3.77.74 \(\int e^{-2 x} (1-2 x) \, dx\)

Optimal. Leaf size=13 \[ 6+e^{-2 x} x+\log \left (\frac {5}{3}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.85, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2176, 2194} \begin {gather*} \frac {e^{-2 x}}{2}-\frac {1}{2} e^{-2 x} (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/E^(2*x),x]

[Out]

1/(2*E^(2*x)) - (1 - 2*x)/(2*E^(2*x))

Rule 2176

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] &&  !$UseGamma === True

Rule 2194

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.54 \begin {gather*} e^{-2 x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)/E^(2*x),x]

[Out]

x/E^(2*x)

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fricas [A]  time = 1.51, size = 6, normalized size = 0.46 \begin {gather*} x e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(-2*x)

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giac [A]  time = 0.14, size = 6, normalized size = 0.46 \begin {gather*} x e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(-2*x)

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maple [A]  time = 0.02, size = 7, normalized size = 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	\({\mathrm e}^{-2 x} x\)	\(7\)
meijerg	\(-\frac {{\mathrm e}^{-2 x}}{2}+\frac {\left (4 x +2\right ) {\mathrm e}^{-2 x}}{4}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(-2*x)*x

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maxima [A]  time = 0.38, size = 18, normalized size = 1.38 \begin {gather*} \frac {1}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {1}{2} \, e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.03, size = 6, normalized size = 0.46 \begin {gather*} x\,{\mathrm {e}}^{-2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-2*x)*(2*x - 1),x)

[Out]

x*exp(-2*x)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.38 \begin {gather*} x e^{- 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(-2*x)

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