\(\int \frac {((40-5 x-20 x^2) \log (x)+(-10+5 x^2) \log ^2(x)) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} (x^2-x^2 \log (x) \log (\log (x))+(-2+x^2) \log (x) \log ^2(\log (x)))}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+(-20 x^2 \log (x)+5 x^2 \log ^2(x)) \log ^2(\log (x))} \, dx\) [8019]

Optimal result

Integrand size = 117, antiderivative size = 27 \[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=\frac {2}{x}+x-\log \left (-4+\frac {1}{5} e^{\frac {x}{\log (\log (x))}}+\log (x)\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=\int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx \]

[In]

Int[(((40 - 5*x - 20*x^2)*Log[x] + (-10 + 5*x^2)*Log[x]^2)*Log[Log[x]]^2 + E^(x/Log[Log[x]])*(x^2 - x^2*Log[x]
*Log[Log[x]] + (-2 + x^2)*Log[x]*Log[Log[x]]^2))/(E^(x/Log[Log[x]])*x^2*Log[x]*Log[Log[x]]^2 + (-20*x^2*Log[x]
+ 5*x^2*Log[x]^2)*Log[Log[x]]^2),x]

[Out]

2/x + x - 5*Defer[Int][1/(x*(-20 + E^(x/Log[Log[x]]) + 5*Log[x])), x] + Defer[Int][1/(Log[x]*Log[Log[x]]^2), x
] - 5*Defer[Int][1/((-20 + E^(x/Log[Log[x]]) + 5*Log[x])*Log[Log[x]]^2), x] + 20*Defer[Int][1/(Log[x]*(-20 + E
^(x/Log[Log[x]]) + 5*Log[x])*Log[Log[x]]^2), x] - Defer[Int][Log[Log[x]]^(-1), x] - 20*Defer[Int][1/((-20 + E^
(x/Log[Log[x]]) + 5*Log[x])*Log[Log[x]]), x] + 5*Defer[Int][Log[x]/((-20 + E^(x/Log[Log[x]]) + 5*Log[x])*Log[L
og[x]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-\left (\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))\right )-e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{x^2 \left (20-e^{\frac {x}{\log (\log (x))}}-5 \log (x)\right ) \log (x) \log ^2(\log (x))} \, dx \\ & = \int \left (\frac {5 \left (4 x-x \log (x)-4 x \log (x) \log (\log (x))+x \log ^2(x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{x \log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))}+\frac {x^2-x^2 \log (x) \log (\log (x))-2 \log (x) \log ^2(\log (x))+x^2 \log (x) \log ^2(\log (x))}{x^2 \log (x) \log ^2(\log (x))}\right ) \, dx \\ & = 5 \int \frac {4 x-x \log (x)-4 x \log (x) \log (\log (x))+x \log ^2(x) \log (\log (x))-\log (x) \log ^2(\log (x))}{x \log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))} \, dx+\int \frac {x^2-x^2 \log (x) \log (\log (x))-2 \log (x) \log ^2(\log (x))+x^2 \log (x) \log ^2(\log (x))}{x^2 \log (x) \log ^2(\log (x))} \, dx \\ & = 5 \int \left (-\frac {1}{x \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right )}-\frac {1}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))}+\frac {4}{\log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))}-\frac {4}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))}+\frac {\log (x)}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))}\right ) \, dx+\int \left (1-\frac {2}{x^2}+\frac {1}{\log (x) \log ^2(\log (x))}-\frac {1}{\log (\log (x))}\right ) \, dx \\ & = \frac {2}{x}+x-5 \int \frac {1}{x \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right )} \, dx-5 \int \frac {1}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))} \, dx+5 \int \frac {\log (x)}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))} \, dx+20 \int \frac {1}{\log (x) \left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log ^2(\log (x))} \, dx-20 \int \frac {1}{\left (-20+e^{\frac {x}{\log (\log (x))}}+5 \log (x)\right ) \log (\log (x))} \, dx+\int \frac {1}{\log (x) \log ^2(\log (x))} \, dx-\int \frac {1}{\log (\log (x))} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=\frac {2}{x}+x-\log \left (20-e^{\frac {x}{\log (\log (x))}}-5 \log (x)\right ) \]

[In]

Integrate[(((40 - 5*x - 20*x^2)*Log[x] + (-10 + 5*x^2)*Log[x]^2)*Log[Log[x]]^2 + E^(x/Log[Log[x]])*(x^2 - x^2*
Log[x]*Log[Log[x]] + (-2 + x^2)*Log[x]*Log[Log[x]]^2))/(E^(x/Log[Log[x]])*x^2*Log[x]*Log[Log[x]]^2 + (-20*x^2*
Log[x] + 5*x^2*Log[x]^2)*Log[Log[x]]^2),x]

[Out]

2/x + x - Log[20 - E^(x/Log[Log[x]]) - 5*Log[x]]

Maple [A] (verified)

Time = 6.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {x^{2}+2}{x}-\ln \left (5 \ln \left (x \right )+{\mathrm e}^{\frac {x}{\ln \left (\ln \left (x \right )\right )}}-20\right )\)	\(28\)
parallelrisch	\(-\frac {\ln \left (\ln \left (x \right )-4+\frac {{\mathrm e}^{\frac {x}{\ln \left (\ln \left (x \right )\right )}}}{5}\right ) x -x^{2}-2}{x}\)	\(30\)

