\(\int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx\) [1820]

Optimal result

Integrand size = 44, antiderivative size = 18 \[ \int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx=-41+e^5+e^{\frac {2 x^2}{\log (\log (x))}}+x \]

[Out]

exp(5)+x-41+exp(x^2/ln(ln(x)))^2

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6874, 6838} \[ \int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx=e^{\frac {2 x^2}{\log (\log (x))}}+x \]

[In]

Int[(Log[x]*Log[Log[x]]^2 + E^((2*x^2)/Log[Log[x]])*(-2*x + 4*x*Log[x]*Log[Log[x]]))/(Log[x]*Log[Log[x]]^2),x]

[Out]

E^((2*x^2)/Log[Log[x]]) + x

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx=e^{\frac {2 x^2}{\log (\log (x))}}+x \]

[In]

Integrate[(Log[x]*Log[Log[x]]^2 + E^((2*x^2)/Log[Log[x]])*(-2*x + 4*x*Log[x]*Log[Log[x]]))/(Log[x]*Log[Log[x]]
^2),x]

[Out]

E^((2*x^2)/Log[Log[x]]) + x

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78

method	result	size
risch	\(x +{\mathrm e}^{\frac {2 x^{2}}{\ln \left (\ln \left (x \right )\right )}}\)	\(14\)
default	\(x +{\mathrm e}^{\frac {2 x^{2}}{\ln \left (\ln \left (x \right )\right )}}\)	\(15\)
parallelrisch	\(x +{\mathrm e}^{\frac {2 x^{2}}{\ln \left (\ln \left (x \right )\right )}}\)	\(15\)
parts	\(x +{\mathrm e}^{\frac {2 x^{2}}{\ln \left (\ln \left (x \right )\right )}}\)	\(15\)

[In]

int(((4*x*ln(x)*ln(ln(x))-2*x)*exp(x^2/ln(ln(x)))^2+ln(x)*ln(ln(x))^2)/ln(x)/ln(ln(x))^2,x,method=_RETURNVERBO
SE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx=x + e^{\left (\frac {2 \, x^{2}}{\log \left (\log \left (x\right )\right )}\right )} \]

[In]

integrate(((4*x*log(x)*log(log(x))-2*x)*exp(x^2/log(log(x)))^2+log(x)*log(log(x))^2)/log(x)/log(log(x))^2,x, a
lgorithm="fricas")

[Out]

x + e^(2*x^2/log(log(x)))

Sympy [F(-2)]

Exception generated. \[ \int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((4*x*ln(x)*ln(ln(x))-2*x)*exp(x**2/ln(ln(x)))**2+ln(x)*ln(ln(x))**2)/ln(x)/ln(ln(x))**2,x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(((4*x*log(x)*log(log(x))-2*x)*exp(x^2/log(log(x)))^2+log(x)*log(log(x))^2)/log(x)/log(log(x))^2,x, a
lgorithm="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

Giac [F]

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

[In]

integrate(((4*x*log(x)*log(log(x))-2*x)*exp(x^2/log(log(x)))^2+log(x)*log(log(x))^2)/log(x)/log(log(x))^2,x, a
lgorithm="giac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 8.87 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {\log (x) \log ^2(\log (x))+e^{\frac {2 x^2}{\log (\log (x))}} (-2 x+4 x \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx=x+{\mathrm {e}}^{\frac {2\,x^2}{\ln \left (\ln \left (x\right )\right )}} \]

[In]

int(-(exp((2*x^2)/log(log(x)))*(2*x - 4*x*log(log(x))*log(x)) - log(log(x))^2*log(x))/(log(log(x))^2*log(x)),x
)

[Out]

x + exp((2*x^2)/log(log(x)))