3.68.27 \(\int \frac {-1+e^{e^2} (1+e^{2 x} (-256-2 x))-x+e^{2 x} (256+257 x+2 x^2)+(-256+256 e^{e^2}-255 x) \log (x)+(-e^{e^2+2 x}+e^{2 x} (1+x)+(-1+e^{e^2}-x) \log (x)) \log (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x})}{-256 e^{e^2+2 x} x+e^{2 x} (256 x+256 x^2)+(-256 x+256 e^{e^2} x-256 x^2) \log (x)+(-e^{e^2+2 x} x+e^{2 x} (x+x^2)+(-x+e^{e^2} x-x^2) \log (x)) \log (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x})} \, dx\)

Optimal. Leaf size=29 \[ \log \left (x \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )\right ) \]

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Rubi [F]  time = 7.14, 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 {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{-256 e^{e^2+2 x} x+e^{2 x} \left (256 x+256 x^2\right )+\left (-256 x+256 e^{e^2} x-256 x^2\right ) \log (x)+\left (-e^{e^2+2 x} x+e^{2 x} \left (x+x^2\right )+\left (-x+e^{e^2} x-x^2\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + E^E^2*(1 + E^(2*x)*(-256 - 2*x)) - x + E^(2*x)*(256 + 257*x + 2*x^2) + (-256 + 256*E^E^2 - 255*x)*Lo
g[x] + (-E^(E^2 + 2*x) + E^(2*x)*(1 + x) + (-1 + E^E^2 - x)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2 - x)])
/(-256*E^(E^2 + 2*x)*x + E^(2*x)*(256*x + 256*x^2) + (-256*x + 256*E^E^2*x - 256*x^2)*Log[x] + (-(E^(E^2 + 2*x
)*x) + E^(2*x)*(x + x^2) + (-x + E^E^2*x - x^2)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2 - x)]),x]

[Out]

Log[x] + 2*Defer[Int][(256 + Log[(E^(2*x) - Log[x])/(1 - E^E^2 + x)])^(-1), x] + Defer[Int][1/((-1 + E^E^2 - x
)*(256 + Log[(E^(2*x) - Log[x])/(1 - E^E^2 + x)])), x] - Defer[Int][1/(x*(E^(2*x) - Log[x])*(256 + Log[(E^(2*x
) - Log[x])/(1 - E^E^2 + x)])), x] + 2*Defer[Int][Log[x]/((E^(2*x) - Log[x])*(256 + Log[(E^(2*x) - Log[x])/(1
- E^E^2 + x)])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+e^{e^2} \left (1+e^{2 x} (-256-2 x)\right )-x+e^{2 x} \left (256+257 x+2 x^2\right )+\left (-256+256 e^{e^2}-255 x\right ) \log (x)+\left (-e^{e^2+2 x}+e^{2 x} (1+x)+\left (-1+e^{e^2}-x\right ) \log (x)\right ) \log \left (\frac {-e^{2 x}+\log (x)}{-1+e^{e^2}-x}\right )}{x \left (1-e^{e^2}+x\right ) \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ &=\int \left (\frac {-1+2 x \log (x)}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}+\frac {256 \left (1-e^{e^2}\right )+257 \left (1-\frac {2 e^{e^2}}{257}\right ) x+2 x^2+\left (1-e^{e^2}\right ) \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )+x \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}{x \left (1-e^{e^2}+x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=\int \frac {-1+2 x \log (x)}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \frac {256 \left (1-e^{e^2}\right )+257 \left (1-\frac {2 e^{e^2}}{257}\right ) x+2 x^2+\left (1-e^{e^2}\right ) \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )+x \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}{x \left (1-e^{e^2}+x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ &=\int \frac {256+257 x+2 x^2-2 e^{e^2} (128+x)+\left (1-e^{e^2}+x\right ) \log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}{x \left (1-e^{e^2}+x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \left (-\frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}+\frac {2 \log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \left (\frac {1}{x}+\frac {-1+2 e^{e^2}-2 x}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=\log (x)+2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \frac {-1+2 e^{e^2}-2 x}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ &=\log (x)+2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \left (\frac {2}{256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )}+\frac {1}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )}\right ) \, dx\\ &=\log (x)+2 \int \frac {1}{256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )} \, dx+2 \int \frac {\log (x)}{\left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx+\int \frac {1}{\left (-1+e^{e^2}-x\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx-\int \frac {1}{x \left (e^{2 x}-\log (x)\right ) \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 30, normalized size = 1.03 \begin {gather*} \log (x)+\log \left (256+\log \left (\frac {e^{2 x}-\log (x)}{1-e^{e^2}+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^E^2*(1 + E^(2*x)*(-256 - 2*x)) - x + E^(2*x)*(256 + 257*x + 2*x^2) + (-256 + 256*E^E^2 - 255
*x)*Log[x] + (-E^(E^2 + 2*x) + E^(2*x)*(1 + x) + (-1 + E^E^2 - x)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2
- x)])/(-256*E^(E^2 + 2*x)*x + E^(2*x)*(256*x + 256*x^2) + (-256*x + 256*E^E^2*x - 256*x^2)*Log[x] + (-(E^(E^2
+ 2*x)*x) + E^(2*x)*(x + x^2) + (-x + E^E^2*x - x^2)*Log[x])*Log[(-E^(2*x) + Log[x])/(-1 + E^E^2 - x)]),x]

