3.71.10 \(\int \frac {e^{1-e^2-x+4 \log ^2(\frac {x}{e^3})} (-x+8 \log (\frac {x}{e^3}))}{x} \, dx\)

Optimal. Leaf size=22 \[ e^{1-e^2-x+4 \log ^2\left (\frac {x}{e^3}\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.27, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6706} \begin {gather*} e^{-x+4 \log ^2\left (\frac {x}{e^3}\right )-e^2+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(1 - E^2 - x + 4*Log[x/E^3]^2)*(-x + 8*Log[x/E^3]))/x,x]

[Out]

E^(1 - E^2 - x + 4*Log[x/E^3]^2)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{1-e^2-x+4 \log ^2\left (\frac {x}{e^3}\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 22, normalized size = 1.00 \begin {gather*} \frac {e^{37-e^2-x+4 \log ^2(x)}}{x^{24}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 - E^2 - x + 4*Log[x/E^3]^2)*(-x + 8*Log[x/E^3]))/x,x]

[Out]

E^(37 - E^2 - x + 4*Log[x]^2)/x^24

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 19, normalized size = 0.86 \begin {gather*} e^{\left (4 \, \log \left (x e^{\left (-3\right )}\right )^{2} - x - e^{2} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(x/exp(3))-x)*exp(4*log(x/exp(3))^2-exp(2)-x+1)/x,x, algorithm="fricas")

[Out]

e^(4*log(x*e^(-3))^2 - x - e^2 + 1)

________________________________________________________________________________________

giac [A]  time = 0.25, size = 19, normalized size = 0.86 \begin {gather*} e^{\left (4 \, \log \left (x e^{\left (-3\right )}\right )^{2} - x - e^{2} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(x/exp(3))-x)*exp(4*log(x/exp(3))^2-exp(2)-x+1)/x,x, algorithm="giac")

[Out]

e^(4*log(x*e^(-3))^2 - x - e^2 + 1)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 20, normalized size = 0.91

method	result	size
risch	\({\mathrm e}^{4 \ln \left (x \,{\mathrm e}^{-3}\right )^{2}-{\mathrm e}^{2}-x +1}\)	\(20\)
norman	\({\mathrm e}^{4 \ln \left (x \,{\mathrm e}^{-3}\right )^{2}-{\mathrm e}^{2}-x +1}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*ln(x/exp(3))-x)*exp(4*ln(x/exp(3))^2-exp(2)-x+1)/x,x,method=_RETURNVERBOSE)

[Out]

exp(4*ln(x*exp(-3))^2-exp(2)-x+1)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 20, normalized size = 0.91 \begin {gather*} \frac {e^{\left (4 \, \log \relax (x)^{2} - x - e^{2} + 37\right )}}{x^{24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*log(x/exp(3))-x)*exp(4*log(x/exp(3))^2-exp(2)-x+1)/x,x, algorithm="maxima")

[Out]

e^(4*log(x)^2 - x - e^2 + 37)/x^24

________________________________________________________________________________________

mupad [B]  time = 4.15, size = 22, normalized size = 1.00 \begin {gather*} \frac {{\mathrm {e}}^{4\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{-{\mathrm {e}}^2}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{37}}{x^{24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*log(x*exp(-3))^2 - exp(2) - x + 1)*(x - 8*log(x*exp(-3))))/x,x)

[Out]

(exp(4*log(x)^2)*exp(-exp(2))*exp(-x)*exp(37))/x^24

________________________________________________________________________________________

sympy [A]  time = 0.32, size = 17, normalized size = 0.77 \begin {gather*} e^{- x + 4 \log {\left (\frac {x}{e^{3}} \right )}^{2} - e^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*ln(x/exp(3))-x)*exp(4*ln(x/exp(3))**2-exp(2)-x+1)/x,x)

[Out]

exp(-x + 4*log(x*exp(-3))**2 - exp(2) + 1)

________________________________________________________________________________________