\(\int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx\) [3813]

Optimal result

Integrand size = 40, antiderivative size = 23 \[ \int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=e^{10} \left (-2 x+\frac {3}{\log (4)}+\frac {x^2}{\log (\log (x))}\right ) \]

[Out]

exp(5)^2*(3/2/ln(2)-2*x+x^2/ln(ln(x)))

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-2 e^{10} x+\frac {e^{10} x^2}{\log (\log (x))} \]

[In]

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

[Out]

-2*E^10*x + (E^10*x^2)/Log[Log[x]]

Maple [A] (verified)

Time = 4.55 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

method	result	size
risch	\(-2 x \,{\mathrm e}^{10}+\frac {x^{2} {\mathrm e}^{10}}{\ln \left (\ln \left (x \right )\right )}\)	\(18\)
norman	\(\frac {x^{2} {\mathrm e}^{10}-2 \,{\mathrm e}^{10} x \ln \left (\ln \left (x \right )\right )}{\ln \left (\ln \left (x \right )\right )}\)	\(26\)
parallelrisch	\(\frac {x^{2} {\mathrm e}^{10}-2 \,{\mathrm e}^{10} x \ln \left (\ln \left (x \right )\right )}{\ln \left (\ln \left (x \right )\right )}\)	\(26\)

[In]

int((-2*exp(5)^2*ln(x)*ln(ln(x))^2+2*x*exp(5)^2*ln(x)*ln(ln(x))-x*exp(5)^2)/ln(x)/ln(ln(x))^2,x,method=_RETURN
VERBOSE)

[Out]

-2*x*exp(10)+x^2*exp(10)/ln(ln(x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=\frac {x^{2} e^{10} - 2 \, x e^{10} \log \left (\log \left (x\right )\right )}{\log \left (\log \left (x\right )\right )} \]

[In]

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

[Out]

(x^2*e^10 - 2*x*e^10*log(log(x)))/log(log(x))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=\frac {x^{2} e^{10}}{\log {\left (\log {\left (x \right )} \right )}} - 2 x e^{10} \]

[In]

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

[Out]

x**2*exp(10)/log(log(x)) - 2*x*exp(10)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-2 \, x e^{10} + \frac {x^{2} e^{10}}{\log \left (\log \left (x\right )\right )} \]

[In]

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

[Out]

-2*x*e^10 + x^2*e^10/log(log(x))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=-2 \, x e^{10} + \frac {x^{2} e^{10}}{\log \left (\log \left (x\right )\right )} \]

[In]

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

[Out]

-2*x*e^10 + x^2*e^10/log(log(x))

Mupad [B] (verification not implemented)

Time = 9.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-e^{10} x+2 e^{10} x \log (x) \log (\log (x))-2 e^{10} \log (x) \log ^2(\log (x))}{\log (x) \log ^2(\log (x))} \, dx=\frac {x^2\,{\mathrm {e}}^{10}}{\ln \left (\ln \left (x\right )\right )}-2\,x\,{\mathrm {e}}^{10} \]

[In]

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

[Out]

(x^2*exp(10))/log(log(x)) - 2*x*exp(10)