3.1.20 \(\int \frac {e^{2 e^x-x} (2-6 x+6 x^2-6 x^3+3 x^4-x^5+e^x (-4+12 x-12 x^2+4 x^3-2 x^4+2 x^5))}{-6+18 x-18 x^2+6 x^3} \, dx\)

Optimal. Leaf size=32 \[ \frac {1}{3} \left (-2+\frac {1}{2} e^{2 e^x-x} \left (2+\frac {x^4}{(-1+x)^2}\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 4.20, 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 {e^{2 e^x-x} \left (2-6 x+6 x^2-6 x^3+3 x^4-x^5+e^x \left (-4+12 x-12 x^2+4 x^3-2 x^4+2 x^5\right )\right )}{-6+18 x-18 x^2+6 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*E^x - x)*(2 - 6*x + 6*x^2 - 6*x^3 + 3*x^4 - x^5 + E^x*(-4 + 12*x - 12*x^2 + 4*x^3 - 2*x^4 + 2*x^5)))
/(-6 + 18*x - 18*x^2 + 6*x^3),x]

[Out]

E^(2*E^x - x)/2 + (2*ExpIntegralEi[2*E^x])/3 - Defer[Int][E^(2*E^x - x)/(-1 + x)^3, x]/3 + Defer[Int][E^(2*E^x
)/(-1 + x)^2, x]/3 - (5*Defer[Int][E^(2*E^x - x)/(-1 + x)^2, x])/6 + (4*Defer[Int][E^(2*E^x)/(-1 + x), x])/3 -
(2*Defer[Int][E^(2*E^x - x)/(-1 + x), x])/3 + (2*Defer[Int][E^(2*E^x)*x, x])/3 + Defer[Int][E^(2*E^x)*x^2, x]
/3 - Defer[Int][E^(2*E^x - x)*x^2, x]/6

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{2 e^x-x}}{3 (-1+x)^3}-\frac {e^{2 e^x-x} x}{(-1+x)^3}+\frac {e^{2 e^x-x} x^2}{(-1+x)^3}-\frac {e^{2 e^x-x} x^3}{(-1+x)^3}+\frac {e^{2 e^x-x} x^4}{2 (-1+x)^3}-\frac {e^{2 e^x-x} x^5}{6 (-1+x)^3}+\frac {e^{2 e^x} \left (2-4 x+2 x^2+x^4\right )}{3 (-1+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{6} \int \frac {e^{2 e^x-x} x^5}{(-1+x)^3} \, dx\right )+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x} \left (2-4 x+2 x^2+x^4\right )}{(-1+x)^2} \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x} x^4}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x} x}{(-1+x)^3} \, dx+\int \frac {e^{2 e^x-x} x^2}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x} x^3}{(-1+x)^3} \, dx\\ &=-\left (\frac {1}{6} \int \left (6 e^{2 e^x-x}+\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {5 e^{2 e^x-x}}{(-1+x)^2}+\frac {10 e^{2 e^x-x}}{-1+x}+3 e^{2 e^x-x} x+e^{2 e^x-x} x^2\right ) \, dx\right )+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \left (5 e^{2 e^x}+\frac {e^{2 e^x}}{(-1+x)^2}+\frac {4 e^{2 e^x}}{-1+x}+2 e^{2 e^x} x+e^{2 e^x} x^2\right ) \, dx+\frac {1}{2} \int \left (3 e^{2 e^x-x}+\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {4 e^{2 e^x-x}}{(-1+x)^2}+\frac {6 e^{2 e^x-x}}{-1+x}+e^{2 e^x-x} x\right ) \, dx-\int \left (\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {e^{2 e^x-x}}{(-1+x)^2}\right ) \, dx+\int \left (\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {2 e^{2 e^x-x}}{(-1+x)^2}+\frac {e^{2 e^x-x}}{-1+x}\right ) \, dx-\int \left (e^{2 e^x-x}+\frac {e^{2 e^x-x}}{(-1+x)^3}+\frac {3 e^{2 e^x-x}}{(-1+x)^2}+\frac {3 e^{2 e^x-x}}{-1+x}\right ) \, dx\\ &=-\left (\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx\right )-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx+\frac {3}{2} \int e^{2 e^x-x} \, dx+\frac {5}{3} \int e^{2 e^x} \, dx-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx-2 \int e^{2 e^x-x} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx\\ &=-\left (\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx\right )-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx+\frac {3}{2} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x^2} \, dx,x,e^x\right )-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+\frac {5}{3} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx-2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x^2} \, dx,x,e^x\right )\\ &=-\frac {3}{2} e^{2 e^x-x}+\frac {5 \text {Ei}\left (2 e^x\right )}{3}-\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-2 \left (-e^{2 e^x-x}+2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+3 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx\\ &=-\frac {3}{2} e^{2 e^x-x}+\frac {14 \text {Ei}\left (2 e^x\right )}{3}-2 \left (-e^{2 e^x-x}+2 \text {Ei}\left (2 e^x\right )\right )-\frac {1}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\frac {1}{6} \int e^{2 e^x-x} x^2 \, dx+\frac {1}{3} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {1}{3} \int \frac {e^{2 e^x}}{(-1+x)^2} \, dx+\frac {1}{3} \int e^{2 e^x} x^2 \, dx+\frac {1}{2} \int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx+\frac {2}{3} \int e^{2 e^x} x \, dx-\frac {5}{6} \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{2 e^x}}{-1+x} \, dx-\frac {5}{3} \int \frac {e^{2 e^x-x}}{-1+x} \, dx+2 \left (2 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx\right )-3 \int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^3} \, dx-\int \frac {e^{2 e^x-x}}{(-1+x)^2} \, dx+\int \frac {e^{2 e^x-x}}{-1+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.94, size = 35, normalized size = 1.09 \begin {gather*} \frac {1}{6} e^{2 e^x-x} \left (5+\frac {1}{(-1+x)^2}+\frac {4}{-1+x}+2 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^x - x)*(2 - 6*x + 6*x^2 - 6*x^3 + 3*x^4 - x^5 + E^x*(-4 + 12*x - 12*x^2 + 4*x^3 - 2*x^4 + 2*
x^5)))/(-6 + 18*x - 18*x^2 + 6*x^3),x]

