3.98.31 \(\int \frac {4 x^3-4 x^4+x^5+(8 x^2-8 x^3+2 x^4) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} (4 x-4 x^2+x^3+(8-8 x+2 x^2) \log (x))+e^{\frac {-9+\log (x)}{-2 x+x^2}} (20-15 x-12 x^2+9 x^3-2 x^4+(-2-14 x+16 x^2-4 x^3) \log (x))}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} (4 x-4 x^2+x^3)+e^{\frac {-9+\log (x)}{-2 x+x^2}} (-8 x^2+8 x^3-2 x^4)} \, dx\)
Optimal. Leaf size=29 \[ x+\frac {x}{e^{\frac {-9+\log (x)}{(-2+x) x}}-x}+\log ^2(x) \]
________________________________________________________________________________________
Rubi [F] time = 168.58, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(4*x^3 - 4*x^4 + x^5 + (8*x^2 - 8*x^3 + 2*x^4)*Log[x] + E^((2*(-9 + Log[x]))/(-2*x + x^2))*(4*x - 4*x^2 +
x^3 + (8 - 8*x + 2*x^2)*Log[x]) + E^((-9 + Log[x])/(-2*x + x^2))*(20 - 15*x - 12*x^2 + 9*x^3 - 2*x^4 + (-2 - 1
4*x + 16*x^2 - 4*x^3)*Log[x]))/(4*x^3 - 4*x^4 + x^5 + E^((2*(-9 + Log[x]))/(-2*x + x^2))*(4*x - 4*x^2 + x^3) +
E^((-9 + Log[x])/(-2*x + x^2))*(-8*x^2 + 8*x^3 - 2*x^4)),x]
[Out]
-1/12*(2 - x)^3 + (4 - x)^2/8 + x + (3*x^2)/8 - x^3/12 + x*Log[x] + ((4*x - x^2)*Log[x])/4 - ((8*x - x^2 - 8*L
og[x])*Log[x])/4 - Log[x]^2 - 18*Defer[Int][E^(18/((-2 + x)*x))/((-2 + x)^2*(E^(9/((-2 + x)*x))*x - x^(-2*x +
x^2)^(-1))^2), x] + 2*Log[x]*Defer[Int][E^(18/((-2 + x)*x))/((-2 + x)^2*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)
^(-1))^2), x] - 19*Defer[Int][E^(18/((-2 + x)*x))/((-2 + x)*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))^2), x
] + 2*Log[x]*Defer[Int][E^(18/((-2 + x)*x))/((-2 + x)*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))^2), x] + De
fer[Int][(E^(18/((-2 + x)*x))*x)/(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))^2, x] + 9*Defer[Int][E^(9/((-2 +
x)*x))/((-2 + x)^2*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))), x] - Log[x]*Defer[Int][E^(9/((-2 + x)*x))/(
(-2 + x)^2*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))), x] + 5*Defer[Int][E^(9/((-2 + x)*x))/((-2 + x)*(E^(9
/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))), x] - (Log[x]*Defer[Int][E^(9/((-2 + x)*x))/((-2 + x)*(E^(9/((-2 + x)
*x))*x - x^(-2*x + x^2)^(-1))), x])/2 - 5*Defer[Int][E^(9/((-2 + x)*x))/(x*(E^(9/((-2 + x)*x))*x - x^(-2*x + x
^2)^(-1))), x] + (Log[x]*Defer[Int][E^(9/((-2 + x)*x))/(x*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))), x])/2
+ Defer[Int][E^(9/((-2 + x)*x))/(-(E^(9/((-2 + x)*x))*x) + x^(-2*x + x^2)^(-1)), x] + Defer[Int][Defer[Int][E
^(9/((-2 + x)*x))/((-2 + x)^2*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))), x]/x, x] + Defer[Int][Defer[Int][
