3.59.1 \(\int e^{-e^{3-x+2 x \log (x)}} (18+e^{3-x+2 x \log (x)} (18-18 x+(36-36 x) \log (x))) \, dx\)

Optimal. Leaf size=21 \[ 18 e^{-e^{3-x+2 x \log (x)}} (-1+x) \]

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Rubi [A]  time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.90, number of steps used = 1, number of rules used = 1, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2288} \begin {gather*} -\frac {18 e^{-e^{3-x} x^{2 x}} (-x+2 (1-x) \log (x)+1)}{2 \log (x)+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(18 + E^(3 - x + 2*x*Log[x])*(18 - 18*x + (36 - 36*x)*Log[x]))/E^E^(3 - x + 2*x*Log[x]),x]

[Out]

(-18*(1 - x + 2*(1 - x)*Log[x]))/(E^(E^(3 - x)*x^(2*x))*(1 + 2*Log[x]))

Rule 2288

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {18 e^{-e^{3-x} x^{2 x}} (1-x+2 (1-x) \log (x))}{1+2 \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 21, normalized size = 1.00 \begin {gather*} 18 e^{-e^{3-x} x^{2 x}} (-1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18 + E^(3 - x + 2*x*Log[x])*(18 - 18*x + (36 - 36*x)*Log[x]))/E^E^(3 - x + 2*x*Log[x]),x]

[Out]

(18*(-1 + x))/E^(E^(3 - x)*x^(2*x))

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fricas [A]  time = 0.66, size = 19, normalized size = 0.90 \begin {gather*} 18 \, {\left (x - 1\right )} e^{\left (-e^{\left (2 \, x \log \relax (x) - x + 3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-36*x+36)*log(x)-18*x+18)*exp(2*x*log(x)+3-x)+18)/exp(exp(2*x*log(x)+3-x)),x, algorithm="fricas")

[Out]

18*(x - 1)*e^(-e^(2*x*log(x) - x + 3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -18 \, {\left ({\left (2 \, {\left (x - 1\right )} \log \relax (x) + x - 1\right )} e^{\left (2 \, x \log \relax (x) - x + 3\right )} - 1\right )} e^{\left (-e^{\left (2 \, x \log \relax (x) - x + 3\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-36*x+36)*log(x)-18*x+18)*exp(2*x*log(x)+3-x)+18)/exp(exp(2*x*log(x)+3-x)),x, algorithm="giac")

[Out]

integrate(-18*((2*(x - 1)*log(x) + x - 1)*e^(2*x*log(x) - x + 3) - 1)*e^(-e^(2*x*log(x) - x + 3)), x)

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maple [A]  time = 0.08, size = 21, normalized size = 1.00

method	result	size
norman	\(\left (18 x -18\right ) {\mathrm e}^{-{\mathrm e}^{2 x \ln \relax (x )+3-x}}\)	\(21\)
risch	\(\left (18 x -18\right ) {\mathrm e}^{-x^{2 x} {\mathrm e}^{3-x}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-36*x+36)*ln(x)-18*x+18)*exp(2*x*ln(x)+3-x)+18)/exp(exp(2*x*ln(x)+3-x)),x,method=_RETURNVERBOSE)

[Out]

(18*x-18)/exp(exp(2*x*ln(x)+3-x))

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maxima [A]  time = 0.41, size = 19, normalized size = 0.90 \begin {gather*} 18 \, {\left (x - 1\right )} e^{\left (-e^{\left (2 \, x \log \relax (x) - x + 3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-36*x+36)*log(x)-18*x+18)*exp(2*x*log(x)+3-x)+18)/exp(exp(2*x*log(x)+3-x)),x, algorithm="maxima")

[Out]

18*(x - 1)*e^(-e^(2*x*log(x) - x + 3))

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mupad [B]  time = 4.26, size = 19, normalized size = 0.90 \begin {gather*} 18\,{\mathrm {e}}^{-x^{2\,x}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3}\,\left (x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-exp(2*x*log(x) - x + 3))*(exp(2*x*log(x) - x + 3)*(18*x + log(x)*(36*x - 36) - 18) - 18),x)

[Out]

18*exp(-x^(2*x)*exp(-x)*exp(3))*(x - 1)

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sympy [A]  time = 1.65, size = 17, normalized size = 0.81 \begin {gather*} \left (18 x - 18\right ) e^{- e^{2 x \log {\relax (x )} - x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-36*x+36)*ln(x)-18*x+18)*exp(2*x*ln(x)+3-x)+18)/exp(exp(2*x*ln(x)+3-x)),x)

[Out]

(18*x - 18)*exp(-exp(2*x*log(x) - x + 3))

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