\(\int \frac {-x+x^2+(x^2-x^3) \log (5)+(x^2-x^3 \log (5)) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+(e^{-e^{e^5 x}+x} (-x+x^2+e^{5+e^5 x} (x-x^2)) \log (x)+(x-x^2+(-2 x^2+2 x^3) \log (5)) \log (x)) \log ((-1+x) \log (x))}{(-x+x^2) \log (x) \log ^2((-1+x) \log (x))} \, dx\) [6485]

Optimal result

Integrand size = 153, antiderivative size = 33 \[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\frac {e^{-e^{e^5 x}+x}-x+x^2 \log (5)}{\log ((-1+x) \log (x))} \]

[Out]

(x^2*ln(5)+exp(x-exp(x*exp(5)))-x)/ln((-1+x)*ln(x))

Rubi [F]

\[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx \]

[In]

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

[Out]

(1 - Log[5])*Defer[Int][Log[(-1 + x)*Log[x]]^(-2), x] + (1 - Log[5])*Defer[Int][1/((-1 + x)*Log[(-1 + x)*Log[x
]]^2), x] - Defer[Int][E^(-E^(E^5*x) + x)/((-1 + x)*Log[(-1 + x)*Log[x]]^2), x] - Log[5]*Defer[Int][x/Log[(-1
+ x)*Log[x]]^2, x] + Defer[Int][1/(Log[x]*Log[(-1 + x)*Log[x]]^2), x] - Defer[Int][E^(-E^(E^5*x) + x)/(x*Log[x
]*Log[(-1 + x)*Log[x]]^2), x] - Log[5]*Defer[Int][x/(Log[x]*Log[(-1 + x)*Log[x]]^2), x] - Defer[Int][Log[(-1 +
x)*Log[x]]^(-1), x] + Defer[Int][E^(-E^(E^5*x) + x)/Log[(-1 + x)*Log[x]], x] - Defer[Int][E^(5 - E^(E^5*x) +
(1 + E^5)*x)/Log[(-1 + x)*Log[x]], x] + Log[25]*Defer[Int][x/Log[(-1 + x)*Log[x]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{(-1+x) x \log (x) \log ^2((-1+x) \log (x))} \, dx \\ & = \int \left (-\frac {x (-1+x \log (5))}{(-1+x) \log ^2((-1+x) \log (x))}-\frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}+\frac {x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}-\frac {x \log (5)}{\log (x) \log ^2((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) x \log (x) \log ^2((-1+x) \log (x))}-\frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ((-1+x) \log (x))}+\frac {-1+x \log (25)}{\log ((-1+x) \log (x))}\right ) \, dx \\ & = -\left (\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx\right )-\int \frac {x (-1+x \log (5))}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ((-1+x) \log (x))} \, dx+\int \frac {-1+x \log (25)}{\log ((-1+x) \log (x))} \, dx \\ & = -\left (\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx\right )-\int \left (-\frac {1-\log (5)}{\log ^2((-1+x) \log (x))}+\frac {-1+\log (5)}{(-1+x) \log ^2((-1+x) \log (x))}+\frac {x \log (5)}{\log ^2((-1+x) \log (x))}\right ) \, dx+\int \left (\frac {1}{\log (x) \log ^2((-1+x) \log (x))}+\frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx-\int \left (\frac {e^{-e^{e^5 x}+x} (1-x-x \log (x))}{x \log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx+\int \left (\frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))}\right ) \, dx+\int \left (-\frac {1}{\log ((-1+x) \log (x))}+\frac {x \log (25)}{\log ((-1+x) \log (x))}\right ) \, dx-\int \frac {1}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ((-1+x) \log (x))} \, dx \\ & = -\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} (1-x-x \log (x))}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} (-1+x+x \log (x))}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx \\ & = -\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx-\int \left (-\frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx-\int \left (\frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ^2((-1+x) \log (x))}-\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx \\ & = -\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x} x}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx \\ & = -\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx-\int \left (\frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ^2((-1+x) \log (x))}\right ) \, dx-\int \left (\frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))}+\frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))}\right ) \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ^2((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx \\ & = -\left ((-1+\log (5)) \int \frac {1}{\log ^2((-1+x) \log (x))} \, dx\right )-(-1+\log (5)) \int \frac {1}{(-1+x) \log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log ^2((-1+x) \log (x))} \, dx-\log (5) \int \frac {x}{\log (x) \log ^2((-1+x) \log (x))} \, dx+\log (25) \int \frac {x}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{(-1+x) \log ^2((-1+x) \log (x))} \, dx+\int \frac {1}{\log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {e^{-e^{e^5 x}+x}}{x \log (x) \log ^2((-1+x) \log (x))} \, dx-\int \frac {1}{\log ((-1+x) \log (x))} \, dx+\int \frac {e^{-e^{e^5 x}+x}}{\log ((-1+x) \log (x))} \, dx-\int \frac {e^{5-e^{e^5 x}+\left (1+e^5\right ) x}}{\log ((-1+x) \log (x))} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\frac {e^{-e^{e^5 x}+x}-x+x^2 \log (5)}{\log ((-1+x) \log (x))} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 186.65 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94