E^(9/((-2 + x)*x))/((-2 + x)*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))), x]/x, x]/2 - Defer[Int][Defer[Int]
[E^(9/((-2 + x)*x))/(x*(E^(9/((-2 + x)*x))*x - x^(-2*x + x^2)^(-1))), x]/x, x]/2 - 2*Defer[Int][Defer[Int][E^(
18/((-2 + x)*x))/((-2 + x)^2*(-(E^(9/((-2 + x)*x))*x) + x^(-2*x + x^2)^(-1))^2), x]/x, x] - 2*Defer[Int][Defer
[Int][E^(18/((-2 + x)*x))/((-2 + x)*(-(E^(9/((-2 + x)*x))*x) + x^(-2*x + x^2)^(-1))^2), x]/x, x]
Rubi steps
Aborted
________________________________________________________________________________________
Mathematica [F] time = 0.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 x^3-4 x^4+x^5+\left (8 x^2-8 x^3+2 x^4\right ) \log (x)+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3+\left (8-8 x+2 x^2\right ) \log (x)\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (20-15 x-12 x^2+9 x^3-2 x^4+\left (-2-14 x+16 x^2-4 x^3\right ) \log (x)\right )}{4 x^3-4 x^4+x^5+e^{\frac {2 (-9+\log (x))}{-2 x+x^2}} \left (4 x-4 x^2+x^3\right )+e^{\frac {-9+\log (x)}{-2 x+x^2}} \left (-8 x^2+8 x^3-2 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(4*x^3 - 4*x^4 + x^5 + (8*x^2 - 8*x^3 + 2*x^4)*Log[x] + E^((2*(-9 + Log[x]))/(-2*x + x^2))*(4*x - 4*
x^2 + x^3 + (8 - 8*x + 2*x^2)*Log[x]) + E^((-9 + Log[x])/(-2*x + x^2))*(20 - 15*x - 12*x^2 + 9*x^3 - 2*x^4 + (
-2 - 14*x + 16*x^2 - 4*x^3)*Log[x]))/(4*x^3 - 4*x^4 + x^5 + E^((2*(-9 + Log[x]))/(-2*x + x^2))*(4*x - 4*x^2 +
x^3) + E^((-9 + Log[x])/(-2*x + x^2))*(-8*x^2 + 8*x^3 - 2*x^4)),x]
[Out]
Integrate[(4*x^3 - 4*x^4 + x^5 + (8*x^2 - 8*x^3 + 2*x^4)*Log[x] + E^((2*(-9 + Log[x]))/(-2*x + x^2))*(4*x - 4*
x^2 + x^3 + (8 - 8*x + 2*x^2)*Log[x]) + E^((-9 + Log[x])/(-2*x + x^2))*(20 - 15*x - 12*x^2 + 9*x^3 - 2*x^4 + (
-2 - 14*x + 16*x^2 - 4*x^3)*Log[x]))/(4*x^3 - 4*x^4 + x^5 + E^((2*(-9 + Log[x]))/(-2*x + x^2))*(4*x - 4*x^2 +
x^3) + E^((-9 + Log[x])/(-2*x + x^2))*(-8*x^2 + 8*x^3 - 2*x^4)), x]
________________________________________________________________________________________
fricas [A] time = 1.43, size = 58, normalized size = 2.00 \begin {gather*} \frac {x \log \relax (x)^{2} + x^{2} - {\left (\log \relax (x)^{2} + x\right )} e^{\left (\frac {\log \relax (x) - 9}{x^{2} - 2 \, x}\right )} - x}{x - e^{\left (\frac {\log \relax (x) - 9}{x^{2} - 2 \, x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((2*x^2-8*x+8)*log(x)+x^3-4*x^2+4*x)*exp((log(x)-9)/(x^2-2*x))^2+((-4*x^3+16*x^2-14*x-2)*log(x)-2*x
^4+9*x^3-12*x^2-15*x+20)*exp((log(x)-9)/(x^2-2*x))+(2*x^4-8*x^3+8*x^2)*log(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x
