3.78.58 \(\int \frac {e^{\frac {-e+\log (3)}{-20-x+x \log (x)}} (e-\log (3)) \log (x)}{400+40 x+x^2+(-40 x-2 x^2) \log (x)+x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {e-\log (3)}{20+x-x \log (x)}} \]

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Rubi [F]  time = 2.28, 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^{\frac {-e+\log (3)}{-20-x+x \log (x)}} (e-\log (3)) \log (x)}{400+40 x+x^2+\left (-40 x-2 x^2\right ) \log (x)+x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-E + Log[3])/(-20 - x + x*Log[x]))*(E - Log[3])*Log[x])/(400 + 40*x + x^2 + (-40*x - 2*x^2)*Log[x] +
x^2*Log[x]^2),x]

[Out]

(E - Log[3])*Defer[Int][(3^(-20 - x + x*Log[x])^(-1)*E^(E/(20 + x - x*Log[x])))/(-20 - x + x*Log[x])^2, x] + 2
0*(E - Log[3])*Defer[Int][(3^(-20 - x + x*Log[x])^(-1)*E^(E/(20 + x - x*Log[x])))/(x*(-20 - x + x*Log[x])^2),
x] + (E - Log[3])*Defer[Int][(3^(-20 - x + x*Log[x])^(-1)*E^(E/(20 + x - x*Log[x])))/(x*(-20 - x + x*Log[x])),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=(e-\log (3)) \int \frac {e^{\frac {-e+\log (3)}{-20-x+x \log (x)}} \log (x)}{400+40 x+x^2+\left (-40 x-2 x^2\right ) \log (x)+x^2 \log ^2(x)} \, dx\\ &=(e-\log (3)) \int \frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}} \log (x)}{(20+x-x \log (x))^2} \, dx\\ &=(e-\log (3)) \int \left (\frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}} (20+x)}{x (-20-x+x \log (x))^2}+\frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{x (-20-x+x \log (x))}\right ) \, dx\\ &=(e-\log (3)) \int \frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}} (20+x)}{x (-20-x+x \log (x))^2} \, dx+(e-\log (3)) \int \frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{x (-20-x+x \log (x))} \, dx\\ &=(e-\log (3)) \int \frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{x (-20-x+x \log (x))} \, dx+(e-\log (3)) \int \left (\frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{(-20-x+x \log (x))^2}+\frac {20\ 3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{x (-20-x+x \log (x))^2}\right ) \, dx\\ &=(e-\log (3)) \int \frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{(-20-x+x \log (x))^2} \, dx+(e-\log (3)) \int \frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{x (-20-x+x \log (x))} \, dx+(20 (e-\log (3))) \int \frac {3^{\frac {1}{-20-x+x \log (x)}} e^{\frac {e}{20+x-x \log (x)}}}{x (-20-x+x \log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {e-\log (3)}{20+x-x \log (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-E + Log[3])/(-20 - x + x*Log[x]))*(E - Log[3])*Log[x])/(400 + 40*x + x^2 + (-40*x - 2*x^2)*Log
[x] + x^2*Log[x]^2),x]

[Out]

E^((E - Log[3])/(20 + x - x*Log[x]))

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fricas [A]  time = 0.61, size = 21, normalized size = 1.11 \begin {gather*} e^{\left (-\frac {e - \log \relax (3)}{x \log \relax (x) - x - 20}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3)+exp(1))*log(x)*exp((log(3)-exp(1))/(x*log(x)-x-20))/(x^2*log(x)^2+(-2*x^2-40*x)*log(x)+x^2+
40*x+400),x, algorithm="fricas")

[Out]

e^(-(e - log(3))/(x*log(x) - x - 20))

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giac [A]  time = 0.13, size = 31, normalized size = 1.63 \begin {gather*} e^{\left (-\frac {e}{x \log \relax (x) - x - 20} + \frac {\log \relax (3)}{x \log \relax (x) - x - 20}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3)+exp(1))*log(x)*exp((log(3)-exp(1))/(x*log(x)-x-20))/(x^2*log(x)^2+(-2*x^2-40*x)*log(x)+x^2+
40*x+400),x, algorithm="giac")

[Out]

e^(-e/(x*log(x) - x - 20) + log(3)/(x*log(x) - x - 20))

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maple [A]  time = 0.19, size = 22, normalized size = 1.16

method	result	size
risch	\({\mathrm e}^{-\frac {-\ln \relax (3)+{\mathrm e}}{x \ln \relax (x )-x -20}}\)	\(22\)
norman	\(\frac {x \ln \relax (x ) {\mathrm e}^{\frac {\ln \relax (3)-{\mathrm e}}{x \ln \relax (x )-x -20}}-x \,{\mathrm e}^{\frac {\ln \relax (3)-{\mathrm e}}{x \ln \relax (x )-x -20}}-20 \,{\mathrm e}^{\frac {\ln \relax (3)-{\mathrm e}}{x \ln \relax (x )-x -20}}}{x \ln \relax (x )-x -20}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(3)+exp(1))*ln(x)*exp((ln(3)-exp(1))/(x*ln(x)-x-20))/(x^2*ln(x)^2+(-2*x^2-40*x)*ln(x)+x^2+40*x+400),x,
method=_RETURNVERBOSE)

[Out]

exp(-(-ln(3)+exp(1))/(x*ln(x)-x-20))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3)+exp(1))*log(x)*exp((log(3)-exp(1))/(x*log(x)-x-20))/(x^2*log(x)^2+(-2*x^2-40*x)*log(x)+x^2+
40*x+400),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [B]  time = 5.97, size = 29, normalized size = 1.53 \begin {gather*} \frac {{\mathrm {e}}^{\frac {\mathrm {e}}{x-x\,\ln \relax (x)+20}}}{3^{\frac {1}{x-x\,\ln \relax (x)+20}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(1) - log(3))/(x - x*log(x) + 20))*log(x)*(exp(1) - log(3)))/(40*x + x^2*log(x)^2 - log(x)*(40*x
+ 2*x^2) + x^2 + 400),x)

[Out]

exp(exp(1)/(x - x*log(x) + 20))/3^(1/(x - x*log(x) + 20))

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sympy [A]  time = 0.97, size = 15, normalized size = 0.79 \begin {gather*} e^{\frac {- e + \log {\relax (3 )}}{x \log {\relax (x )} - x - 20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(3)+exp(1))*ln(x)*exp((ln(3)-exp(1))/(x*ln(x)-x-20))/(x**2*ln(x)**2+(-2*x**2-40*x)*ln(x)+x**2+40
*x+400),x)

[Out]

exp((-E + log(3))/(x*log(x) - x - 20))

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