[In]

int((((x^2-2)*ln(x)*ln(ln(x))^2-x^2*ln(x)*ln(ln(x))+x^2)*exp(x/ln(ln(x)))+((5*x^2-10)*ln(x)^2+(-20*x^2-5*x+40)
*ln(x))*ln(ln(x))^2)/(x^2*ln(x)*ln(ln(x))^2*exp(x/ln(ln(x)))+(5*x^2*ln(x)^2-20*x^2*ln(x))*ln(ln(x))^2),x,metho
d=_RETURNVERBOSE)

[Out]

(x^2+2)/x-ln(5*ln(x)+exp(x/ln(ln(x)))-20)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=\frac {x^{2} - x \log \left (e^{\frac {x}{\log \left (\log \left (x\right )\right )}} + 5 \, \log \left (x\right ) - 20\right ) + 2}{x} \]

[In]

integrate((((x^2-2)*log(x)*log(log(x))^2-x^2*log(x)*log(log(x))+x^2)*exp(x/log(log(x)))+((5*x^2-10)*log(x)^2+(
-20*x^2-5*x+40)*log(x))*log(log(x))^2)/(x^2*log(x)*log(log(x))^2*exp(x/log(log(x)))+(5*x^2*log(x)^2-20*x^2*log
(x))*log(log(x))^2),x, algorithm="fricas")

[Out]

(x^2 - x*log(e^(x/log(log(x))) + 5*log(x) - 20) + 2)/x

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=x - \log {\left (e^{\frac {x}{\log {\left (\log {\left (x \right )} \right )}}} + 5 \log {\left (x \right )} - 20 \right )} + \frac {2}{x} \]

[In]

integrate((((x**2-2)*ln(x)*ln(ln(x))**2-x**2*ln(x)*ln(ln(x))+x**2)*exp(x/ln(ln(x)))+((5*x**2-10)*ln(x)**2+(-20
*x**2-5*x+40)*ln(x))*ln(ln(x))**2)/(x**2*ln(x)*ln(ln(x))**2*exp(x/ln(ln(x)))+(5*x**2*ln(x)**2-20*x**2*ln(x))*l
n(ln(x))**2),x)

[Out]

x - log(exp(x/log(log(x))) + 5*log(x) - 20) + 2/x

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=\frac {x^{2} + 2}{x} - \log \left (e^{\frac {x}{\log \left (\log \left (x\right )\right )}} + 5 \, \log \left (x\right ) - 20\right ) \]

[In]

integrate((((x^2-2)*log(x)*log(log(x))^2-x^2*log(x)*log(log(x))+x^2)*exp(x/log(log(x)))+((5*x^2-10)*log(x)^2+(
-20*x^2-5*x+40)*log(x))*log(log(x))^2)/(x^2*log(x)*log(log(x))^2*exp(x/log(log(x)))+(5*x^2*log(x)^2-20*x^2*log
(x))*log(log(x))^2),x, algorithm="maxima")

[Out]

(x^2 + 2)/x - log(e^(x/log(log(x))) + 5*log(x) - 20)

Giac [F]

\[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=\int { \frac {5 \, {\left ({\left (x^{2} - 2\right )} \log \left (x\right )^{2} - {\left (4 \, x^{2} + x - 8\right )} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{2} - {\left (x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - {\left (x^{2} - 2\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - x^{2}\right )} e^{\frac {x}{\log \left (\log \left (x\right )\right )}}}{x^{2} e^{\frac {x}{\log \left (\log \left (x\right )\right )}} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 5 \, {\left (x^{2} \log \left (x\right )^{2} - 4 \, x^{2} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{2}} \,d x } \]

[In]

integrate((((x^2-2)*log(x)*log(log(x))^2-x^2*log(x)*log(log(x))+x^2)*exp(x/log(log(x)))+((5*x^2-10)*log(x)^2+(
-20*x^2-5*x+40)*log(x))*log(log(x))^2)/(x^2*log(x)*log(log(x))^2*exp(x/log(log(x)))+(5*x^2*log(x)^2-20*x^2*log
(x))*log(log(x))^2),x, algorithm="giac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 13.85 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (\left (40-5 x-20 x^2\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)\right ) \log ^2(\log (x))+e^{\frac {x}{\log (\log (x))}} \left (x^2-x^2 \log (x) \log (\log (x))+\left (-2+x^2\right ) \log (x) \log ^2(\log (x))\right )}{e^{\frac {x}{\log (\log (x))}} x^2 \log (x) \log ^2(\log (x))+\left (-20 x^2 \log (x)+5 x^2 \log ^2(x)\right ) \log ^2(\log (x))} \, dx=x-\ln \left (5\,\ln \left (x\right )+{\mathrm {e}}^{\frac {x}{\ln \left (\ln \left (x\right )\right )}}-20\right )+\frac {2}{x} \]

[In]

int(-(exp(x/log(log(x)))*(x^2 + log(log(x))^2*log(x)*(x^2 - 2) - x^2*log(log(x))*log(x)) + log(log(x))^2*(log(
x)^2*(5*x^2 - 10) - log(x)*(5*x + 20*x^2 - 40)))/(log(log(x))^2*(20*x^2*log(x) - 5*x^2*log(x)^2) - x^2*log(log
(x))^2*exp(x/log(log(x)))*log(x)),x)

[Out]

x - log(5*log(x) + exp(x/log(log(x))) - 20) + 2/x