3.22.66 \(\int \frac {e^{2-e^{2 x+2 x^2+4 x \log (\frac {3}{2} x \log (x))+2 \log ^2(\frac {3}{2} x \log (x))}-x} (-x \log (x)+e^{2 x+2 x^2+4 x \log (\frac {3}{2} x \log (x))+2 \log ^2(\frac {3}{2} x \log (x))} (-4 x+(-6 x-4 x^2) \log (x)+(-4+(-4-4 x) \log (x)) \log (\frac {3}{2} x \log (x))))}{x \log (x)} \, dx\)

Optimal. Leaf size=27 \[ e^{2-e^{2 \left (x+\left (x+\log \left (\frac {3}{2} x \log (x)\right )\right )^2\right )}-x} \]

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Rubi [A]  time = 6.34, antiderivative size = 47, normalized size of antiderivative = 1.74, number of steps used = 1, number of rules used = 1, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6706} \begin {gather*} \exp \left (-\left (\frac {3}{2}\right )^{4 x} e^{2 x^2+2 x+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} (x \log (x))^{4 x}-x+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2 - E^(2*x + 2*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log[(3*x*Log[x])/2]^2) - x)*(-(x*Log[x]) + E^(2*x + 2
*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log[(3*x*Log[x])/2]^2)*(-4*x + (-6*x - 4*x^2)*Log[x] + (-4 + (-4 - 4*x)*Log
[x])*Log[(3*x*Log[x])/2])))/(x*Log[x]),x]

[Out]

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

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

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Mathematica [A]  time = 0.41, size = 39, normalized size = 1.44 \begin {gather*} e^{2-e^{2 \left (x+x^2+2 x \log \left (\frac {3}{2} x \log (x)\right )+\log ^2\left (\frac {3}{2} x \log (x)\right )\right )}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2 - E^(2*x + 2*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log[(3*x*Log[x])/2]^2) - x)*(-(x*Log[x]) + E^(2
*x + 2*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log[(3*x*Log[x])/2]^2)*(-4*x + (-6*x - 4*x^2)*Log[x] + (-4 + (-4 - 4*
x)*Log[x])*Log[(3*x*Log[x])/2])))/(x*Log[x]),x]

[Out]

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

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fricas [A]  time = 0.57, size = 37, normalized size = 1.37 \begin {gather*} e^{\left (-x - e^{\left (2 \, x^{2} + 4 \, x \log \left (\frac {3}{2} \, x \log \relax (x)\right ) + 2 \, \log \left (\frac {3}{2} \, x \log \relax (x)\right )^{2} + 2 \, x\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x-4)*log(x)-4)*log(3/2*x*log(x))+(-4*x^2-6*x)*log(x)-4*x)*exp(2*log(3/2*x*log(x))^2+4*x*log(3
/2*x*log(x))+2*x^2+2*x)-x*log(x))*exp(-exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)+2-x)/x/log(x
),x, algorithm="fricas")

[Out]

e^(-x - e^(2*x^2 + 4*x*log(3/2*x*log(x)) + 2*log(3/2*x*log(x))^2 + 2*x) + 2)

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giac [A]  time = 81.81, size = 37, normalized size = 1.37 \begin {gather*} e^{\left (-x - e^{\left (2 \, x^{2} + 4 \, x \log \left (\frac {3}{2} \, x \log \relax (x)\right ) + 2 \, \log \left (\frac {3}{2} \, x \log \relax (x)\right )^{2} + 2 \, x\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x-4)*log(x)-4)*log(3/2*x*log(x))+(-4*x^2-6*x)*log(x)-4*x)*exp(2*log(3/2*x*log(x))^2+4*x*log(3
/2*x*log(x))+2*x^2+2*x)-x*log(x))*exp(-exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)+2-x)/x/log(x
),x, algorithm="giac")

[Out]

e^(-x - e^(2*x^2 + 4*x*log(3/2*x*log(x)) + 2*log(3/2*x*log(x))^2 + 2*x) + 2)

