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

Optimal. Leaf size=21 \[ 4-e^{3-4 x}+\frac {3}{x}-\frac {x}{2} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2194} \begin {gather*} -\frac {x}{2}-e^{3-4 x}+\frac {3}{x} \end {gather*}

Antiderivative was successfully verified.

[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 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} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.95 \begin {gather*} -e^{3-4 x}+\frac {3}{x}-\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 1.12, size = 19, normalized size = 0.90 \begin {gather*} -\frac {x^{2} + 2 \, x e^{\left (-4 \, x + 3\right )} - 6}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.13, size = 19, normalized size = 0.90 \begin {gather*} -\frac {x^{2} + 2 \, x e^{\left (-4 \, x + 3\right )} - 6}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.14, size = 18, normalized size = 0.86

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.38, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{2} \, x + \frac {3}{x} - e^{\left (-4 \, x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.03, size = 17, normalized size = 0.81 \begin {gather*} \frac {3}{x}-{\mathrm {e}}^{3-4\,x}-\frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.10, size = 12, normalized size = 0.57 \begin {gather*} - \frac {x}{2} - e^{3 - 4 x} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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