[Out]

(E^(2*E^x - x)*(5 + (-1 + x)^(-2) + 4/(-1 + x) + 2*x + x^2))/6

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 34, normalized size = 1.06 \begin {gather*} \frac {{\left (x^{4} + 2 \, x^{2} - 4 \, x + 2\right )} e^{\left (-x + 2 \, e^{x}\right )}}{6 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-2*x^4+4*x^3-12*x^2+12*x-4)*exp(x)-x^5+3*x^4-6*x^3+6*x^2-6*x+2)*exp(2*exp(x)-x)/(6*x^3-18*x^2
+18*x-6),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.49, size = 53, normalized size = 1.66 \begin {gather*} \frac {x^{4} e^{\left (2 \, e^{x}\right )} + 2 \, x^{2} e^{\left (2 \, e^{x}\right )} - 4 \, x e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (2 \, e^{x}\right )}}{6 \, {\left (x^{2} e^{x} - 2 \, x e^{x} + e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-2*x^4+4*x^3-12*x^2+12*x-4)*exp(x)-x^5+3*x^4-6*x^3+6*x^2-6*x+2)*exp(2*exp(x)-x)/(6*x^3-18*x^2
+18*x-6),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 30, normalized size = 0.94

method	result	size
risch	\(\frac {\left (x^{4}+2 x^{2}-4 x +2\right ) {\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{6 \left (x -1\right )^{2}}\)	\(30\)
norman	\(\frac {-\frac {2 x \,{\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{3}+\frac {x^{2} {\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{3}+\frac {x^{4} {\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{6}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}-x}}{3}}{\left (x -1\right )^{2}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5-2*x^4+4*x^3-12*x^2+12*x-4)*exp(x)-x^5+3*x^4-6*x^3+6*x^2-6*x+2)*exp(2*exp(x)-x)/(6*x^3-18*x^2+18*x-
6),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^4+2*x^2-4*x+2)/(x-1)^2*exp(2*exp(x)-x)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 34, normalized size = 1.06 \begin {gather*} \frac {{\left (x^{4} + 2 \, x^{2} - 4 \, x + 2\right )} e^{\left (-x + 2 \, e^{x}\right )}}{6 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-2*x^4+4*x^3-12*x^2+12*x-4)*exp(x)-x^5+3*x^4-6*x^3+6*x^2-6*x+2)*exp(2*exp(x)-x)/(6*x^3-18*x^2
+18*x-6),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.22, size = 35, normalized size = 1.09 \begin {gather*} \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^x-x}\,\left (\frac {x^4}{6}+\frac {x^2}{3}-\frac {2\,x}{3}+\frac {1}{3}\right )}{x^2-2\,x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*exp(x) - x)*(exp(x)*(12*x - 12*x^2 + 4*x^3 - 2*x^4 + 2*x^5 - 4) - 6*x + 6*x^2 - 6*x^3 + 3*x^4 - x^5
+ 2))/(18*x - 18*x^2 + 6*x^3 - 6),x)

[Out]

(exp(2*exp(x) - x)*(x^2/3 - (2*x)/3 + x^4/6 + 1/3))/(x^2 - 2*x + 1)

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 31, normalized size = 0.97 \begin {gather*} \frac {\left (x^{4} + 2 x^{2} - 4 x + 2\right ) e^{- x + 2 e^{x}}}{6 x^{2} - 12 x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**5-2*x**4+4*x**3-12*x**2+12*x-4)*exp(x)-x**5+3*x**4-6*x**3+6*x**2-6*x+2)*exp(2*exp(x)-x)/(6*x*
*3-18*x**2+18*x-6),x)

[Out]

(x**4 + 2*x**2 - 4*x + 2)*exp(-x + 2*exp(x))/(6*x**2 - 12*x + 6)

________________________________________________________________________________________