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maple [C]  time = 1.19, size = 631, normalized size = 23.37

method	result	size
risch	\({\mathrm e}^{-\ln \relax (x )^{-4 \ln \relax (2)} \ln \relax (x )^{4 x} 3^{-2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )} 2^{2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )} x^{-2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )} x^{-4 \ln \relax (2)} \ln \relax (x )^{4 \ln \relax (x )} x^{4 \ln \relax (3)} 81^{x} \left (\frac {1}{16}\right )^{x} x^{4 x} 3^{-2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right )} 2^{2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right )} x^{-2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right )} \ln \relax (x )^{-2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right )} \left (\frac {1}{16}\right )^{\ln \relax (3)} \ln \relax (x )^{4 \ln \relax (3)} \ln \relax (x )^{-2 i \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )} 3^{2 i \pi \,\mathrm {csgn}\left (i x \right )} 3^{2 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right )} 2^{-2 i \pi \,\mathrm {csgn}\left (i x \right )} 2^{-2 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right )} x^{2 i \pi \,\mathrm {csgn}\left (i x \right )} x^{2 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right )} \ln \relax (x )^{2 i \pi \,\mathrm {csgn}\left (i x \right )} \ln \relax (x )^{2 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right )} {\mathrm e}^{2 x +2 \ln \relax (3)^{2}+2 \ln \relax (2)^{2}+2 \ln \left (\ln \relax (x )\right )^{2}+2 \ln \relax (x )^{2}+2 x^{2}} {\mathrm e}^{2 i x \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )} {\mathrm e}^{2 i x \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )} {\mathrm e}^{-\frac {\pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}}{2}} {\mathrm e}^{\pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}} {\mathrm e}^{-\frac {\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}}{2}} {\mathrm e}^{\pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}} {\mathrm e}^{-\frac {\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}}{2}} {\mathrm e}^{\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}} {\mathrm e}^{\pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}} {\mathrm e}^{-2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}} {\mathrm e}^{-2 i x \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}} {\mathrm e}^{-\frac {\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{6}}{2}} {\mathrm e}^{-2 i x \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )}+2-x}\)	\(631\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-4*x-4)*ln(x)-4)*ln(3/2*x*ln(x))+(-4*x^2-6*x)*ln(x)-4*x)*exp(2*ln(3/2*x*ln(x))^2+4*x*ln(3/2*x*ln(x))+2
*x^2+2*x)-x*ln(x))*exp(-exp(2*ln(3/2*x*ln(x))^2+4*x*ln(3/2*x*ln(x))+2*x^2+2*x)+2-x)/x/ln(x),x,method=_RETURNVE
RBOSE)

[Out]

exp(-ln(x)^(-4*ln(2))*ln(x)^(4*x)*3^(-2*I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x)))*2^(2*I*Pi*csgn(I*x*ln(x)
)*csgn(I*x)*csgn(I*ln(x)))*x^(-2*I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x)))*x^(-4*ln(2))*ln(x)^(4*ln(x))*x^
(4*ln(3))*81^x*(1/16)^x*x^(4*x)*3^(-2*I*Pi*csgn(I*x*ln(x)))*2^(2*I*Pi*csgn(I*x*ln(x)))*x^(-2*I*Pi*csgn(I*x*ln(
x)))*ln(x)^(-2*I*Pi*csgn(I*x*ln(x)))*(1/16)^ln(3)*ln(x)^(4*ln(3))*ln(x)^(-2*I*Pi*csgn(I*x*ln(x))*csgn(I*x)*csg
n(I*ln(x)))*3^(2*I*Pi*csgn(I*x))*3^(2*I*Pi*csgn(I*ln(x)))*2^(-2*I*Pi*csgn(I*x))*2^(-2*I*Pi*csgn(I*ln(x)))*x^(2
*I*Pi*csgn(I*x))*x^(2*I*Pi*csgn(I*ln(x)))*ln(x)^(2*I*Pi*csgn(I*x))*ln(x)^(2*I*Pi*csgn(I*ln(x)))*exp(2*x+2*ln(3
)^2+2*ln(2)^2+2*ln(ln(x))^2+2*ln(x)^2+2*x^2)*exp(2*I*x*Pi*csgn(I*x*ln(x))^2*csgn(I*ln(x)))*exp(2*I*x*Pi*csgn(I
*x*ln(x))^2*csgn(I*x))*exp(-1/2*Pi^2*csgn(I*ln(x))^2*csgn(I*x*ln(x))^4)*exp(Pi^2*csgn(I*ln(x))*csgn(I*x*ln(x))
^5)*exp(-1/2*Pi^2*csgn(I*x)^2*csgn(I*x*ln(x))^4)*exp(Pi^2*csgn(I*x)*csgn(I*x*ln(x))^5)*exp(-1/2*Pi^2*csgn(I*x)
^2*csgn(I*ln(x))^2*csgn(I*x*ln(x))^2)*exp(Pi^2*csgn(I*x)^2*csgn(I*ln(x))*csgn(I*x*ln(x))^3)*exp(Pi^2*csgn(I*x)
*csgn(I*ln(x))^2*csgn(I*x*ln(x))^3)*exp(-2*Pi^2*csgn(I*x)*csgn(I*ln(x))*csgn(I*x*ln(x))^4)*exp(-2*I*x*Pi*csgn(
I*x*ln(x))^3)*exp(-1/2*Pi^2*csgn(I*x*ln(x))^6)*exp(-2*I*x*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x)))+2-x)

