\(\int e^{-2 x} (1-2 x) \, dx\) [7674]

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int e^{-2 x} (1-2 x) \, dx=6+e^{-2 x} x+\log \left (\frac {5}{3}\right ) \]

[Out]

6-ln(3/5)+exp(x)*exp(-3*x)*x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2207, 2225} \[ \int e^{-2 x} (1-2 x) \, dx=\frac {e^{-2 x}}{2}-\frac {1}{2} e^{-2 x} (1-2 x) \]

[In]

Int[(1 - 2*x)/E^(2*x),x]

[Out]

1/(2*E^(2*x)) - (1 - 2*x)/(2*E^(2*x))

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

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}& = -\frac {1}{2} e^{-2 x} (1-2 x)-\int e^{-2 x} \, dx \\ & = \frac {e^{-2 x}}{2}-\frac {1}{2} e^{-2 x} (1-2 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54 \[ \int e^{-2 x} (1-2 x) \, dx=e^{-2 x} x \]

[In]

Integrate[(1 - 2*x)/E^(2*x),x]

[Out]

x/E^(2*x)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54

method	result	size
gosper	\({\mathrm e}^{-2 x} x\)	\(7\)
default	\({\mathrm e}^{-2 x} x\)	\(7\)
norman	\({\mathrm e}^{-2 x} x\)	\(7\)
risch	\({\mathrm e}^{-2 x} x\)	\(7\)
parallelrisch	\({\mathrm e}^{x} {\mathrm e}^{-3 x} x\)	\(9\)
meijerg	\(-\frac {{\mathrm e}^{-2 x}}{2}+\frac {\left (4 x +2\right ) {\mathrm e}^{-2 x}}{4}\)	\(19\)

[In]

int((1-2*x)*exp(-3*x)*exp(x),x,method=_RETURNVERBOSE)

[Out]

exp(-2*x)*x

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int e^{-2 x} (1-2 x) \, dx=x e^{\left (-2 \, x\right )} \]

[In]

integrate((1-2*x)*exp(-3*x)*exp(x),x, algorithm="fricas")

[Out]

x*e^(-2*x)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.38 \[ \int e^{-2 x} (1-2 x) \, dx=x e^{- 2 x} \]

[In]

integrate((1-2*x)*exp(-3*x)*exp(x),x)

[Out]

x*exp(-2*x)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int e^{-2 x} (1-2 x) \, dx=\frac {1}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {1}{2} \, e^{\left (-2 \, x\right )} \]

[In]

integrate((1-2*x)*exp(-3*x)*exp(x),x, algorithm="maxima")

[Out]

1/2*(2*x + 1)*e^(-2*x) - 1/2*e^(-2*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int e^{-2 x} (1-2 x) \, dx=x e^{\left (-2 \, x\right )} \]

[In]

integrate((1-2*x)*exp(-3*x)*exp(x),x, algorithm="giac")

[Out]

x*e^(-2*x)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int e^{-2 x} (1-2 x) \, dx=x\,{\mathrm {e}}^{-2\,x} \]

[In]

int(-exp(-2*x)*(2*x - 1),x)

[Out]

x*exp(-2*x)