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

Optimal. Leaf size=25 \[ \left (-4^{4 e^{-x}}+\log (-1+x)\right ) \log \left (\frac {2 x}{\log (x)}\right ) \]

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Rubi [F]  time = 5.97, 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^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{\left (-x+x^2\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (e^x (1-x) \log (-1+x)+e^x (-1+x) \log (-1+x) \log (x)+4^{4 e^{-x}} \left (e^x (-1+x)+e^x (1-x) \log (x)\right )+\left (e^x x \log (x)+4^{4 e^{-x}} \left (-4 x+4 x^2\right ) \log (4) \log (x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right )}{(-1+x) x \log (x)} \, dx\\ &=\int \frac {-256^{e^{-x}}+\frac {256^{e^{-x}}}{\log (x)}+\frac {\log (-1+x) (-1+\log (x))}{\log (x)}+\frac {e^{-x} x \left (e^x+4^{1+4 e^{-x}} (-1+x) \log (4)\right ) \log \left (\frac {2 x}{\log (x)}\right )}{-1+x}}{x} \, dx\\ &=\int \left (4^{e^{-x} \left (4+e^x\right )} e^{-x} \log (4) \log \left (\frac {2 x}{\log (x)}\right )-\frac {256^{e^{-x}}-256^{e^{-x}} x-\log (-1+x)+x \log (-1+x)-256^{e^{-x}} \log (x)+256^{e^{-x}} x \log (x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)}\right ) \, dx\\ &=\log (4) \int 4^{e^{-x} \left (4+e^x\right )} e^{-x} \log \left (\frac {2 x}{\log (x)}\right ) \, dx-\int \frac {256^{e^{-x}}-256^{e^{-x}} x-\log (-1+x)+x \log (-1+x)-256^{e^{-x}} \log (x)+256^{e^{-x}} x \log (x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)} \, dx\\ &=-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\log (4) \int \frac {256^{e^{-x}} (1-\log (x))}{x \log (4) \log (x)} \, dx-\int \frac {256^{e^{-x}} (-1+x)+(-1+x) \log (-1+x) (-1+\log (x))-\log (x) \left (256^{e^{-x}} (-1+x)-x \log \left (\frac {2 x}{\log (x)}\right )\right )}{(1-x) x \log (x)} \, dx\\ &=-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\int \frac {256^{e^{-x}} (1-\log (x))}{x \log (x)} \, dx-\int \left (\frac {256^{e^{-x}} (-1+\log (x))}{x \log (x)}+\frac {-\log (-1+x)+x \log (-1+x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)}\right ) \, dx\\ &=-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\int \left (-\frac {256^{e^{-x}}}{x}+\frac {256^{e^{-x}}}{x \log (x)}\right ) \, dx-\int \frac {256^{e^{-x}} (-1+\log (x))}{x \log (x)} \, dx-\int \frac {-\log (-1+x)+x \log (-1+x)+\log (-1+x) \log (x)-x \log (-1+x) \log (x)-x \log (x) \log \left (\frac {2 x}{\log (x)}\right )}{(-1+x) x \log (x)} \, dx\\ &=-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \frac {256^{e^{-x}}}{x} \, dx-\int \left (\frac {256^{e^{-x}}}{x}-\frac {256^{e^{-x}}}{x \log (x)}\right ) \, dx-\int \frac {256^{e^{-x}}}{x \log (x)} \, dx-\int \left (-\frac {\log (-1+x) (-1+\log (x))}{x \log (x)}-\frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x}\right ) \, dx\\ &=-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \frac {\log (-1+x) (-1+\log (x))}{x \log (x)} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx\\ &=-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \left (\frac {\log (-1+x)}{x}-\frac {\log (-1+x)}{x \log (x)}\right ) \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx\\ &=-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\int \frac {\log (-1+x)}{x} \, dx-\int \frac {\log (-1+x)}{x \log (x)} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx\\ &=\log (-1+x) \log (x)-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )-\int \frac {\log (-1+x)}{x \log (x)} \, dx-\int \frac {\log (x)}{-1+x} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx\\ &=\log (-1+x) \log (x)-4^{4 e^{-x}} \log \left (\frac {2 x}{\log (x)}\right )+\text {Li}_2(1-x)-\int \frac {\log (-1+x)}{x \log (x)} \, dx+\int \frac {\log \left (\frac {2 x}{\log (x)}\right )}{-1+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 24, normalized size = 0.96 \begin {gather*} -\left (\left (256^{e^{-x}}-\log (-1+x)\right ) \log \left (\frac {2 x}{\log (x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 25, normalized size = 1.00 \begin {gather*} -{\left (2^{8 \, e^{\left (-x\right )}} - \log \left (x - 1\right )\right )} \log \left (\frac {2 \, x}{\log \relax (x)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2^(8*e^(-x)) - log(x - 1))*log(2*x/log(x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x - 1\right )} e^{x} \log \left (x - 1\right ) \log \relax (x) - {\left (x - 1\right )} e^{x} \log \left (x - 1\right ) - {\left ({\left (x - 1\right )} e^{x} \log \relax (x) - {\left (x - 1\right )} e^{x}\right )} 2^{8 \, e^{\left (-x\right )}} + {\left (8 \, {\left (x^{2} - x\right )} 2^{8 \, e^{\left (-x\right )}} \log \relax (2) \log \relax (x) + x e^{x} \log \relax (x)\right )} \log \left (\frac {2 \, x}{\log \relax (x)}\right )\right )} e^{\left (-x\right )}}{{\left (x^{2} - x\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((x - 1)*e^x*log(x - 1)*log(x) - (x - 1)*e^x*log(x - 1) - ((x - 1)*e^x*log(x) - (x - 1)*e^x)*2^(8*e^
(-x)) + (8*(x^2 - x)*2^(8*e^(-x))*log(2)*log(x) + x*e^x*log(x))*log(2*x/log(x)))*e^(-x)/((x^2 - x)*log(x)), x)