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maxima [B]  time = 1.17, size = 105, normalized size = 3.89 \begin {gather*} e^{\left (-x - \frac {e^{\left (2 \, x^{2} + 4 \, x \log \relax (3) + 2 \, \log \relax (3)^{2} - 4 \, x \log \relax (2) + 2 \, \log \relax (2)^{2} + 4 \, x \log \relax (x) + 4 \, \log \relax (3) \log \relax (x) - 4 \, \log \relax (2) \log \relax (x) + 2 \, \log \relax (x)^{2} + 4 \, x \log \left (\log \relax (x)\right ) + 4 \, \log \relax (3) \log \left (\log \relax (x)\right ) - 4 \, \log \relax (2) \log \left (\log \relax (x)\right ) + 4 \, \log \relax (x) \log \left (\log \relax (x)\right ) + 2 \, \log \left (\log \relax (x)\right )^{2} + 2 \, x\right )}}{2^{4 \, \log \relax (3)}} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x-4)*log(x)-4)*log(3/2*x*log(x))+(-4*x^2-6*x)*log(x)-4*x)*exp(2*log(3/2*x*log(x))^2+4*x*log(3
/2*x*log(x))+2*x^2+2*x)-x*log(x))*exp(-exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)+2-x)/x/log(x
),x, algorithm="maxima")

[Out]

e^(-x - e^(2*x^2 + 4*x*log(3) + 2*log(3)^2 - 4*x*log(2) + 2*log(2)^2 + 4*x*log(x) + 4*log(3)*log(x) - 4*log(2)
*log(x) + 2*log(x)^2 + 4*x*log(log(x)) + 4*log(3)*log(log(x)) - 4*log(2)*log(log(x)) + 4*log(x)*log(log(x)) +
2*log(log(x))^2 + 2*x)/2^(4*log(3)) + 2)

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mupad [B]  time = 1.50, size = 76, normalized size = 2.81 \begin {gather*} {\mathrm {e}}^{-\frac {1}{2^{4\,x+4\,\ln \relax (3)}}\,{81}^x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,{\ln \left (x\,\ln \relax (x)\right )}^2}\,{\mathrm {e}}^{2\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{2\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{2\,x^2}\,{\left (x\,\ln \relax (x)\right )}^{4\,x-4\,\ln \relax (2)+4\,\ln \relax (3)}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2 - exp(2*x + 2*log((3*x*log(x))/2)^2 + 2*x^2 + 4*x*log((3*x*log(x))/2)) - x)*(exp(2*x + 2*log((3*x*
log(x))/2)^2 + 2*x^2 + 4*x*log((3*x*log(x))/2))*(4*x + log((3*x*log(x))/2)*(log(x)*(4*x + 4) + 4) + log(x)*(6*
x + 4*x^2)) + x*log(x)))/(x*log(x)),x)

[Out]

exp(-1/2^(4*x + 4*log(3))*81^x*exp(2*x)*exp(2*log(x*log(x))^2)*exp(2*log(2)^2)*exp(2*log(3)^2)*exp(2*x^2)*(x*l
og(x))^(4*x - 4*log(2) + 4*log(3)))*exp(-x)*exp(2)

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sympy [A]  time = 13.18, size = 41, normalized size = 1.52 \begin {gather*} e^{- x - e^{2 x^{2} + 4 x \log {\left (\frac {3 x \log {\relax (x )}}{2} \right )} + 2 x + 2 \log {\left (\frac {3 x \log {\relax (x )}}{2} \right )}^{2}} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x-4)*ln(x)-4)*ln(3/2*x*ln(x))+(-4*x**2-6*x)*ln(x)-4*x)*exp(2*ln(3/2*x*ln(x))**2+4*x*ln(3/2*x*
ln(x))+2*x**2+2*x)-x*ln(x))*exp(-exp(2*ln(3/2*x*ln(x))**2+4*x*ln(3/2*x*ln(x))+2*x**2+2*x)+2-x)/x/ln(x),x)

[Out]

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

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