[Out]

Log[x] + Log[256 + Log[(E^(2*x) - Log[x])/(1 - E^E^2 + x)]]

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fricas [A]  time = 0.76, size = 42, normalized size = 1.45 \begin {gather*} \log \relax (x) + \log \left (\log \left (-\frac {e^{\left (e^{2}\right )} \log \relax (x) - e^{\left (2 \, x + e^{2}\right )}}{{\left (x + 1\right )} e^{\left (e^{2}\right )} - e^{\left (2 \, e^{2}\right )}}\right ) + 256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(exp(2))-x-1)*log(x)-exp(2*x)*exp(exp(2))+(x+1)*exp(2*x))*log((log(x)-exp(2*x))/(exp(exp(2))-x
-1))+(256*exp(exp(2))-255*x-256)*log(x)+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/((
(x*exp(exp(2))-x^2-x)*log(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*log((log(x)-exp(2*x))/(exp(exp(2))-x-1))
+(256*x*exp(exp(2))-256*x^2-256*x)*log(x)-256*x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x, algorithm="f
ricas")

[Out]

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

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giac [A]  time = 2.81, size = 27, normalized size = 0.93 \begin {gather*} \log \relax (x) + \log \left (\log \left (x - e^{\left (e^{2}\right )} + 1\right ) - \log \left (e^{\left (2 \, x\right )} - \log \relax (x)\right ) - 256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(exp(2))-x-1)*log(x)-exp(2*x)*exp(exp(2))+(x+1)*exp(2*x))*log((log(x)-exp(2*x))/(exp(exp(2))-x
-1))+(256*exp(exp(2))-255*x-256)*log(x)+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/((
(x*exp(exp(2))-x^2-x)*log(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*log((log(x)-exp(2*x))/(exp(exp(2))-x-1))
+(256*x*exp(exp(2))-256*x^2-256*x)*log(x)-256*x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x, algorithm="g
iac")

[Out]

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

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maple [C]  time = 0.20, size = 228, normalized size = 7.86

method	result	size
risch	\(\ln \relax (x )+\ln \left (\ln \left ({\mathrm e}^{{\mathrm e}^{2}}-x -1\right )-\frac {i \left (-2 \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )-{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (\frac {i}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )-\pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )-{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-{\mathrm e}^{2 x}\right )}{1-{\mathrm e}^{{\mathrm e}^{2}}+x}\right )^{3}+2 \pi -2 i \ln \left ({\mathrm e}^{2 x}-\ln \relax (x )\right )-512 i\right )}{2}\right )\)	\(228\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((exp(exp(2))-x-1)*ln(x)-exp(2*x)*exp(exp(2))+(x+1)*exp(2*x))*ln((ln(x)-exp(2*x))/(exp(exp(2))-x-1))+(256
*exp(exp(2))-255*x-256)*ln(x)+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/(((x*exp(exp
(2))-x^2-x)*ln(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*ln((ln(x)-exp(2*x))/(exp(exp(2))-x-1))+(256*x*exp(e
xp(2))-256*x^2-256*x)*ln(x)-256*x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(ln(exp(exp(2))-x-1)-1/2*I*(-2*Pi*csgn(I*(ln(x)-exp(2*x))/(1-exp(exp(2))+x))^2-Pi*csgn(I*(ln(x)-exp(2*
x)))*csgn(I/(1-exp(exp(2))+x))*csgn(I*(ln(x)-exp(2*x))/(1-exp(exp(2))+x))-Pi*csgn(I*(ln(x)-exp(2*x)))*csgn(I*(
ln(x)-exp(2*x))/(1-exp(exp(2))+x))^2-Pi*csgn(I/(1-exp(exp(2))+x))*csgn(I*(ln(x)-exp(2*x))/(1-exp(exp(2))+x))^2
+Pi*csgn(I*(ln(x)-exp(2*x))/(1-exp(exp(2))+x))^3+2*Pi-2*I*ln(exp(2*x)-ln(x))-512*I))

