3.45.61 \(\int \frac {e^{-e^x x+3 e^{x+\frac {4 x}{(-25+10 x) \log (x)}} x} (e^x (-125-25 x+80 x^2-20 x^3) \log ^2(x)+e^{\frac {4 x}{(-25+10 x) \log (x)}} (e^x (60 x-24 x^2)-60 e^x x \log (x)+e^x (375+75 x-240 x^2+60 x^3) \log ^2(x)))}{(125-100 x+20 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ e^{e^x \left (-x+3 e^{\frac {4}{5 \left (2-\frac {5}{x}\right ) \log (x)}} x\right )} \]

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Rubi [F]  time = 79.39, 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 {\exp \left (-e^x x+3 e^{x+\frac {4 x}{(-25+10 x) \log (x)}} x\right ) \left (e^x \left (-125-25 x+80 x^2-20 x^3\right ) \log ^2(x)+e^{\frac {4 x}{(-25+10 x) \log (x)}} \left (e^x \left (60 x-24 x^2\right )-60 e^x x \log (x)+e^x \left (375+75 x-240 x^2+60 x^3\right ) \log ^2(x)\right )\right )}{\left (125-100 x+20 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-(E^x*x) + 3*E^(x + (4*x)/((-25 + 10*x)*Log[x]))*x)*(E^x*(-125 - 25*x + 80*x^2 - 20*x^3)*Log[x]^2 + E^
((4*x)/((-25 + 10*x)*Log[x]))*(E^x*(60*x - 24*x^2) - 60*E^x*x*Log[x] + E^x*(375 + 75*x - 240*x^2 + 60*x^3)*Log
[x]^2)))/((125 - 100*x + 20*x^2)*Log[x]^2),x]

[Out]