method	result	size
parallelrisch	\(\frac {x^{2} \ln \left (5\right )+{\mathrm e}^{x -{\mathrm e}^{x \,{\mathrm e}^{5}}}-x}{\ln \left (\left (-1+x \right ) \ln \left (x \right )\right )}\)	\(31\)
risch	\(\frac {2 i \left (x^{2} \ln \left (5\right )+{\mathrm e}^{x -{\mathrm e}^{x \,{\mathrm e}^{5}}}-x \right )}{\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left (-1+x \right )\right )-\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left (-1+x \right )\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left (-1+x \right )\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x \right ) \left (-1+x \right )\right )^{3}+2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (-1+x \right )}\)	\(118\)

[In]

int(((((-x^2+x)*exp(5)*exp(x*exp(5))+x^2-x)*ln(x)*exp(x-exp(x*exp(5)))+((2*x^3-2*x^2)*ln(5)-x^2+x)*ln(x))*ln((
-1+x)*ln(x))+(-x*ln(x)-x+1)*exp(x-exp(x*exp(5)))+(-x^3*ln(5)+x^2)*ln(x)+(-x^3+x^2)*ln(5)+x^2-x)/(x^2-x)/ln(x)/
ln((-1+x)*ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

(x^2*ln(5)+exp(x-exp(x*exp(5)))-x)/ln((-1+x)*ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\frac {x^{2} \log \left (5\right ) - x + e^{\left ({\left (x e^{5} - e^{\left (x e^{5} + 5\right )}\right )} e^{\left (-5\right )}\right )}}{\log \left ({\left (x - 1\right )} \log \left (x\right )\right )} \]

[In]

integrate(((((-x^2+x)*exp(5)*exp(x*exp(5))+x^2-x)*log(x)*exp(x-exp(x*exp(5)))+((2*x^3-2*x^2)*log(5)-x^2+x)*log
(x))*log((-1+x)*log(x))+(-x*log(x)-x+1)*exp(x-exp(x*exp(5)))+(-x^3*log(5)+x^2)*log(x)+(-x^3+x^2)*log(5)+x^2-x)
/(x^2-x)/log(x)/log((-1+x)*log(x))^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\frac {x^{2} \log {\left (5 \right )} - x}{\log {\left (\left (x - 1\right ) \log {\left (x \right )} \right )}} + \frac {e^{x - e^{x e^{5}}}}{\log {\left (\left (x - 1\right ) \log {\left (x \right )} \right )}} \]

[In]

integrate(((((-x**2+x)*exp(5)*exp(x*exp(5))+x**2-x)*ln(x)*exp(x-exp(x*exp(5)))+((2*x**3-2*x**2)*ln(5)-x**2+x)*
ln(x))*ln((-1+x)*ln(x))+(-x*ln(x)-x+1)*exp(x-exp(x*exp(5)))+(-x**3*ln(5)+x**2)*ln(x)+(-x**3+x**2)*ln(5)+x**2-x
)/(x**2-x)/ln(x)/ln((-1+x)*ln(x))**2,x)

[Out]

(x**2*log(5) - x)/log((x - 1)*log(x)) + exp(x - exp(x*exp(5)))/log((x - 1)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\frac {x^{2} \log \left (5\right ) - x + e^{\left (x - e^{\left (x e^{5}\right )}\right )}}{\log \left (x - 1\right ) + \log \left (\log \left (x\right )\right )} \]