)*exp((log(x)-9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((log(x)-9)/(x^2-2*x))+x^5-4*x^4+4*x^3),x, algorithm="fr
icas")
[Out]
(x*log(x)^2 + x^2 - (log(x)^2 + x)*e^((log(x) - 9)/(x^2 - 2*x)) - x)/(x - e^((log(x) - 9)/(x^2 - 2*x)))
________________________________________________________________________________________
giac [B] time = 0.84, size = 74, normalized size = 2.55 \begin {gather*} \frac {x \log \relax (x)^{2} - e^{\left (\frac {\log \relax (x) - 9}{x^{2} - 2 \, x}\right )} \log \relax (x)^{2} + x^{2} - x e^{\left (\frac {\log \relax (x) - 9}{x^{2} - 2 \, x}\right )} - x}{x - e^{\left (\frac {\log \relax (x) - 9}{x^{2} - 2 \, x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((2*x^2-8*x+8)*log(x)+x^3-4*x^2+4*x)*exp((log(x)-9)/(x^2-2*x))^2+((-4*x^3+16*x^2-14*x-2)*log(x)-2*x
^4+9*x^3-12*x^2-15*x+20)*exp((log(x)-9)/(x^2-2*x))+(2*x^4-8*x^3+8*x^2)*log(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x
)*exp((log(x)-9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((log(x)-9)/(x^2-2*x))+x^5-4*x^4+4*x^3),x, algorithm="gi
ac")
[Out]
(x*log(x)^2 - e^((log(x) - 9)/(x^2 - 2*x))*log(x)^2 + x^2 - x*e^((log(x) - 9)/(x^2 - 2*x)) - x)/(x - e^((log(x
) - 9)/(x^2 - 2*x)))
________________________________________________________________________________________
maple [A] time = 0.06, size = 30, normalized size = 1.03
|
|
|
| method |
result |
size |
|
|
|
| risch |
\(\ln \relax (x )^{2}+x -\frac {x}{x -{\mathrm e}^{\frac {\ln \relax (x )-9}{\left (x -2\right ) x}}}\) |
\(30\) |
|
|
|
| |
| |
| |
| |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((2*x^2-8*x+8)*ln(x)+x^3-4*x^2+4*x)*exp((ln(x)-9)/(x^2-2*x))^2+((-4*x^3+16*x^2-14*x-2)*ln(x)-2*x^4+9*x^3-
12*x^2-15*x+20)*exp((ln(x)-9)/(x^2-2*x))+(2*x^4-8*x^3+8*x^2)*ln(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x)*exp((ln(x
)-9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((ln(x)-9)/(x^2-2*x))+x^5-4*x^4+4*x^3),x,method=_RETURNVERBOSE)
[Out]
ln(x)^2+x-x/(x-exp((ln(x)-9)/(x-2)/x))
________________________________________________________________________________________
maxima [B] time = 0.46, size = 96, normalized size = 3.31 \begin {gather*} -\frac {{\left (\log \relax (x)^{2} + x\right )} e^{\left (\frac {\log \relax (x)}{2 \, {\left (x - 2\right )}} + \frac {9}{2 \, x}\right )} - {\left (x \log \relax (x)^{2} + x^{2} - x\right )} e^{\left (\frac {\log \relax (x)}{2 \, x} + \frac {9}{2 \, {\left (x - 2\right )}}\right )}}{x e^{\left (\frac {\log \relax (x)}{2 \, x} + \frac {9}{2 \, {\left (x - 2\right )}}\right )} - e^{\left (\frac {\log \relax (x)}{2 \, {\left (x - 2\right )}} + \frac {9}{2 \, x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((2*x^2-8*x+8)*log(x)+x^3-4*x^2+4*x)*exp((log(x)-9)/(x^2-2*x))^2+((-4*x^3+16*x^2-14*x-2)*log(x)-2*x
