3.24.28 \(\int \frac {-4 e^{22}+x^2}{2 e^{22} x^2} \, dx\)

Optimal. Leaf size=18 \[ -2+\frac {2}{x}+\frac {1}{2} \left (1+\frac {x}{e^{22}}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 14} \begin {gather*} \frac {x}{2 e^{22}}+\frac {2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*E^22 + x^2)/(2*E^22*x^2),x]

[Out]

2/x + x/(2*E^22)

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 {-4 e^{22}+x^2}{x^2} \, dx}{2 e^{22}}\\ &=\frac {\int \left (1-\frac {4 e^{22}}{x^2}\right ) \, dx}{2 e^{22}}\\ &=\frac {2}{x}+\frac {x}{2 e^{22}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{x}+\frac {x}{2 e^{22}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*E^22 + x^2)/(2*E^22*x^2),x]

[Out]

2/x + x/(2*E^22)

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fricas [A]  time = 0.73, size = 15, normalized size = 0.83 \begin {gather*} \frac {{\left (x^{2} + 4 \, e^{22}\right )} e^{\left (-22\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*exp(22)+x^2)/x^2/exp(22),x, algorithm="fricas")

[Out]

1/2*(x^2 + 4*e^22)*e^(-22)/x

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giac [A]  time = 0.20, size = 13, normalized size = 0.72 \begin {gather*} \frac {1}{2} \, {\left (x + \frac {4 \, e^{22}}{x}\right )} e^{\left (-22\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*exp(22)+x^2)/x^2/exp(22),x, algorithm="giac")

[Out]

1/2*(x + 4*e^22/x)*e^(-22)

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maple [A]  time = 0.06, size = 12, normalized size = 0.67




method result size



risch \(\frac {x \,{\mathrm e}^{-22}}{2}+\frac {2}{x}\) \(12\)
default \(\frac {{\mathrm e}^{-22} \left (x +\frac {4 \,{\mathrm e}^{22}}{x}\right )}{2}\) \(16\)
norman \(\frac {2+\frac {{\mathrm e}^{-22} x^{2}}{2}}{x}\) \(16\)
gosper \(\frac {\left (x^{2}+4 \,{\mathrm e}^{22}\right ) {\mathrm e}^{-22}}{2 x}\) \(18\)



Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-4*exp(22)+x^2)/x^2/exp(22),x,method=_RETURNVERBOSE)

[Out]

1/2*x*exp(-22)+2/x

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maxima [A]  time = 0.43, size = 13, normalized size = 0.72 \begin {gather*} \frac {1}{2} \, {\left (x + \frac {4 \, e^{22}}{x}\right )} e^{\left (-22\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*exp(22)+x^2)/x^2/exp(22),x, algorithm="maxima")

[Out]

1/2*(x + 4*e^22/x)*e^(-22)

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mupad [B]  time = 1.40, size = 11, normalized size = 0.61 \begin {gather*} \frac {x\,{\mathrm {e}}^{-22}}{2}+\frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-22)*(2*exp(22) - x^2/2))/x^2,x)

[Out]

(x*exp(-22))/2 + 2/x

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sympy [A]  time = 0.08, size = 12, normalized size = 0.67 \begin {gather*} \frac {x + \frac {4 e^{22}}{x}}{2 e^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-4*exp(22)+x**2)/x**2/exp(22),x)

[Out]

(x + 4*exp(22)/x)*exp(-22)/2

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