3.19.6 \(\int \frac {e^{-e-2 x+x^2+2 x \log ^2(\frac {x-\log (x)}{\log (x)})} ((2 x-2 x^2) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log (\frac {x-\log (x)}{\log (x)})+(-2 x \log (x)+2 \log ^2(x)) \log ^2(\frac {x-\log (x)}{\log (x)}))}{-x \log (x)+\log ^2(x)} \, dx\) [1806]

Optimal. Leaf size=26 \[ e^{-e-2 x+x^2+2 x \log ^2\left (-1+\frac {x}{\log (x)}\right )} \]

[Out]

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

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Rubi [F]
time = 2.05, 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 \frac {e^{-e-2 x+x^2+2 x \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )} \left (\left (2 x-2 x^2\right ) \log (x)+(-2+2 x) \log ^2(x)+(4 x-4 x \log (x)) \log \left (\frac {x-\log (x)}{\log (x)}\right )+\left (-2 x \log (x)+2 \log ^2(x)\right ) \log ^2\left (\frac {x-\log (x)}{\log (x)}\right )\right )}{-x \log (x)+\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*Defer[Int][E^(-E - 2*x + x^2 + 2*x*Log[(x - Log[x])/Log[x]]^2), x] + 2*Defer[Int][E^(-E - 2*x + x^2 + 2*x*L
og[(x - Log[x])/Log[x]]^2)*x, x] + 4*Defer[Int][(E^(-E - 2*x + x^2 + 2*x*Log[(x - Log[x])/Log[x]]^2)*x*Log[-1
+ x/Log[x]])/(x - Log[x]), x] - 4*Defer[Int][(E^(-E - 2*x + x^2 + 2*x*Log[(x - Log[x])/Log[x]]^2)*x*Log[-1 + x
/Log[x]])/((x - Log[x])*Log[x]), x] + 2*Defer[Int][E^(-E - 2*x + x^2 + 2*x*Log[(x - Log[x])/Log[x]]^2)*Log[-1
+ x/Log[x]]^2, x]

Rubi steps

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

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Mathematica [A]
time = 0.13, size = 26, normalized size = 1.00 \begin {gather*} e^{-e-2 x+x^2+2 x \log ^2\left (-1+\frac {x}{\log (x)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 571, normalized size = 21.96

method	result	size
risch	\(\left (x -\ln \left (x \right )\right )^{-2 i x \pi \,\mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )} \left (x -\ln \left (x \right )\right )^{-2 i x \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )} \ln \left (x \right )^{-2 i x \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right )} \ln \left (x \right )^{-2 i x \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )} \ln \left (x \right )^{-4 x \ln \left (x -\ln \left (x \right )\right )} \left (x -\ln \left (x \right )\right )^{2 i x \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )} \left (x -\ln \left (x \right )\right )^{2 i x \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right )} \ln \left (x \right )^{2 i x \pi \,\mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )} \ln \left (x \right )^{2 i x \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )} {\mathrm e}^{-\frac {x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{6}}{2}-x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{5} \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right )+x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{5} \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )-\frac {x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{4} \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right )^{2}}{2}-\frac {x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{4} \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )^{2}}{2}+2 x \ln \left (\ln \left (x \right )\right )^{2}+2 x \ln \left (x -\ln \left (x \right )\right )^{2}+2 x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{4} \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )+x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{3} \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right )^{2} \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )-x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{3} \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right )-\frac {x \,\pi ^{2} \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )-x \right )}{\ln \left (x \right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right )^{2} \mathrm {csgn}\left (i \left (\ln \left (x \right )-x \right )\right )^{2}}{2}-{\mathrm e}+x^{2}-2 x}\)	\(571\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.43, size = 45, normalized size = 1.73 \begin {gather*} e^{\left (2 \, x \log \left (x - \log \left (x\right )\right )^{2} - 4 \, x \log \left (x - \log \left (x\right )\right ) \log \left (\log \left (x\right )\right ) + 2 \, x \log \left (\log \left (x\right )\right )^{2} + x^{2} - 2 \, x - e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4*x)*log((x-log(x))/log(x))+(-2+2*x)*
log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*log((x-log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x, algorith
m="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 29, normalized size = 1.12 \begin {gather*} e^{\left (2 \, x \log \left (\frac {x - \log \left (x\right )}{\log \left (x\right )}\right )^{2} + x^{2} - 2 \, x - e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4*x)*log((x-log(x))/log(x))+(-2+2*x)*
log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*log((x-log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x, algorith
m="fricas")

[Out]

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

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Sympy [A]
time = 0.84, size = 26, normalized size = 1.00 \begin {gather*} e^{x^{2} + 2 x \log {\left (\frac {x - \log {\left (x \right )}}{\log {\left (x \right )}} \right )}^{2} - 2 x - e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x**2 + 2*x*log((x - log(x))/log(x))**2 - 2*x - E)

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Giac [A]
time = 3.99, size = 26, normalized size = 1.00 \begin {gather*} e^{\left (2 \, x \log \left (\frac {x}{\log \left (x\right )} - 1\right )^{2} + x^{2} - 2 \, x - e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(x)^2-2*x*log(x))*log((x-log(x))/log(x))^2+(-4*x*log(x)+4*x)*log((x-log(x))/log(x))+(-2+2*x)*
log(x)^2+(-2*x^2+2*x)*log(x))*exp(2*x*log((x-log(x))/log(x))^2-exp(1)+x^2-2*x)/(log(x)^2-x*log(x)),x, algorith
m="giac")

[Out]

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

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Mupad [B]
time = 1.45, size = 32, normalized size = 1.23 \begin {gather*} {\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{2\,x\,{\ln \left (\frac {x-\ln \left (x\right )}{\ln \left (x\right )}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x*log((x - log(x))/log(x))^2 - exp(1) - 2*x + x^2)*(log((x - log(x))/log(x))*(4*x - 4*x*log(x)) + l
og((x - log(x))/log(x))^2*(2*log(x)^2 - 2*x*log(x)) + log(x)*(2*x - 2*x^2) + log(x)^2*(2*x - 2)))/(log(x)^2 -
x*log(x)),x)

[Out]

exp(-exp(1))*exp(-2*x)*exp(x^2)*exp(2*x*log((x - log(x))/log(x))^2)

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