3.50.100 \(\int \frac {1-2 \log (x)}{2 e} \, dx\)

Optimal. Leaf size=22 \[ \frac {x+\frac {1}{2} (-2+x+4 \log (2))-x \log (x)}{e} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 2295} \begin {gather*} \frac {3 x}{2 e}-\frac {x \log (x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*Log[x])/(2*E),x]

[Out]

(3*x)/(2*E) - (x*Log[x])/E

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.73 \begin {gather*} \frac {3 x-2 x \log (x)}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*Log[x])/(2*E),x]

[Out]

(3*x - 2*x*Log[x])/(2*E)

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fricas [A]  time = 0.79, size = 13, normalized size = 0.59 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x \log \relax (x) - 3 \, x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*x*log(x) - 3*x)*e^(-1)

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giac [A]  time = 0.21, size = 13, normalized size = 0.59 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x \log \relax (x) - 3 \, x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*x*log(x) - 3*x)*e^(-1)

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maple [A]  time = 0.03, size = 14, normalized size = 0.64

method	result	size
risch	\(\frac {3 \,{\mathrm e}^{-1} x}{2}-x \,{\mathrm e}^{-1} \ln \relax (x )\)	\(14\)
default	\(\frac {{\mathrm e}^{-1} \left (3 x -2 x \ln \relax (x )\right )}{2}\)	\(16\)
norman	\(\frac {3 \,{\mathrm e}^{-1} x}{2}-x \,{\mathrm e}^{-1} \ln \relax (x )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-2*ln(x)+1)/exp(1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.34, size = 13, normalized size = 0.59 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x \log \relax (x) - 3 \, x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*x*log(x) - 3*x)*e^(-1)

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mupad [B]  time = 3.30, size = 11, normalized size = 0.50 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-1}\,\left (2\,\ln \relax (x)-3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-1)*(log(x) - 1/2),x)

[Out]

-(x*exp(-1)*(2*log(x) - 3))/2

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sympy [A]  time = 0.09, size = 15, normalized size = 0.68 \begin {gather*} - \frac {x \log {\relax (x )}}{e} + \frac {3 x}{2 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*exp(-1)*log(x) + 3*x*exp(-1)/2

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