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maple [C]  time = 0.23, size = 261, normalized size = 10.44

method	result	size
risch	\(\left (-\ln \left (x -1\right )+256^{{\mathrm e}^{-x}}\right ) \ln \left (\ln \relax (x )\right )+\ln \left (x -1\right ) \ln \relax (x )-\frac {i 256^{{\mathrm e}^{-x}} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{2}}{2}+\frac {i 256^{{\mathrm e}^{-x}} \pi \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{3}}{2}+\frac {i 256^{{\mathrm e}^{-x}} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )}{2}-\frac {i 256^{{\mathrm e}^{-x}} \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{2}}{2}-\frac {i \pi \ln \left (x -1\right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{3}}{2}+\frac {i \pi \ln \left (x -1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{2}}{2}+\frac {i \pi \ln \left (x -1\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{2}}{2}-\frac {i \pi \ln \left (x -1\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )}{2}+\ln \relax (2) \ln \left (x -1\right )-256^{{\mathrm e}^{-x}} \ln \relax (2)-256^{{\mathrm e}^{-x}} \ln \relax (x )\)	\(261\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*(4*x^2-4*x)*ln(2)*ln(x)*exp(8*ln(2)/exp(x))+x*exp(x)*ln(x))*ln(2*x/ln(x))+((1-x)*exp(x)*ln(x)+(x-1)*ex
p(x))*exp(8*ln(2)/exp(x))+(x-1)*exp(x)*ln(x-1)*ln(x)+(1-x)*exp(x)*ln(x-1))/(x^2-x)/exp(x)/ln(x),x,method=_RETU
RNVERBOSE)

[Out]

(-ln(x-1)+256^exp(-x))*ln(ln(x))+ln(x-1)*ln(x)-1/2*I*256^exp(-x)*Pi*csgn(I*x)*csgn(I*x/ln(x))^2+1/2*I*256^exp(
-x)*Pi*csgn(I*x/ln(x))^3+1/2*I*256^exp(-x)*Pi*csgn(I*x)*csgn(I/ln(x))*csgn(I*x/ln(x))-1/2*I*256^exp(-x)*Pi*csg
n(I/ln(x))*csgn(I*x/ln(x))^2-1/2*I*Pi*ln(x-1)*csgn(I*x/ln(x))^3+1/2*I*Pi*ln(x-1)*csgn(I*x)*csgn(I*x/ln(x))^2+1
/2*I*Pi*ln(x-1)*csgn(I/ln(x))*csgn(I*x/ln(x))^2-1/2*I*Pi*ln(x-1)*csgn(I*x)*csgn(I/ln(x))*csgn(I*x/ln(x))+ln(2)
*ln(x-1)-256^exp(-x)*ln(2)-256^exp(-x)*ln(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} {\left (\log \relax (2) + \log \relax (x) - \log \left (\log \relax (x)\right )\right )} \log \left (x - 1\right ) - \int -\frac {{\left (8 \, x \log \relax (2)^{2} \log \relax (x) + 8 \, x \log \relax (2) \log \relax (x)^{2} - 8 \, x \log \relax (2) \log \relax (x) \log \left (\log \relax (x)\right ) - {\left (\log \relax (x) - 1\right )} e^{x}\right )} e^{\left (8 \, e^{\left (-x\right )} \log \relax (2) - x\right )}}{x \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(2) + log(x) - log(log(x)))*log(x - 1) - integrate(-(8*x*log(2)^2*log(x) + 8*x*log(2)*log(x)^2 - 8*x*log(2
)*log(x)*log(log(x)) - (log(x) - 1)*e^x)*e^(8*e^(-x)*log(2) - x)/(x*log(x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{8\,{\mathrm {e}}^{-x}\,\ln \relax (2)}\,\left ({\mathrm {e}}^x\,\left (x-1\right )-{\mathrm {e}}^x\,\ln \relax (x)\,\left (x-1\right )\right )+\ln \left (\frac {2\,x}{\ln \relax (x)}\right )\,\left (x\,{\mathrm {e}}^x\,\ln \relax (x)-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^{-x}\,\ln \relax (2)}\,\ln \relax (2)\,\ln \relax (x)\,\left (4\,x-4\,x^2\right )\right )-\ln \left (x-1\right )\,{\mathrm {e}}^x\,\left (x-1\right )+\ln \left (x-1\right )\,{\mathrm {e}}^x\,\ln \relax (x)\,\left (x-1\right )\right )}{\ln \relax (x)\,\left (x-x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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