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

Optimal. Leaf size=18 \[ -41+e^5+e^{\frac {2 x^2}{\log (\log (x))}}+x \]

________________________________________________________________________________________

Rubi [A]  time = 0.61, antiderivative size = 14, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6742, 6706} \begin {gather*} e^{\frac {2 x^2}{\log (\log (x))}}+x \end {gather*}

Antiderivative was successfully verified.

[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 6706

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 14, normalized size = 0.78 \begin {gather*} e^{\frac {2 x^2}{\log (\log (x))}}+x \end {gather*}

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

fricas [A]  time = 0.76, size = 13, normalized size = 0.72 \begin {gather*} x + e^{\left (\frac {2 \, x^{2}}{\log \left (\log \relax (x)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

________________________________________________________________________________________

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(((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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 14, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

[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)))

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 1.20, size = 13, normalized size = 0.72 \begin {gather*} x+{\mathrm {e}}^{\frac {2\,x^2}{\ln \left (\ln \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

________________________________________________________________________________________

sympy [A]  time = 0.51, size = 12, normalized size = 0.67 \begin {gather*} x + e^{\frac {2 x^{2}}{\log {\left (\log {\relax (x )} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

x + exp(2*x**2/log(log(x)))

________________________________________________________________________________________