3.30.91 \(\int \frac {-2-2 e^x+(1+e^x (1-x)) \log (x)+(1+e^x+(-1-e^x) \log (x)) \log (\frac {5}{x+e^x x})}{(-10 x-10 e^x x) \log (x)+(5 x+5 e^x x) \log (x) \log (\frac {5}{x+e^x x})+((-2 x-2 e^x x) \log (x)+(x+e^x x) \log (x) \log (\frac {5}{x+e^x x})) \log (\frac {2 \log (x)-\log (x) \log (\frac {5}{x+e^x x})}{x})} \, dx\)

Optimal. Leaf size=26 \[ \log \left (5+\log \left (\frac {\log (x) \left (2-\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right ) \]

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Rubi [F]  time = 6.45, 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 {-2-2 e^x+\left (1+e^x (1-x)\right ) \log (x)+\left (1+e^x+\left (-1-e^x\right ) \log (x)\right ) \log \left (\frac {5}{x+e^x x}\right )}{\left (-10 x-10 e^x x\right ) \log (x)+\left (5 x+5 e^x x\right ) \log (x) \log \left (\frac {5}{x+e^x x}\right )+\left (\left (-2 x-2 e^x x\right ) \log (x)+\left (x+e^x x\right ) \log (x) \log \left (\frac {5}{x+e^x x}\right )\right ) \log \left (\frac {2 \log (x)-\log (x) \log \left (\frac {5}{x+e^x x}\right )}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2 - 2*E^x + (1 + E^x*(1 - x))*Log[x] + (1 + E^x + (-1 - E^x)*Log[x])*Log[5/(x + E^x*x)])/((-10*x - 10*E^
x*x)*Log[x] + (5*x + 5*E^x*x)*Log[x]*Log[5/(x + E^x*x)] + ((-2*x - 2*E^x*x)*Log[x] + (x + E^x*x)*Log[x]*Log[5/
(x + E^x*x)])*Log[(2*Log[x] - Log[x]*Log[5/(x + E^x*x)])/x]),x]

[Out]

-Defer[Int][1/((-2 + Log[5/(x + E^x*x)])*(5 + Log[-((Log[x]*(-2 + Log[5/(x + E^x*x)]))/x)])), x] + Defer[Int][
1/((1 + E^x)*(-2 + Log[5/(x + E^x*x)])*(5 + Log[-((Log[x]*(-2 + Log[5/(x + E^x*x)]))/x)])), x] + Defer[Int][1/
(x*(-2 + Log[5/(x + E^x*x)])*(5 + Log[-((Log[x]*(-2 + Log[5/(x + E^x*x)]))/x)])), x] - 2*Defer[Int][1/(x*Log[x
]*(-2 + Log[5/(x + E^x*x)])*(5 + Log[-((Log[x]*(-2 + Log[5/(x + E^x*x)]))/x)])), x] - Defer[Int][Log[5/(x + E^
x*x)]/(x*(-2 + Log[5/(x + E^x*x)])*(5 + Log[-((Log[x]*(-2 + Log[5/(x + E^x*x)]))/x)])), x] + Defer[Int][Log[5/
(x + E^x*x)]/(x*Log[x]*(-2 + Log[5/(x + E^x*x)])*(5 + Log[-((Log[x]*(-2 + Log[5/(x + E^x*x)]))/x)])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\left (\left (1+e^x\right ) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )\right )+\log (x) \left (-1+e^x (-1+x)+\left (1+e^x\right ) \log \left (\frac {5}{x+e^x x}\right )\right )}{\left (1+e^x\right ) x \log (x) \left (2-\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx\\ &=\int \left (\frac {1}{\left (1+e^x\right ) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )}+\frac {-2+\log (x)-x \log (x)+\log \left (\frac {5}{x+e^x x}\right )-\log (x) \log \left (\frac {5}{x+e^x x}\right )}{x \log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )}\right ) \, dx\\ &=\int \frac {1}{\left (1+e^x\right ) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx+\int \frac {-2+\log (x)-x \log (x)+\log \left (\frac {5}{x+e^x x}\right )-\log (x) \log \left (\frac {5}{x+e^x x}\right )}{x \log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx\\ &=\int \frac {1}{\left (1+e^x\right ) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx+\int \left (-\frac {1}{\left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )}+\frac {1}{x \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )}-\frac {2}{x \log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )}-\frac {\log \left (\frac {5}{x+e^x x}\right )}{x \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )}+\frac {\log \left (\frac {5}{x+e^x x}\right )}{x \log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x \log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx\right )-\int \frac {1}{\left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx+\int \frac {1}{\left (1+e^x\right ) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx+\int \frac {1}{x \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {5}{x+e^x x}\right )}{x \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx+\int \frac {\log \left (\frac {5}{x+e^x x}\right )}{x \log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right ) \left (5+\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 27, normalized size = 1.04 \begin {gather*} \log \left (-5-\log \left (-\frac {\log (x) \left (-2+\log \left (\frac {5}{x+e^x x}\right )\right )}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 2*E^x + (1 + E^x*(1 - x))*Log[x] + (1 + E^x + (-1 - E^x)*Log[x])*Log[5/(x + E^x*x)])/((-10*x -
10*E^x*x)*Log[x] + (5*x + 5*E^x*x)*Log[x]*Log[5/(x + E^x*x)] + ((-2*x - 2*E^x*x)*Log[x] + (x + E^x*x)*Log[x]*
Log[5/(x + E^x*x)])*Log[(2*Log[x] - Log[x]*Log[5/(x + E^x*x)])/x]),x]

[Out]

Log[-5 - Log[-((Log[x]*(-2 + Log[5/(x + E^x*x)]))/x)]]

