\(\int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx\) [6272]

Optimal result

Integrand size = 26, antiderivative size = 21 \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=4-e^{3-4 x}+\frac {3}{x}-\frac {x}{2} \]

[Out]

-1/2*x-exp(3-4*x)+4+3/x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2225} \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=-\frac {x}{2}-e^{3-4 x}+\frac {3}{x} \]

[In]

Int[(-6 - x^2 + 8*E^(3 - 4*x)*x^2)/(2*x^2),x]

[Out]

-E^(3 - 4*x) + 3/x - 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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} \int \frac {-6-x^2+8 e^{3-4 x} x^2}{x^2} \, dx \\ & = \frac {1}{2} \int \left (8 e^{3-4 x}+\frac {-6-x^2}{x^2}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-6-x^2}{x^2} \, dx+4 \int e^{3-4 x} \, dx \\ & = -e^{3-4 x}+\frac {1}{2} \int \left (-1-\frac {6}{x^2}\right ) \, dx \\ & = -e^{3-4 x}+\frac {3}{x}-\frac {x}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=-e^{3-4 x}+\frac {3}{x}-\frac {x}{2} \]

[In]

Integrate[(-6 - x^2 + 8*E^(3 - 4*x)*x^2)/(2*x^2),x]

[Out]

-E^(3 - 4*x) + 3/x - x/2

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
risch	\(-\frac {x}{2}+\frac {3}{x}-{\mathrm e}^{3-4 x}\)	\(18\)
parts	\(-\frac {x}{2}+\frac {3}{x}-{\mathrm e}^{3-4 x}\)	\(18\)
derivativedivides	\(\frac {3}{x}+\frac {3}{8}-\frac {x}{2}-{\mathrm e}^{3-4 x}\)	\(19\)
default	\(\frac {3}{x}+\frac {3}{8}-\frac {x}{2}-{\mathrm e}^{3-4 x}\)	\(19\)
parallelrisch	\(-\frac {2 x \,{\mathrm e}^{3-4 x}+x^{2}-6}{2 x}\)	\(20\)
norman	\(\frac {3-\frac {x^{2}}{2}-x \,{\mathrm e}^{3-4 x}}{x}\)	\(21\)

[In]

int(1/2*(8*x^2*exp(3-4*x)-x^2-6)/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x+3/x-exp(3-4*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=-\frac {x^{2} + 2 \, x e^{\left (-4 \, x + 3\right )} - 6}{2 \, x} \]

[In]

integrate(1/2*(8*x^2*exp(3-4*x)-x^2-6)/x^2,x, algorithm="fricas")

[Out]

-1/2*(x^2 + 2*x*e^(-4*x + 3) - 6)/x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=- \frac {x}{2} - e^{3 - 4 x} + \frac {3}{x} \]

[In]

integrate(1/2*(8*x**2*exp(3-4*x)-x**2-6)/x**2,x)

[Out]

-x/2 - exp(3 - 4*x) + 3/x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=-\frac {1}{2} \, x + \frac {3}{x} - e^{\left (-4 \, x + 3\right )} \]

[In]

integrate(1/2*(8*x^2*exp(3-4*x)-x^2-6)/x^2,x, algorithm="maxima")

[Out]

-1/2*x + 3/x - e^(-4*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=-\frac {x^{2} + 2 \, x e^{\left (-4 \, x + 3\right )} - 6}{2 \, x} \]

[In]

integrate(1/2*(8*x^2*exp(3-4*x)-x^2-6)/x^2,x, algorithm="giac")

[Out]

-1/2*(x^2 + 2*x*e^(-4*x + 3) - 6)/x

Mupad [B] (verification not implemented)

Time = 11.80 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-6-x^2+8 e^{3-4 x} x^2}{2 x^2} \, dx=\frac {3}{x}-{\mathrm {e}}^{3-4\,x}-\frac {x}{2} \]

[In]

int(-(x^2/2 - 4*x^2*exp(3 - 4*x) + 3)/x^2,x)

[Out]

3/x - exp(3 - 4*x) - x/2