^4+9*x^3-12*x^2-15*x+20)*exp((log(x)-9)/(x^2-2*x))+(2*x^4-8*x^3+8*x^2)*log(x)+x^5-4*x^4+4*x^3)/((x^3-4*x^2+4*x
)*exp((log(x)-9)/(x^2-2*x))^2+(-2*x^4+8*x^3-8*x^2)*exp((log(x)-9)/(x^2-2*x))+x^5-4*x^4+4*x^3),x, algorithm="ma
xima")
[Out]
-((log(x)^2 + x)*e^(1/2*log(x)/(x - 2) + 9/2/x) - (x*log(x)^2 + x^2 - x)*e^(1/2*log(x)/x + 9/2/(x - 2)))/(x*e^
(1/2*log(x)/x + 9/2/(x - 2)) - e^(1/2*log(x)/(x - 2) + 9/2/x))
________________________________________________________________________________________
mupad [B] time = 6.46, size = 123, normalized size = 4.24 \begin {gather*} \frac {{\mathrm {e}}^{\frac {9}{2\,x-x^2}}\,{\ln \relax (x)}^2+x\,{\mathrm {e}}^{\frac {9}{2\,x-x^2}}+x\,x^{\frac {1}{2\,x-x^2}}-x^{\frac {1}{2\,x-x^2}}\,x^2-x\,x^{\frac {1}{2\,x-x^2}}\,{\ln \relax (x)}^2}{{\mathrm {e}}^{\frac {9}{2\,x-x^2}}-x\,x^{\frac {1}{2\,x-x^2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(-(2*(log(x) - 9))/(2*x - x^2))*(4*x + log(x)*(2*x^2 - 8*x + 8) - 4*x^2 + x^3) + log(x)*(8*x^2 - 8*x^3
+ 2*x^4) + 4*x^3 - 4*x^4 + x^5 - exp(-(log(x) - 9)/(2*x - x^2))*(15*x + 12*x^2 - 9*x^3 + 2*x^4 + log(x)*(14*x
- 16*x^2 + 4*x^3 + 2) - 20))/(exp(-(2*(log(x) - 9))/(2*x - x^2))*(4*x - 4*x^2 + x^3) - exp(-(log(x) - 9)/(2*x
- x^2))*(8*x^2 - 8*x^3 + 2*x^4) + 4*x^3 - 4*x^4 + x^5),x)
[Out]
(exp(9/(2*x - x^2))*log(x)^2 + x*exp(9/(2*x - x^2)) + x*x^(1/(2*x - x^2)) - x^(1/(2*x - x^2))*x^2 - x*x^(1/(2*
x - x^2))*log(x)^2)/(exp(9/(2*x - x^2)) - x*x^(1/(2*x - x^2)))
________________________________________________________________________________________
sympy [A] time = 0.69, size = 22, normalized size = 0.76 \begin {gather*} x + \frac {x}{- x + e^{\frac {\log {\relax (x )} - 9}{x^{2} - 2 x}}} + \log {\relax (x )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((2*x**2-8*x+8)*ln(x)+x**3-4*x**2+4*x)*exp((ln(x)-9)/(x**2-2*x))**2+((-4*x**3+16*x**2-14*x-2)*ln(x)
-2*x**4+9*x**3-12*x**2-15*x+20)*exp((ln(x)-9)/(x**2-2*x))+(2*x**4-8*x**3+8*x**2)*ln(x)+x**5-4*x**4+4*x**3)/((x
**3-4*x**2+4*x)*exp((ln(x)-9)/(x**2-2*x))**2+(-2*x**4+8*x**3-8*x**2)*exp((ln(x)-9)/(x**2-2*x))+x**5-4*x**4+4*x
**3),x)
[Out]
x + x/(-x + exp((log(x) - 9)/(x**2 - 2*x))) + log(x)**2
________________________________________________________________________________________