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fricas [A]  time = 0.62, size = 28, normalized size = 1.08 \begin {gather*} \log \left (\log \left (-\frac {\log \relax (x) \log \left (\frac {5}{x e^{x} + x}\right ) - 2 \, \log \relax (x)}{x}\right ) + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)-1)*log(x)+exp(x)+1)*log(5/(exp(x)*x+x))+((-x+1)*exp(x)+1)*log(x)-2*exp(x)-2)/(((exp(x)*x+
x)*log(x)*log(5/(exp(x)*x+x))+(-2*exp(x)*x-2*x)*log(x))*log((-log(x)*log(5/(exp(x)*x+x))+2*log(x))/x)+(5*exp(x
)*x+5*x)*log(x)*log(5/(exp(x)*x+x))+(-10*exp(x)*x-10*x)*log(x)),x, algorithm="fricas")

[Out]

log(log(-(log(x)*log(5/(x*e^x + x)) - 2*log(x))/x) + 5)

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giac [A]  time = 1.25, size = 31, normalized size = 1.19 \begin {gather*} \log \left (\log \left (-\log \relax (5) \log \relax (x) + \log \relax (x)^{2} + \log \relax (x) \log \left (e^{x} + 1\right ) + 2 \, \log \relax (x)\right ) - \log \relax (x) + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)-1)*log(x)+exp(x)+1)*log(5/(exp(x)*x+x))+((-x+1)*exp(x)+1)*log(x)-2*exp(x)-2)/(((exp(x)*x+
x)*log(x)*log(5/(exp(x)*x+x))+(-2*exp(x)*x-2*x)*log(x))*log((-log(x)*log(5/(exp(x)*x+x))+2*log(x))/x)+(5*exp(x
)*x+5*x)*log(x)*log(5/(exp(x)*x+x))+(-10*exp(x)*x-10*x)*log(x)),x, algorithm="giac")

[Out]

log(log(-log(5)*log(x) + log(x)^2 + log(x)*log(e^x + 1) + 2*log(x)) - log(x) + 5)

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maple [C]  time = 4.12, size = 15838, normalized size = 609.15

method	result	size
risch	\(\text {Expression too large to display}\)	\(15838\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(x)-1)*ln(x)+exp(x)+1)*ln(5/(exp(x)*x+x))+((1-x)*exp(x)+1)*ln(x)-2*exp(x)-2)/(((exp(x)*x+x)*ln(x)*l
n(5/(exp(x)*x+x))+(-2*exp(x)*x-2*x)*ln(x))*ln((-ln(x)*ln(5/(exp(x)*x+x))+2*ln(x))/x)+(5*exp(x)*x+5*x)*ln(x)*ln
(5/(exp(x)*x+x))+(-10*exp(x)*x-10*x)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [A]  time = 0.88, size = 24, normalized size = 0.92 \begin {gather*} \log \left (-\log \relax (x) + \log \left (-\log \relax (5) + \log \relax (x) + \log \left (e^{x} + 1\right ) + 2\right ) + \log \left (\log \relax (x)\right ) + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)-1)*log(x)+exp(x)+1)*log(5/(exp(x)*x+x))+((-x+1)*exp(x)+1)*log(x)-2*exp(x)-2)/(((exp(x)*x+
x)*log(x)*log(5/(exp(x)*x+x))+(-2*exp(x)*x-2*x)*log(x))*log((-log(x)*log(5/(exp(x)*x+x))+2*log(x))/x)+(5*exp(x
)*x+5*x)*log(x)*log(5/(exp(x)*x+x))+(-10*exp(x)*x-10*x)*log(x)),x, algorithm="maxima")

[Out]

log(-log(x) + log(-log(5) + log(x) + log(e^x + 1) + 2) + log(log(x)) + 5)

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mupad [B]  time = 2.50, size = 28, normalized size = 1.08 \begin {gather*} \ln \left (\ln \left (\frac {2\,\ln \relax (x)-\ln \left (\frac {5}{x+x\,{\mathrm {e}}^x}\right )\,\ln \relax (x)}{x}\right )+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x) + log(x)*(exp(x)*(x - 1) - 1) - log(5/(x + x*exp(x)))*(exp(x) - log(x)*(exp(x) + 1) + 1) + 2)/(l
og(x)*(10*x + 10*x*exp(x)) + log((2*log(x) - log(5/(x + x*exp(x)))*log(x))/x)*(log(x)*(2*x + 2*x*exp(x)) - log
(5/(x + x*exp(x)))*log(x)*(x + x*exp(x))) - log(5/(x + x*exp(x)))*log(x)*(5*x + 5*x*exp(x))),x)

[Out]

log(log((2*log(x) - log(5/(x + x*exp(x)))*log(x))/x) + 5)

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sympy [A]  time = 143.71, size = 24, normalized size = 0.92 \begin {gather*} \log {\left (\log {\left (\frac {- \log {\relax (x )} \log {\left (\frac {5}{x e^{x} + x} \right )} + 2 \log {\relax (x )}}{x} \right )} + 5 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(x)-1)*ln(x)+exp(x)+1)*ln(5/(exp(x)*x+x))+((-x+1)*exp(x)+1)*ln(x)-2*exp(x)-2)/(((exp(x)*x+x)*
ln(x)*ln(5/(exp(x)*x+x))+(-2*exp(x)*x-2*x)*ln(x))*ln((-ln(x)*ln(5/(exp(x)*x+x))+2*ln(x))/x)+(5*exp(x)*x+5*x)*l
n(x)*ln(5/(exp(x)*x+x))+(-10*exp(x)*x-10*x)*ln(x)),x)

[Out]

log(log((-log(x)*log(5/(x*exp(x) + x)) + 2*log(x))/x) + 5)

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