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

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

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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 = {} $$\begin{array}{c}\int \frac{e^{2e^x-x}(2-6x+6x^2-6x^3+3x^4-x^5+e^x(-4+12x-12x^2+4x^3-2x^4+2x^5))}{-6+18x-18x^2+6x^3}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{e^{2e^x-x}}{3(-1+x{)}^3}-\frac{e^{2e^x-x}x}{(-1+x{)}^3}+\frac{e^{2e^x-x}{x}^2}{(-1+x{)}^3}-\frac{e^{2e^x-x}{x}^3}{(-1+x{)}^3}+\frac{e^{2e^x-x}{x}^4}{2(-1+x{)}^3}-\frac{e^{2e^x-x}{x}^5}{6(-1+x{)}^3}+\frac{e^{2e^x}(2-4x+2x^2+x^4)}{3(-1+x{)}^2})dx\\ & =-(\frac{1}{6}\int \frac{e^{2e^x-x}{x}^5}{(-1+x{)}^3}dx)+\frac{1}{3}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{1}{3}\int \frac{e^{2e^x}(2-4x+2x^2+x^4)}{(-1+x{)}^2}dx+\frac{1}{2}\int \frac{e^{2e^x-x}{x}^4}{(-1+x{)}^3}dx-\int \frac{e^{2e^x-x}x}{(-1+x{)}^3}dx+\int \frac{e^{2e^x-x}{x}^2}{(-1+x{)}^3}dx-\int \frac{e^{2e^x-x}{x}^3}{(-1+x{)}^3}dx\\ & =-(\frac{1}{6}\int (6e^{2e^x-x}+\frac{e^{2e^x-x}}{(-1+x{)}^3}+\frac{5e^{2e^x-x}}{(-1+x{)}^2}+\frac{10e^{2e^x-x}}{-1+x}+3e^{2e^x-x}x+e^{2e^x-x}{x}^2)dx)+\frac{1}{3}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{1}{3}\int (5e^{2e^x}+\frac{e^{2e^x}}{(-1+x{)}^2}+\frac{4e^{2e^x}}{-1+x}+2e^{2e^x}x+e^{2e^x}{x}^2)dx+\frac{1}{2}\int (3e^{2e^x-x}+\frac{e^{2e^x-x}}{(-1+x{)}^3}+\frac{4e^{2e^x-x}}{(-1+x{)}^2}+\frac{6e^{2e^x-x}}{-1+x}+e^{2e^x-x}x)dx-\int (\frac{e^{2e^x-x}}{(-1+x{)}^3}+\frac{e^{2e^x-x}}{(-1+x{)}^2})dx+\int (\frac{e^{2e^x-x}}{(-1+x{)}^3}+\frac{2e^{2e^x-x}}{(-1+x{)}^2}+\frac{e^{2e^x-x}}{-1+x})dx-\int (e^{2e^x-x}+\frac{e^{2e^x-x}}{(-1+x{)}^3}+\frac{3e^{2e^x-x}}{(-1+x{)}^2}+\frac{3e^{2e^x-x}}{-1+x})dx\\ & =-(\frac{1}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx)-\frac{1}{6}\int e^{2e^x-x}{x}^{2}dx+\frac{1}{3}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{1}{3}\int \frac{e^{2e^x}}{(-1+x{)}^2}dx+\frac{1}{3}\int e^{2e^x}{x}^{2}dx+\frac{1}{2}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{2}{3}\int e^{2e^x}xdx-\frac{5}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\frac{4}{3}\int \frac{e^{2e^x}}{-1+x}dx+\frac{3}{2}\int e^{2e^x-x}dx+\frac{5}{3}\int e^{2e^x}dx-\frac{5}{3}\int \frac{e^{2e^x-x}}{-1+x}dx+2(2\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx)-3\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx-2\int e^{2e^x-x}dx-\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx-\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\int \frac{e^{2e^x-x}}{-1+x}dx\\ & =-(\frac{1}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx)-\frac{1}{6}\int e^{2e^x-x}{x}^{2}dx+\frac{1}{3}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{1}{3}\int \frac{e^{2e^x}}{(-1+x{)}^2}dx+\frac{1}{3}\int e^{2e^x}{x}^{2}dx+\frac{1}{2}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{2}{3}\int e^{2e^x}xdx-\frac{5}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\frac{4}{3}\int \frac{e^{2e^x}}{-1+x}dx+\frac{3}{2}\mathrm{Subst}(\int \frac{e^{2x}}{x^2}dx,x,e^x)-\frac{5}{3}\int \frac{e^{2e^x-x}}{-1+x}dx+\frac{5}{3}\mathrm{Subst}(\int \frac{e^{2x}}{x}dx,x,e^x)+2(2\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx)-3\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx-\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx-\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\int \frac{e^{2e^x-x}}{-1+x}dx-2\mathrm{Subst}(\int \frac{e^{2x}}{x^2}dx,x,e^x)\\ & =-\frac{3}{2}{e}^{2e^x-x}+\frac{5\text{Ei}\left(2e^x\right)}{3}-\frac{1}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx-\frac{1}{6}\int e^{2e^x-x}{x}^{2}dx+\frac{1}{3}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{1}{3}\int \frac{e^{2e^x}}{(-1+x{)}^2}dx+\frac{1}{3}\int e^{2e^x}{x}^{2}dx+\frac{1}{2}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{2}{3}\int e^{2e^x}xdx-\frac{5}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\frac{4}{3}\int \frac{e^{2e^x}}{-1+x}dx-\frac{5}{3}\int \frac{e^{2e^x-x}}{-1+x}dx+2(2\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx)-2(-e^{2e^x-x}+2\mathrm{Subst}(\int \frac{e^{2x}}{x}dx,x,e^x))-3\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+3\mathrm{Subst}(\int \frac{e^{2x}}{x}dx,x,e^x)-\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx-\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\int \frac{e^{2e^x-x}}{-1+x}dx\\ & =-\frac{3}{2}{e}^{2e^x-x}+\frac{14\text{Ei}\left(2e^x\right)}{3}-2(-e^{2e^x-x}+2\text{Ei}\left(2e^x\right))-\frac{1}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx-\frac{1}{6}\int e^{2e^x-x}{x}^{2}dx+\frac{1}{3}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{1}{3}\int \frac{e^{2e^x}}{(-1+x{)}^2}dx+\frac{1}{3}\int e^{2e^x}{x}^{2}dx+\frac{1}{2}\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx+\frac{2}{3}\int e^{2e^x}xdx-\frac{5}{6}\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\frac{4}{3}\int \frac{e^{2e^x}}{-1+x}dx-\frac{5}{3}\int \frac{e^{2e^x-x}}{-1+x}dx+2(2\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx)-3\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx-\int \frac{e^{2e^x-x}}{(-1+x{)}^3}dx-\int \frac{e^{2e^x-x}}{(-1+x{)}^2}dx+\int \frac{e^{2e^x-x}}{-1+x}dx\end{array}\end{array}$$

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

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

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

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)

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

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)

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

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

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)

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

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)

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

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)

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

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)

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