3.65.6 \(\int \frac {\log (x) \log ^2(\log (x))+e^{\frac {3 x}{\log (\log (x))}} (-3+3 \log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=12 \[ e^{\frac {3 x}{\log (\log (x))}}+x \]

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Rubi [A]  time = 0.38, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6742, 6706} \begin {gather*} x+e^{\frac {3 x}{\log (\log (x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^((3*x)/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 {3 e^{\frac {3 x}{\log (\log (x))}} (-1+\log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))}\right ) \, dx\\ &=x+3 \int \frac {e^{\frac {3 x}{\log (\log (x))}} (-1+\log (x) \log (\log (x)))}{\log (x) \log ^2(\log (x))} \, dx\\ &=e^{\frac {3 x}{\log (\log (x))}}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 12, normalized size = 1.00 \begin {gather*} e^{\frac {3 x}{\log (\log (x))}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^((3*x)/Log[Log[x]]) + x

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fricas [A]  time = 0.81, size = 11, normalized size = 0.92 \begin {gather*} x + e^{\left (\frac {3 \, x}{\log \left (\log \relax (x)\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(x)*log(log(x))-3)*exp(3/2*x/log(log(x)))^2+log(x)*log(log(x))^2)/log(x)/log(log(x))^2,x, alg
orithm="fricas")

[Out]

x + e^(3*x/log(log(x)))

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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(((3*log(x)*log(log(x))-3)*exp(3/2*x/log(log(x)))^2+log(x)*log(log(x))^2)/log(x)/log(log(x))^2,x, alg
orithm="giac")

[Out]

undef

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maple [A]  time = 0.06, size = 12, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{\frac {3 x}{\ln \left (\ln \relax (x )\right )}}+x\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(3*x/ln(ln(x)))+x

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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(((3*log(x)*log(log(x))-3)*exp(3/2*x/log(log(x)))^2+log(x)*log(log(x))^2)/log(x)/log(log(x))^2,x, alg
orithm="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

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mupad [B]  time = 4.24, size = 11, normalized size = 0.92 \begin {gather*} x+{\mathrm {e}}^{\frac {3\,x}{\ln \left (\ln \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp((3*x)/log(log(x)))

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sympy [A]  time = 0.48, size = 10, normalized size = 0.83 \begin {gather*} x + e^{\frac {3 x}{\log {\left (\log {\relax (x )} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(3*x/log(log(x)))

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