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

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

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Rubi [F]  time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int x^{-3+x} \left (e^2 (4-2 x)-2 e^2 x \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^(-3 + x)*(E^2*(4 - 2*x) - 2*E^2*x*Log[x]),x]

[Out]

4*E^2*Defer[Int][x^(-3 + x), x] - 2*E^2*Defer[Int][x^(-2 + x), x] - 2*E^2*Log[x]*Defer[Int][x^(-2 + x), x] + 2
*E^2*Defer[Int][Defer[Int][x^(-2 + x), x]/x, x]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 10, normalized size = 0.71 \begin {gather*} -2 e^2 x^{-2+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-3 + x)*(E^2*(4 - 2*x) - 2*E^2*x*Log[x]),x]

[Out]

-2*E^2*x^(-2 + x)

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fricas [A]  time = 0.65, size = 9, normalized size = 0.64 \begin {gather*} -2 \, x^{x - 2} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x^(x - 2)*e^2

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giac [A]  time = 0.12, size = 9, normalized size = 0.64 \begin {gather*} -2 \, x^{x - 2} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x^(x - 2)*e^2

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maple [A]  time = 0.04, size = 10, normalized size = 0.71

method	result	size
risch	\(-2 \,{\mathrm e}^{2} x^{x -2}\)	\(10\)
norman	\(-2 \,{\mathrm e}^{2} {\mathrm e}^{\left (x -2\right ) \ln \relax (x )}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*exp(2)*x^(x-2)

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maxima [A]  time = 0.39, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2 \, e^{\left (x \log \relax (x) + 2\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*e^(x*log(x) + 2)/x^2

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mupad [B]  time = 3.05, size = 9, normalized size = 0.64 \begin {gather*} -2\,x^{x-2}\,{\mathrm {e}}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(x)*(x - 2))*(exp(2)*(2*x - 4) + 2*x*exp(2)*log(x)))/x,x)

[Out]

-2*x^(x - 2)*exp(2)

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sympy [A]  time = 0.28, size = 14, normalized size = 1.00 \begin {gather*} - 2 e^{2} e^{\left (x - 2\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*exp(2)*exp((x - 2)*log(x))

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