[In]

integrate(((((-x^2+x)*exp(5)*exp(x*exp(5))+x^2-x)*log(x)*exp(x-exp(x*exp(5)))+((2*x^3-2*x^2)*log(5)-x^2+x)*log
(x))*log((-1+x)*log(x))+(-x*log(x)-x+1)*exp(x-exp(x*exp(5)))+(-x^3*log(5)+x^2)*log(x)+(-x^3+x^2)*log(5)+x^2-x)
/(x^2-x)/log(x)/log((-1+x)*log(x))^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\frac {x^{2} \log \left (5\right ) - x + e^{\left (x - e^{\left (x e^{5}\right )}\right )}}{\log \left (x \log \left (x\right ) - \log \left (x\right )\right )} \]

[In]

integrate(((((-x^2+x)*exp(5)*exp(x*exp(5))+x^2-x)*log(x)*exp(x-exp(x*exp(5)))+((2*x^3-2*x^2)*log(5)-x^2+x)*log
(x))*log((-1+x)*log(x))+(-x*log(x)-x+1)*exp(x-exp(x*exp(5)))+(-x^3*log(5)+x^2)*log(x)+(-x^3+x^2)*log(5)+x^2-x)
/(x^2-x)/log(x)/log((-1+x)*log(x))^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.18 \[ \int \frac {-x+x^2+\left (x^2-x^3\right ) \log (5)+\left (x^2-x^3 \log (5)\right ) \log (x)+e^{-e^{e^5 x}+x} (1-x-x \log (x))+\left (e^{-e^{e^5 x}+x} \left (-x+x^2+e^{5+e^5 x} \left (x-x^2\right )\right ) \log (x)+\left (x-x^2+\left (-2 x^2+2 x^3\right ) \log (5)\right ) \log (x)\right ) \log ((-1+x) \log (x))}{\left (-x+x^2\right ) \log (x) \log ^2((-1+x) \log (x))} \, dx=\frac {{\mathrm {e}}^{x-{\mathrm {e}}^{x\,{\mathrm {e}}^5}}}{\ln \left (\ln \left (x\right )\,\left (x-1\right )\right )}-x\,\left (\ln \left (25\right )+1\right )+2\,x^2\,\ln \left (5\right )+\frac {x\,\left (x\,\ln \left (5\right )-1\right )-\frac {x\,\ln \left (\ln \left (x\right )\,\left (x-1\right )\right )\,\ln \left (x\right )\,\left (2\,x\,\ln \left (5\right )-1\right )\,\left (x-1\right )}{x+x\,\ln \left (x\right )-1}}{\ln \left (\ln \left (x\right )\,\left (x-1\right )\right )}-\frac {x+2\,x\,\ln \left (5\right )-2\,x^2\,\ln \left (5\right )-2\,x^3\,\ln \left (5\right )+2\,x^4\,\ln \left (5\right )+x^2-x^3-1}{\left (x+1\right )\,\left (x+x\,\ln \left (x\right )-1\right )} \]

[In]

int((x + exp(x - exp(x*exp(5)))*(x + x*log(x) - 1) + log(log(x)*(x - 1))*(log(x)*(log(5)*(2*x^2 - 2*x^3) - x +
x^2) - exp(x - exp(x*exp(5)))*log(x)*(x^2 - x + exp(5)*exp(x*exp(5))*(x - x^2))) - log(5)*(x^2 - x^3) + log(x
)*(x^3*log(5) - x^2) - x^2)/(log(log(x)*(x - 1))^2*log(x)*(x - x^2)),x)

[Out]

exp(x - exp(x*exp(5)))/log(log(x)*(x - 1)) - x*(log(25) + 1) + 2*x^2*log(5) + (x*(x*log(5) - 1) - (x*log(log(x
)*(x - 1))*log(x)*(2*x*log(5) - 1)*(x - 1))/(x + x*log(x) - 1))/log(log(x)*(x - 1)) - (x + 2*x*log(5) - 2*x^2*
log(5) - 2*x^3*log(5) + 2*x^4*log(5) + x^2 - x^3 - 1)/((x + 1)*(x + x*log(x) - 1))