\(\int \frac {1}{128} e^{-2 x} (x \log (x)+(x-x^2) \log ^2(x)) \, dx\) [1343]

Optimal result

Integrand size = 26, antiderivative size = 16 \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {1}{256} e^{-2 x} x^2 \log ^2(x) \]

[Out]

1/256*x^2*ln(x)^2/exp(x)^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 6873, 2326} \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {1}{256} e^{-2 x} x^2 \log ^2(x) \]

[In]

Int[(x*Log[x] + (x - x^2)*Log[x]^2)/(128*E^(2*x)),x]

[Out]

(x^2*Log[x]^2)/(256*E^(2*x))

Rule 12

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

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{128} \int e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx \\ & = \frac {1}{128} \int e^{-2 x} x \log (x) (1+\log (x)-x \log (x)) \, dx \\ & = \frac {1}{256} e^{-2 x} x^2 \log ^2(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {1}{256} e^{-2 x} x^2 \log ^2(x) \]

[In]

Integrate[(x*Log[x] + (x - x^2)*Log[x]^2)/(128*E^(2*x)),x]

[Out]

(x^2*Log[x]^2)/(256*E^(2*x))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {x^{2} \ln \left (x \right )^{2} {\mathrm e}^{-2 x}}{256}\)	\(14\)
parallelrisch	\(\frac {x^{2} \ln \left (x \right )^{2} {\mathrm e}^{-2 x}}{256}\)	\(14\)

[In]

int(1/128*((-x^2+x)*ln(x)^2+x*ln(x))/exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

1/256*x^2*ln(x)^2*exp(-2*x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {1}{256} \, x^{2} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} \]

[In]

integrate(1/128*((-x^2+x)*log(x)^2+x*log(x))/exp(x)^2,x, algorithm="fricas")

[Out]

1/256*x^2*e^(-2*x)*log(x)^2

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {x^{2} e^{- 2 x} \log {\left (x \right )}^{2}}{256} \]

[In]

integrate(1/128*((-x**2+x)*ln(x)**2+x*ln(x))/exp(x)**2,x)

[Out]

x**2*exp(-2*x)*log(x)**2/256

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {1}{256} \, x^{2} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} \]

[In]

integrate(1/128*((-x^2+x)*log(x)^2+x*log(x))/exp(x)^2,x, algorithm="maxima")

[Out]

1/256*x^2*e^(-2*x)*log(x)^2

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {1}{256} \, x^{2} e^{\left (-2 \, x\right )} \log \left (x\right )^{2} \]

[In]

integrate(1/128*((-x^2+x)*log(x)^2+x*log(x))/exp(x)^2,x, algorithm="giac")

[Out]

1/256*x^2*e^(-2*x)*log(x)^2

Mupad [B] (verification not implemented)

Time = 9.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1}{128} e^{-2 x} \left (x \log (x)+\left (x-x^2\right ) \log ^2(x)\right ) \, dx=\frac {x^2\,{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^2}{256} \]

[In]

int(exp(-2*x)*((log(x)^2*(x - x^2))/128 + (x*log(x))/128),x)

[Out]

(x^2*exp(-2*x)*log(x)^2)/256