3.73.17 $\int (1-2x+e^{4+e^2+e^x+2x-x^2}(-1-2x-e^{x}x+2x^2))dx$

Optimal. Leaf size=25 $$x-x(e^{4+e^2+e^x+2x-x^2}+x)$$

________________________________________________________________________________________

Rubi [B]  time = 0.12, antiderivative size = 51, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 1, integrand size = 40, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.025, Rules used = {2288} $$\begin{array}{c}-x^2-\frac{e^{-x^2+2x+e^x+e^2+4}(-2x^2+e^{x}x+2x)}{-2x+e^x+2}+x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[1 - 2*x + E^(4 + E^2 + E^x + 2*x - x^2)*(-1 - 2*x - E^x*x + 2*x^2),x]

[Out]

x - x^2 - (E^(4 + E^2 + E^x + 2*x - x^2)*(2*x + E^x*x - 2*x^2))/(2 + E^x - 2*x)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =x-x^2+\int e^{4+e^2+e^x+2x-x^2}(-1-2x-e^{x}x+2x^2)dx\\ & =x-x^2-\frac{e^{4+e^2+e^x+2x-x^2}(2x+e^{x}x-2x^2)}{2+e^x-2x}\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 27, normalized size = 1.08 $$\begin{array}{c}(1-e^{4+e^2+e^x+2x-x^2}-x)x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[1 - 2*x + E^(4 + E^2 + E^x + 2*x - x^2)*(-1 - 2*x - E^x*x + 2*x^2),x]

[Out]

(1 - E^(4 + E^2 + E^x + 2*x - x^2) - x)*x

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 25, normalized size = 1.00 $$\begin{array}{c}-x^2-x{e}^{(-x^2+2x+e^2+e^x+4)}+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x, algorithm="fricas")

[Out]

-x^2 - x*e^(-x^2 + 2*x + e^2 + e^x + 4) + x

________________________________________________________________________________________

giac [A]  time = 0.22, size = 25, normalized size = 1.00 $$\begin{array}{c}-x^2-x{e}^{(-x^2+2x+e^2+e^x+4)}+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x, algorithm="giac")

[Out]

-x^2 - x*e^(-x^2 + 2*x + e^2 + e^x + 4) + x

________________________________________________________________________________________

maple [A]  time = 0.05, size = 26, normalized size = 1.04

method	result	size
default	$x-{\mathrm{e}}^{{\mathrm{e}}^x+{\mathrm{e}}^2-x^2+2x+4}x-x^2$	$26$
norman	$x-{\mathrm{e}}^{{\mathrm{e}}^x+{\mathrm{e}}^2-x^2+2x+4}x-x^2$	$26$
risch	$x-{\mathrm{e}}^{{\mathrm{e}}^x+{\mathrm{e}}^2-x^2+2x+4}x-x^2$	$26$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x,method=_RETURNVERBOSE)

[Out]

x-exp(exp(x)+exp(2)-x^2+2*x+4)*x-x^2

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 25, normalized size = 1.00 $$\begin{array}{c}-x^2-x{e}^{(-x^2+2x+e^2+e^x+4)}+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x^2-2*x-1)*exp(exp(x)+exp(2)-x^2+2*x+4)+1-2*x,x, algorithm="maxima")

[Out]

-x^2 - x*e^(-x^2 + 2*x + e^2 + e^x + 4) + x

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 28, normalized size = 1.12 $$\begin{array}{c}x-x^2-x{\mathrm{e}}^{2x}{\mathrm{e}}^{{\mathrm{e}}^x}{\mathrm{e}}^4{\mathrm{e}}^{-x^2}{\mathrm{e}}^{{\mathrm{e}}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - exp(2*x + exp(2) + exp(x) - x^2 + 4)*(2*x + x*exp(x) - 2*x^2 + 1) - 2*x,x)

[Out]

x - x^2 - x*exp(2*x)*exp(exp(x))*exp(4)*exp(-x^2)*exp(exp(2))

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 22, normalized size = 0.88 $$\begin{array}{c}-x^2-x{e}^{-x^2+2x+e^x+4+e^2}+x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x+2*x**2-2*x-1)*exp(exp(x)+exp(2)-x**2+2*x+4)+1-2*x,x)

[Out]

-x**2 - x*exp(-x**2 + 2*x + exp(x) + 4 + exp(2)) + x

________________________________________________________________________________________