {9 e^{-2 x} x^7 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4}-\frac {24 e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )}+\frac {4 e^{-2 x} x \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )}-\frac {36 e^{-2 x} x^3 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )}\right ) \, dx+14 \int e^{-2 x} x \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx+18 \int \left (\frac {e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x}+e^{-2 x} x^3 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )-\frac {4 e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \log \left (2+\frac {2}{x^4}\right )}\right ) \, dx-24 \int e^{-2 x} \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx \\ & = 2 \int \frac {e^{-2 x} x \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4} \, dx+2 \int \frac {e^{-2 x} x^5 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4} \, dx-2 \int e^{-2 x} x^2 \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx-8 \int \frac {e^{-2 x} x \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx-12 \int \frac {e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4} \, dx-12 \int \frac {e^{-2 x} x^4 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4} \, dx+14 \int e^{-2 x} x \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx+18 \int \frac {e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x} \, dx+18 \int e^{-2 x} x^3 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx-18 \int \frac {e^{-2 x} x^3 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4} \, dx-18 \int \frac {e^{-2 x} x^7 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{1+x^4} \, dx-24 \int e^{-2 x} \log ^2\left (x \log \left (2+\frac {2}{x^4}\right )\right ) \, dx+48 \int \frac {e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx-72 \int \frac {e^{-2 x} \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{x \log \left (2+\frac {2}{x^4}\right )} \, dx+72 \int \frac {e^{-2 x} x^3 \log \left (x \log \left (2+\frac {2}{x^4}\right )\right )}{\left (1+x^4\right ) \log \left (2+\frac {2}{x^4}\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx \]

[In]

Integrate[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2*x^6)*Log[(2 + 2*x^4)/x^4])*Log[x*Log
[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*
x^4)/x^4]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]),x]

[Out]

Integrate[((-72 + 48*x - 8*x^2 + (18 - 12*x + 2*x^2 + 18*x^4 - 12*x^5 + 2*x^6)*Log[(2 + 2*x^4)/x^4])*Log[x*Log
[(2 + 2*x^4)/x^4]] + (-24*x + 14*x^2 - 2*x^3 - 24*x^5 + 14*x^6 - 2*x^7)*Log[(2 + 2*x^4)/x^4]*Log[x*Log[(2 + 2*
x^4)/x^4]]^2)/(E^(2*x)*(x + x^5)*Log[(2 + 2*x^4)/x^4]), x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(25)=50\).

Time = 39.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54

method	result	size
parallelrisch	\(\frac {\left (2 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2} x^{2}-12 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2} x +18 {\ln \left (x \ln \left (\frac {2 x^{4}+2}{x^{4}}\right )\right )}^{2}\right ) {\mathrm e}^{-2 x}}{2}\)	\(66\)
risch	\(\text {Expression too large to display}\)	\(110816\)

[In]

int(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*ln((2*x^4+2)/x^4)*ln(x*ln((2*x^4+2)/x^4))^2+((2*x^6-12*x^5+18*x^
4+2*x^2-12*x+18)*ln((2*x^4+2)/x^4)-8*x^2+48*x-72)*ln(x*ln((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/ln((2*x^4+2)/x^4),
x,method=_RETURNVERBOSE)

[Out]

1/2*(2*ln(x*ln(2*(x^4+1)/x^4))^2*x^2-12*ln(x*ln(2*(x^4+1)/x^4))^2*x+18*ln(x*ln(2*(x^4+1)/x^4))^2)/exp(x)^2

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x \log \left (\frac {2 \, {\left (x^{4} + 1\right )}}{x^{4}}\right )\right )^{2} \]

[In]

integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log(x*log((2*x^4+2)/x^4))^2+((2*x^6-12*
x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2)/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2
*x^4+2)/x^4),x, algorithm="fricas")

[Out]

(x^2 - 6*x + 9)*e^(-2*x)*log(x*log(2*(x^4 + 1)/x^4))^2

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\text {Timed out} \]

[In]

integrate(((-2*x**7+14*x**6-24*x**5-2*x**3+14*x**2-24*x)*ln((2*x**4+2)/x**4)*ln(x*ln((2*x**4+2)/x**4))**2+((2*
x**6-12*x**5+18*x**4+2*x**2-12*x+18)*ln((2*x**4+2)/x**4)-8*x**2+48*x-72)*ln(x*ln((2*x**4+2)/x**4)))/(x**5+x)/e
xp(x)**2/ln((2*x**4+2)/x**4),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (25) = 50\).

Time = 0.38 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (x\right ) \log \left (\log \left (2\right ) + \log \left (x^{4} + 1\right ) - 4 \, \log \left (x\right )\right ) + {\left (x^{2} - 6 \, x + 9\right )} e^{\left (-2 \, x\right )} \log \left (\log \left (2\right ) + \log \left (x^{4} + 1\right ) - 4 \, \log \left (x\right )\right )^{2} \]

[In]

integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log(x*log((2*x^4+2)/x^4))^2+((2*x^6-12*
x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2)/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2
*x^4+2)/x^4),x, algorithm="maxima")

[Out]

(x^2 - 6*x + 9)*e^(-2*x)*log(x)^2 + 2*(x^2 - 6*x + 9)*e^(-2*x)*log(x)*log(log(2) + log(x^4 + 1) - 4*log(x)) +
(x^2 - 6*x + 9)*e^(-2*x)*log(log(2) + log(x^4 + 1) - 4*log(x))^2

Giac [F(-1)]

Timed out. \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx=\text {Timed out} \]

[In]

integrate(((-2*x^7+14*x^6-24*x^5-2*x^3+14*x^2-24*x)*log((2*x^4+2)/x^4)*log(x*log((2*x^4+2)/x^4))^2+((2*x^6-12*
x^5+18*x^4+2*x^2-12*x+18)*log((2*x^4+2)/x^4)-8*x^2+48*x-72)*log(x*log((2*x^4+2)/x^4)))/(x^5+x)/exp(x)^2/log((2
*x^4+2)/x^4),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 12.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 x} \left (\left (-72+48 x-8 x^2+\left (18-12 x+2 x^2+18 x^4-12 x^5+2 x^6\right ) \log \left (\frac {2+2 x^4}{x^4}\right )\right ) \log \left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )+\left (-24 x+14 x^2-2 x^3-24 x^5+14 x^6-2 x^7\right ) \log \left (\frac {2+2 x^4}{x^4}\right ) \log ^2\left (x \log \left (\frac {2+2 x^4}{x^4}\right )\right )\right )}{\left (x+x^5\right ) \log \left (\frac {2+2 x^4}{x^4}\right )} \, dx={\ln \left (x\,\ln \left (\frac {2\,\left (x^4+1\right )}{x^4}\right )\right )}^2\,{\mathrm {e}}^{-2\,x}\,{\left (x-3\right )}^2 \]

[In]

int((exp(-2*x)*(log(x*log((2*x^4 + 2)/x^4))*(48*x + log((2*x^4 + 2)/x^4)*(2*x^2 - 12*x + 18*x^4 - 12*x^5 + 2*x
^6 + 18) - 8*x^2 - 72) - log((2*x^4 + 2)/x^4)*log(x*log((2*x^4 + 2)/x^4))^2*(24*x - 14*x^2 + 2*x^3 + 24*x^5 -
14*x^6 + 2*x^7)))/(log((2*x^4 + 2)/x^4)*(x + x^5)),x)

[Out]

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