3.73.42 $\int \frac{e^2-3x^2}{e^2{x}^2}dx$

Optimal. Leaf size=14 $$17-(\frac{3}{e^2}+\frac{1}{x^2})x$$

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 2, integrand size = 16, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.125, Rules used = {12, 14} $$\begin{array}{c}-\frac{3x}{e^2}-\frac{1}{x}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^2 - 3*x^2)/(E^2*x^2),x]

[Out]

-x^(-1) - (3*x)/E^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]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int \frac{e^2-3x^2}{x^2}dx}{e^2}\\ & =\frac{\int (-3+\frac{e^2}{x^2})dx}{e^2}\\ & =-\frac{1}{x}-\frac{3x}{e^2}\end{array}\end{array}$$

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.86 $$\begin{array}{c}-\frac{1}{x}-\frac{3x}{e^2}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^2 - 3*x^2)/(E^2*x^2),x]

[Out]

-x^(-1) - (3*x)/E^2

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fricas [A]  time = 0.65, size = 15, normalized size = 1.07 $$\begin{array}{c}-\frac{(3x^2+e^2)e^{(-2)}}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x^2 + e^2)*e^(-2)/x

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giac [A]  time = 0.15, size = 14, normalized size = 1.00 $$\begin{array}{c}-(3x+\frac{e^2}{x})e^{(-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x + e^2/x)*e^(-2)

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maple [A]  time = 0.02, size = 12, normalized size = 0.86

method	result	size
risch	$-3x{\mathrm{e}}^{-2}-\frac{1}{x}$	$12$
norman	$\frac{-1-3x^2{\mathrm{e}}^{-2}}{x}$	$16$
default	${\mathrm{e}}^{-2}(-3x-\frac{{\mathrm{e}}^2}{x})$	$17$
gosper	$-\frac{(3x^2+{\mathrm{e}}^2){\mathrm{e}}^{-2}}{x}$	$18$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 14, normalized size = 1.00 $$\begin{array}{c}-(3x+\frac{e^2}{x})e^{(-2)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x + e^2/x)*e^(-2)

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mupad [B]  time = 0.03, size = 11, normalized size = 0.79 $$\begin{array}{c}-3x{\mathrm{e}}^{-2}-\frac{1}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2)*(exp(2) - 3*x^2))/x^2,x)

[Out]

- 3*x*exp(-2) - 1/x

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sympy [A]  time = 0.08, size = 12, normalized size = 0.86 $$\begin{array}{c}\frac{-3x-\frac{e^2}{x}}{e^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*x - exp(2)/x)*exp(-2)

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