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

Optimal. Leaf size=19 \[ 10+\frac {e^4}{x}+3 \log \left (\frac {e^x}{5}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.58, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} 3 x+\frac {e^4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^4 + 3*x^2)/x^2,x]

[Out]

E^4/x + 3*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} &=\int \left (3-\frac {e^4}{x^2}\right ) \, dx\\ &=\frac {e^4}{x}+3 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.58 \begin {gather*} \frac {e^4}{x}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^4 + 3*x^2)/x^2,x]

[Out]

E^4/x + 3*x

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fricas [A]  time = 0.52, size = 12, normalized size = 0.63 \begin {gather*} \frac {3 \, x^{2} + e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x^2 + e^4)/x

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giac [A]  time = 0.13, size = 10, normalized size = 0.53 \begin {gather*} 3 \, x + \frac {e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x + e^4/x

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maple [A]  time = 0.02, size = 11, normalized size = 0.58

method	result	size
default	\(3 x +\frac {{\mathrm e}^{4}}{x}\)	\(11\)
risch	\(3 x +\frac {{\mathrm e}^{4}}{x}\)	\(11\)
gosper	\(\frac {3 x^{2}+{\mathrm e}^{4}}{x}\)	\(13\)
norman	\(\frac {3 x^{2}+{\mathrm e}^{4}}{x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x+exp(4)/x

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maxima [A]  time = 0.37, size = 10, normalized size = 0.53 \begin {gather*} 3 \, x + \frac {e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x + e^4/x

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mupad [B]  time = 0.03, size = 10, normalized size = 0.53 \begin {gather*} 3\,x+\frac {{\mathrm {e}}^4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4) - 3*x^2)/x^2,x)

[Out]

3*x + exp(4)/x

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sympy [A]  time = 0.07, size = 7, normalized size = 0.37 \begin {gather*} 3 x + \frac {e^{4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x + exp(4)/x

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