-Defer[Int][E^(x + E^x*(-1 + 3*E^((4*x)/(5*(-5 + 2*x)*Log[x])))*x), x] + 3*Defer[Int][E^(x + E^x*(-1 + 3*E^((4
*x)/(5*(-5 + 2*x)*Log[x])))*x + (4*x)/(5*(-5 + 2*x)*Log[x])), x] - Defer[Int][E^(x + E^x*(-1 + 3*E^((4*x)/(5*(
-5 + 2*x)*Log[x])))*x)*x, x] + 3*Defer[Int][E^(x + E^x*(-1 + 3*E^((4*x)/(5*(-5 + 2*x)*Log[x])))*x + (4*x)/(5*(
-5 + 2*x)*Log[x]))*x, x] - (6*Defer[Int][E^(x + E^x*(-1 + 3*E^((4*x)/(5*(-5 + 2*x)*Log[x])))*x + (4*x)/(5*(-5
+ 2*x)*Log[x]))/Log[x]^2, x])/5 - 6*Defer[Int][E^(x + E^x*(-1 + 3*E^((4*x)/(5*(-5 + 2*x)*Log[x])))*x + (4*x)/(
5*(-5 + 2*x)*Log[x]))/((-5 + 2*x)*Log[x]^2), x] - 30*Defer[Int][E^(x + E^x*(-1 + 3*E^((4*x)/(5*(-5 + 2*x)*Log[
x])))*x + (4*x)/(5*(-5 + 2*x)*Log[x]))/((-5 + 2*x)^2*Log[x]), x] - 6*Defer[Int][E^(x + E^x*(-1 + 3*E^((4*x)/(5
*(-5 + 2*x)*Log[x])))*x + (4*x)/(5*(-5 + 2*x)*Log[x]))/((-5 + 2*x)*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-e^x x+3 e^{x+\frac {4 x}{(-25+10 x) \log (x)}} x\right ) \left (e^x \left (-125-25 x+80 x^2-20 x^3\right ) \log ^2(x)+e^{\frac {4 x}{(-25+10 x) \log (x)}} \left (e^x \left (60 x-24 x^2\right )-60 e^x x \log (x)+e^x \left (375+75 x-240 x^2+60 x^3\right ) \log ^2(x)\right )\right )}{5 (-5+2 x)^2 \log ^2(x)} \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (-e^x x+3 e^{x+\frac {4 x}{(-25+10 x) \log (x)}} x\right ) \left (e^x \left (-125-25 x+80 x^2-20 x^3\right ) \log ^2(x)+e^{\frac {4 x}{(-25+10 x) \log (x)}} \left (e^x \left (60 x-24 x^2\right )-60 e^x x \log (x)+e^x \left (375+75 x-240 x^2+60 x^3\right ) \log ^2(x)\right )\right )}{(-5+2 x)^2 \log ^2(x)} \, dx\\ &=\frac {1}{5} \int \frac {\exp \left (e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) \left (e^x \left (-125-25 x+80 x^2-20 x^3\right ) \log ^2(x)+e^{\frac {4 x}{(-25+10 x) \log (x)}} \left (e^x \left (60 x-24 x^2\right )-60 e^x x \log (x)+e^x \left (375+75 x-240 x^2+60 x^3\right ) \log ^2(x)\right )\right )}{(5-2 x)^2 \log ^2(x)} \, dx\\ &=\frac {1}{5} \int \left (-5 \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) (1+x)+\frac {3 \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) \left (20 x-8 x^2-20 x \log (x)+125 \log ^2(x)+25 x \log ^2(x)-80 x^2 \log ^2(x)+20 x^3 \log ^2(x)\right )}{(-5+2 x)^2 \log ^2(x)}\right ) \, dx\\ &=\frac {3}{5} \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) \left (20 x-8 x^2-20 x \log (x)+125 \log ^2(x)+25 x \log ^2(x)-80 x^2 \log ^2(x)+20 x^3 \log ^2(x)\right )}{(-5+2 x)^2 \log ^2(x)} \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) (1+x) \, dx\\ &=\frac {3}{5} \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) \left (4 (5-2 x) x-20 x \log (x)+5 (5-2 x)^2 (1+x) \log ^2(x)\right )}{(5-2 x)^2 \log ^2(x)} \, dx-\int \left (\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right )+\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) x\right ) \, dx\\ &=\frac {3}{5} \int \left (5 \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) (1+x)-\frac {4 \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) x}{(-5+2 x) \log ^2(x)}-\frac {20 \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) x}{(-5+2 x)^2 \log (x)}\right ) \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) x \, dx\\ &=-\left (\frac {12}{5} \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) x}{(-5+2 x) \log ^2(x)} \, dx\right )+3 \int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) (1+x) \, dx-12 \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) x}{(-5+2 x)^2 \log (x)} \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) x \, dx\\ &=-\left (\frac {12}{5} \int \left (\frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{2 \log ^2(x)}+\frac {5 \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{2 (-5+2 x) \log ^2(x)}\right ) \, dx\right )+3 \int \left (\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )+\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) x\right ) \, dx-12 \int \left (\frac {5 \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{2 (-5+2 x)^2 \log (x)}+\frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{2 (-5+2 x) \log (x)}\right ) \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) x \, dx\\ &=-\left (\frac {6}{5} \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{\log ^2(x)} \, dx\right )+3 \int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) \, dx+3 \int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right ) x \, dx-6 \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{(-5+2 x) \log ^2(x)} \, dx-6 \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{(-5+2 x) \log (x)} \, dx-30 \int \frac {\exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x+\frac {4 x}{5 (-5+2 x) \log (x)}\right )}{(-5+2 x)^2 \log (x)} \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) \, dx-\int \exp \left (x+e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x\right ) x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 29, normalized size = 0.91 \begin {gather*} e^{e^x \left (-1+3 e^{\frac {4 x}{5 (-5+2 x) \log (x)}}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-(E^x*x) + 3*E^(x + (4*x)/((-25 + 10*x)*Log[x]))*x)*(E^x*(-125 - 25*x + 80*x^2 - 20*x^3)*Log[x]^
2 + E^((4*x)/((-25 + 10*x)*Log[x]))*(E^x*(60*x - 24*x^2) - 60*E^x*x*Log[x] + E^x*(375 + 75*x - 240*x^2 + 60*x^
3)*Log[x]^2)))/((125 - 100*x + 20*x^2)*Log[x]^2),x]

[Out]

E^(E^x*(-1 + 3*E^((4*x)/(5*(-5 + 2*x)*Log[x])))*x)

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fricas [A]  time = 0.67, size = 41, normalized size = 1.28 \begin {gather*} e^{\left (-x e^{x} + 3 \, x e^{\left (\frac {5 \, {\left (2 \, x^{2} - 5 \, x\right )} \log \relax (x) + 4 \, x}{5 \, {\left (2 \, x - 5\right )} \log \relax (x)}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((60*x^3-240*x^2+75*x+375)*exp(x)*log(x)^2-60*x*exp(x)*log(x)+(-24*x^2+60*x)*exp(x))*exp(4*x/(10*x-
25)/log(x))+(-20*x^3+80*x^2-25*x-125)*exp(x)*log(x)^2)*exp(3*x*exp(x)*exp(4*x/(10*x-25)/log(x))-exp(x)*x)/(20*
x^2-100*x+125)/log(x)^2,x, algorithm="fricas")