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maxima [A]  time = 0.53, size = 27, normalized size = 0.93 \begin {gather*} \log \relax (x) + \log \left (-\log \left (-x + e^{\left (e^{2}\right )} - 1\right ) + \log \left (-e^{\left (2 \, x\right )} + \log \relax (x)\right ) + 256\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(exp(2))-x-1)*log(x)-exp(2*x)*exp(exp(2))+(x+1)*exp(2*x))*log((log(x)-exp(2*x))/(exp(exp(2))-x
-1))+(256*exp(exp(2))-255*x-256)*log(x)+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x^2+257*x+256)*exp(2*x)-x-1)/((
(x*exp(exp(2))-x^2-x)*log(x)-x*exp(2*x)*exp(exp(2))+(x^2+x)*exp(2*x))*log((log(x)-exp(2*x))/(exp(exp(2))-x-1))
+(256*x*exp(exp(2))-256*x^2-256*x)*log(x)-256*x*exp(2*x)*exp(exp(2))+(256*x^2+256*x)*exp(2*x)),x, algorithm="m
axima")

[Out]

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

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mupad [B]  time = 4.70, size = 27, normalized size = 0.93 \begin {gather*} \ln \left (\ln \left (\frac {{\mathrm {e}}^{2\,x}-\ln \relax (x)}{x-{\mathrm {e}}^{{\mathrm {e}}^2}+1}\right )+256\right )+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - exp(2*x)*(257*x + 2*x^2 + 256) + exp(exp(2))*(exp(2*x)*(2*x + 256) - 1) + log(x)*(255*x - 256*exp(exp
(2)) + 256) + log((exp(2*x) - log(x))/(x - exp(exp(2)) + 1))*(log(x)*(x - exp(exp(2)) + 1) + exp(2*x)*exp(exp(
2)) - exp(2*x)*(x + 1)) + 1)/(log(x)*(256*x - 256*x*exp(exp(2)) + 256*x^2) - exp(2*x)*(256*x + 256*x^2) + log(
(exp(2*x) - log(x))/(x - exp(exp(2)) + 1))*(log(x)*(x - x*exp(exp(2)) + x^2) - exp(2*x)*(x + x^2) + x*exp(2*x)
*exp(exp(2))) + 256*x*exp(2*x)*exp(exp(2))),x)

[Out]

log(log((exp(2*x) - log(x))/(x - exp(exp(2)) + 1)) + 256) + log(x)

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sympy [A]  time = 4.53, size = 24, normalized size = 0.83 \begin {gather*} \log {\relax (x )} + \log {\left (\log {\left (\frac {- e^{2 x} + \log {\relax (x )}}{- x - 1 + e^{e^{2}}} \right )} + 256 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(exp(2))-x-1)*ln(x)-exp(2*x)*exp(exp(2))+(x+1)*exp(2*x))*ln((ln(x)-exp(2*x))/(exp(exp(2))-x-1)
)+(256*exp(exp(2))-255*x-256)*ln(x)+((-2*x-256)*exp(2*x)+1)*exp(exp(2))+(2*x**2+257*x+256)*exp(2*x)-x-1)/(((x*
exp(exp(2))-x**2-x)*ln(x)-x*exp(2*x)*exp(exp(2))+(x**2+x)*exp(2*x))*ln((ln(x)-exp(2*x))/(exp(exp(2))-x-1))+(25
6*x*exp(exp(2))-256*x**2-256*x)*ln(x)-256*x*exp(2*x)*exp(exp(2))+(256*x**2+256*x)*exp(2*x)),x)

[Out]

log(x) + log(log((-exp(2*x) + log(x))/(-x - 1 + exp(exp(2)))) + 256)

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