3.55.41 \(\int \frac {1}{4} (-2-2 e^{2 x}+e^x (-1+x)) \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{4} \left (6-e^2-e^x \left (2+e^x-x\right )-2 x\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 2194, 2176} \begin {gather*} -\frac {1}{4} e^x (1-x)-\frac {e^x}{4}-\frac {e^{2 x}}{4}-\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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}{4} \int \left (-2-2 e^{2 x}+e^x (-1+x)\right ) \, dx\\ &=-\frac {x}{2}+\frac {1}{4} \int e^x (-1+x) \, dx-\frac {1}{2} \int e^{2 x} \, dx\\ &=-\frac {e^{2 x}}{4}-\frac {1}{4} e^x (1-x)-\frac {x}{2}-\frac {\int e^x \, dx}{4}\\ &=-\frac {e^x}{4}-\frac {e^{2 x}}{4}-\frac {1}{4} e^x (1-x)-\frac {x}{2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{4} \left (-e^{2 x}+e^x (-2+x)-2 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-E^(2*x) + E^x*(-2 + x) - 2*x)/4

________________________________________________________________________________________

fricas [A]  time = 1.14, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{4} \, {\left (x - 2\right )} e^{x} - \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x - 2)*e^x - 1/2*x - 1/4*e^(2*x)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{4} \, {\left (x - 2\right )} e^{x} - \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x - 2)*e^x - 1/2*x - 1/4*e^(2*x)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 18, normalized size = 0.67

method	result	size
risch	\(-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{x} \left (x -2\right )}{4}-\frac {x}{2}\)	\(18\)
default	\(-\frac {x}{2}-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{x} x}{4}-\frac {{\mathrm e}^{x}}{2}\)	\(20\)
norman	\(-\frac {x}{2}-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{x} x}{4}-\frac {{\mathrm e}^{x}}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{4} \, {\left (x - 1\right )} e^{x} - \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (2 \, x\right )} - \frac {1}{4} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x - 1)*e^x - 1/2*x - 1/4*e^(2*x) - 1/4*e^x

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 19, normalized size = 0.70 \begin {gather*} \frac {x\,{\mathrm {e}}^x}{4}-\frac {{\mathrm {e}}^{2\,x}}{4}-\frac {{\mathrm {e}}^x}{2}-\frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 19, normalized size = 0.70 \begin {gather*} - \frac {x}{2} + \frac {\left (4 x - 8\right ) e^{x}}{16} - \frac {e^{2 x}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/2 + (4*x - 8)*exp(x)/16 - exp(2*x)/4

________________________________________________________________________________________