3.73.42 \(\int \frac {e^2-3 x^2}{e^2 x^2} \, dx\)

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

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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 {gather*} -\frac {3 x}{e^2}-\frac {1}{x} \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^2-3 x^2}{x^2} \, dx}{e^2}\\ &=\frac {\int \left (-3+\frac {e^2}{x^2}\right ) \, dx}{e^2}\\ &=-\frac {1}{x}-\frac {3 x}{e^2}\\ \end {aligned} \end {gather*}

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

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 {gather*} -\frac {{\left (3 \, x^{2} + e^{2}\right )} e^{\left (-2\right )}}{x} \end {gather*}

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 {gather*} -{\left (3 \, x + \frac {e^{2}}{x}\right )} e^{\left (-2\right )} \end {gather*}

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	\(-3 x \,{\mathrm e}^{-2}-\frac {1}{x}\)	\(12\)
norman	\(\frac {-1-3 x^{2} {\mathrm e}^{-2}}{x}\)	\(16\)
default	\({\mathrm e}^{-2} \left (-3 x -\frac {{\mathrm e}^{2}}{x}\right )\)	\(17\)
gosper	\(-\frac {\left (3 x^{2}+{\mathrm e}^{2}\right ) {\mathrm e}^{-2}}{x}\)	\(18\)

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 {gather*} -{\left (3 \, x + \frac {e^{2}}{x}\right )} e^{\left (-2\right )} \end {gather*}

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 {gather*} -3\,x\,{\mathrm {e}}^{-2}-\frac {1}{x} \end {gather*}

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 {gather*} \frac {- 3 x - \frac {e^{2}}{x}}{e^{2}} \end {gather*}

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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