\(\int (1-2 e^x-2 e^{2+2 x}+4 x) \, dx\) [6491]

Optimal result

Integrand size = 19, antiderivative size = 30 \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=-1-e^{2+2 x}+x+2 \left (3+x \left (-\frac {4+e^x}{x}+x\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2225} \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=2 x^2+x-2 e^x-e^{2 x+2} \]

[In]

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

[Out]

-2*E^x - E^(2 + 2*x) + x + 2*x^2

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = x+2 x^2-2 \int e^x \, dx-2 \int e^{2+2 x} \, dx \\ & = -2 e^x-e^{2+2 x}+x+2 x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=-2 e^x-e^{2+2 x}+x+2 x^2 \]

[In]

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

[Out]

-2*E^x - E^(2 + 2*x) + x + 2*x^2

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67

method	result	size
default	\(2 x^{2}+x -{\mathrm e}^{2+2 x}-2 \,{\mathrm e}^{x}\)	\(20\)
risch	\(2 x^{2}+x -{\mathrm e}^{2+2 x}-2 \,{\mathrm e}^{x}\)	\(20\)
parallelrisch	\(2 x^{2}+x -{\mathrm e}^{2+2 x}-2 \,{\mathrm e}^{x}\)	\(20\)
parts	\(2 x^{2}+x -{\mathrm e}^{2+2 x}-2 \,{\mathrm e}^{x}\)	\(20\)
norman	\(x +2 x^{2}-{\mathrm e}^{2 x} {\mathrm e}^{2}-2 \,{\mathrm e}^{x}\)	\(22\)

[In]

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

[Out]

2*x^2+x-exp(1+x)^2-2*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=2 \, x^{2} + x - e^{\left (2 \, x + 2\right )} - 2 \, e^{x} \]

[In]

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

[Out]

2*x^2 + x - e^(2*x + 2) - 2*e^x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=2 x^{2} + x - e^{2} e^{2 x} - 2 e^{x} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=2 \, x^{2} + x - e^{\left (2 \, x + 2\right )} - 2 \, e^{x} \]

[In]

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

[Out]

2*x^2 + x - e^(2*x + 2) - 2*e^x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=2 \, x^{2} + x - e^{\left (2 \, x + 2\right )} - 2 \, e^{x} \]

[In]

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

[Out]

2*x^2 + x - e^(2*x + 2) - 2*e^x

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \left (1-2 e^x-2 e^{2+2 x}+4 x\right ) \, dx=x-{\mathrm {e}}^{2\,x+2}-2\,{\mathrm {e}}^x+2\,x^2 \]

[In]

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

[Out]

x - exp(2*x + 2) - 2*exp(x) + 2*x^2