[Out]

e^(-x*e^x + 3*x*e^(1/5*(5*(2*x^2 - 5*x)*log(x) + 4*x)/((2*x - 5)*log(x))))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((60*x^3-240*x^2+75*x+375)*exp(x)*log(x)^2-60*x*exp(x)*log(x)+(-24*x^2+60*x)*exp(x))*exp(4*x/(10*x-
25)/log(x))+(-20*x^3+80*x^2-25*x-125)*exp(x)*log(x)^2)*exp(3*x*exp(x)*exp(4*x/(10*x-25)/log(x))-exp(x)*x)/(20*
x^2-100*x+125)/log(x)^2,x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.16, size = 36, normalized size = 1.12

method	result	size
risch	\({\mathrm e}^{-x \left ({\mathrm e}^{x}-3 \,{\mathrm e}^{\frac {x \left (10 x \ln \relax (x )-25 \ln \relax (x )+4\right )}{5 \left (2 x -5\right ) \ln \relax (x )}}\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((60*x^3-240*x^2+75*x+375)*exp(x)*ln(x)^2-60*x*exp(x)*ln(x)+(-24*x^2+60*x)*exp(x))*exp(4*x/(10*x-25)/ln(x
))+(-20*x^3+80*x^2-25*x-125)*exp(x)*ln(x)^2)*exp(3*x*exp(x)*exp(4*x/(10*x-25)/ln(x))-exp(x)*x)/(20*x^2-100*x+1
25)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(-x*(exp(x)-3*exp(1/5*x*(10*x*ln(x)-25*ln(x)+4)/(2*x-5)/ln(x))))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{\left (3 \, x e^{\left (x + \frac {2}{5 \, \log \relax (x)} + \frac {2}{{\left (2 \, x - 5\right )} \log \relax (x)}\right )} - x e^{x}\right )} - \frac {1}{5} \, \int 0\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((60*x^3-240*x^2+75*x+375)*exp(x)*log(x)^2-60*x*exp(x)*log(x)+(-24*x^2+60*x)*exp(x))*exp(4*x/(10*x-
25)/log(x))+(-20*x^3+80*x^2-25*x-125)*exp(x)*log(x)^2)*exp(3*x*exp(x)*exp(4*x/(10*x-25)/log(x))-exp(x)*x)/(20*
x^2-100*x+125)/log(x)^2,x, algorithm="maxima")

[Out]

e^(3*x*e^(x + 2/5/log(x) + 2/((2*x - 5)*log(x))) - x*e^x) - 1/5*integrate(0, x)

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mupad [B]  time = 3.39, size = 29, normalized size = 0.91 \begin {gather*} {\mathrm {e}}^{3\,x\,{\mathrm {e}}^x\,{\mathrm {e}}^{-\frac {4\,x}{25\,\ln \relax (x)-10\,x\,\ln \relax (x)}}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x*exp((4*x)/(log(x)*(10*x - 25)))*exp(x) - x*exp(x))*(exp((4*x)/(log(x)*(10*x - 25)))*(exp(x)*(60*x
- 24*x^2) + exp(x)*log(x)^2*(75*x - 240*x^2 + 60*x^3 + 375) - 60*x*exp(x)*log(x)) - exp(x)*log(x)^2*(25*x - 8
0*x^2 + 20*x^3 + 125)))/(log(x)^2*(20*x^2 - 100*x + 125)),x)

[Out]

exp(3*x*exp(x)*exp(-(4*x)/(25*log(x) - 10*x*log(x))))*exp(-x*exp(x))

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sympy [A]  time = 63.09, size = 26, normalized size = 0.81 \begin {gather*} e^{3 x e^{x} e^{\frac {4 x}{\left (10 x - 25\right ) \log {\relax (x )}}} - x e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((60*x**3-240*x**2+75*x+375)*exp(x)*ln(x)**2-60*x*exp(x)*ln(x)+(-24*x**2+60*x)*exp(x))*exp(4*x/(10*
x-25)/ln(x))+(-20*x**3+80*x**2-25*x-125)*exp(x)*ln(x)**2)*exp(3*x*exp(x)*exp(4*x/(10*x-25)/ln(x))-exp(x)*x)/(2
0*x**2-100*x+125)/ln(x)**2,x)

[Out]

exp(3*x*exp(x)*exp(4*x/((10*x - 25)*log(x))) - x*exp(x))

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