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

Optimal. Leaf size=19 \[ 9-e^x+\log (3)-\log \left (\frac {x^2}{e^6}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.53, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {14, 2194} \begin {gather*} -e^x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 - E^x*x)/x,x]

[Out]

-E^x - 2*Log[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} &=\int \left (-e^x-\frac {2}{x}\right ) \, dx\\ &=-2 \log (x)-\int e^x \, dx\\ &=-e^x-2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 0.53 \begin {gather*} -e^x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - E^x*x)/x,x]

[Out]

-E^x - 2*Log[x]

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fricas [A]  time = 0.50, size = 9, normalized size = 0.47 \begin {gather*} -e^{x} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^x - 2*log(x)

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giac [A]  time = 0.15, size = 9, normalized size = 0.47 \begin {gather*} -e^{x} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^x - 2*log(x)

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maple [A]  time = 0.02, size = 10, normalized size = 0.53

method	result	size
default	\(-2 \ln \relax (x )-{\mathrm e}^{x}\)	\(10\)
norman	\(-2 \ln \relax (x )-{\mathrm e}^{x}\)	\(10\)
risch	\(-2 \ln \relax (x )-{\mathrm e}^{x}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(x)-exp(x)

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maxima [A]  time = 0.37, size = 9, normalized size = 0.47 \begin {gather*} -e^{x} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^x - 2*log(x)

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mupad [B]  time = 5.05, size = 9, normalized size = 0.47 \begin {gather*} -{\mathrm {e}}^x-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*exp(x) + 2)/x,x)

[Out]

- exp(x) - 2*log(x)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.42 \begin {gather*} - e^{x} - 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x) - 2*log(x)

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