3.39.54 \(\int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}} (-1+(1-3 x) \log (4 x))}{\log ^2(4 x)} \, dx\)

Optimal. Leaf size=19 \[ 3+e^{\frac {e^{-3-3 x} x}{\log (4 x)}} \]

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Rubi [F]  time = 1.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}} (-1+(1-3 x) \log (4 x))}{\log ^2(4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-3 - 3*x + (E^(-3 - 3*x)*x)/Log[4*x])*(-1 + (1 - 3*x)*Log[4*x]))/Log[4*x]^2,x]

[Out]

-Defer[Int][E^(-3 - 3*x + (E^(-3 - 3*x)*x)/Log[4*x])/Log[4*x]^2, x] + Defer[Int][E^(-3 - 3*x + (E^(-3 - 3*x)*x
)/Log[4*x])/Log[4*x], x] - 3*Defer[Int][(E^(-3 - 3*x + (E^(-3 - 3*x)*x)/Log[4*x])*x)/Log[4*x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}}}{\log ^2(4 x)}+\frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}} (1-3 x)}{\log (4 x)}\right ) \, dx\\ &=-\int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}}}{\log ^2(4 x)} \, dx+\int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}} (1-3 x)}{\log (4 x)} \, dx\\ &=\int \left (\frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}}}{\log (4 x)}-\frac {3 e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}} x}{\log (4 x)}\right ) \, dx-\int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}}}{\log ^2(4 x)} \, dx\\ &=-\left (3 \int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}} x}{\log (4 x)} \, dx\right )-\int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}}}{\log ^2(4 x)} \, dx+\int \frac {e^{-3-3 x+\frac {e^{-3-3 x} x}{\log (4 x)}}}{\log (4 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.86, size = 17, normalized size = 0.89 \begin {gather*} e^{\frac {e^{-3-3 x} x}{\log (4 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-3 - 3*x + (E^(-3 - 3*x)*x)/Log[4*x])*(-1 + (1 - 3*x)*Log[4*x]))/Log[4*x]^2,x]

[Out]

E^((E^(-3 - 3*x)*x)/Log[4*x])

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fricas [A]  time = 0.54, size = 19, normalized size = 1.00 \begin {gather*} e^{\left (e^{\left (-3 \, x + \log \relax (x) - \log \left (2 \, \log \relax (2) + \log \relax (x)\right ) - 3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+1)*log(4*x)-1)*exp(-log(log(4*x))+log(x)-3*x-3)*exp(exp(-log(log(4*x))+log(x)-3*x-3))/x/log(4
*x),x, algorithm="fricas")

[Out]

e^(e^(-3*x + log(x) - log(2*log(2) + log(x)) - 3))

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giac [A]  time = 0.14, size = 25, normalized size = 1.32 \begin {gather*} e^{\left (\frac {x}{2 \, e^{\left (3 \, x + 3\right )} \log \relax (2) + e^{\left (3 \, x + 3\right )} \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+1)*log(4*x)-1)*exp(-log(log(4*x))+log(x)-3*x-3)*exp(exp(-log(log(4*x))+log(x)-3*x-3))/x/log(4
*x),x, algorithm="giac")

[Out]

e^(x/(2*e^(3*x + 3)*log(2) + e^(3*x + 3)*log(x)))

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maple [A]  time = 0.08, size = 19, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{\frac {x \,{\mathrm e}^{-3 x -3}}{\ln \relax (x )+2 \ln \relax (2)}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x+1)*ln(4*x)-1)*exp(-ln(ln(4*x))+ln(x)-3*x-3)*exp(exp(-ln(ln(4*x))+ln(x)-3*x-3))/x/ln(4*x),x,method=_
RETURNVERBOSE)

[Out]

exp(1/(ln(x)+2*ln(2))*x*exp(-3*x-3))

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maxima [A]  time = 0.87, size = 25, normalized size = 1.32 \begin {gather*} e^{\left (\frac {x}{2 \, e^{\left (3 \, x + 3\right )} \log \relax (2) + e^{\left (3 \, x + 3\right )} \log \relax (x)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+1)*log(4*x)-1)*exp(-log(log(4*x))+log(x)-3*x-3)*exp(exp(-log(log(4*x))+log(x)-3*x-3))/x/log(4
*x),x, algorithm="maxima")

[Out]

e^(x/(2*e^(3*x + 3)*log(2) + e^(3*x + 3)*log(x)))

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mupad [B]  time = 2.40, size = 15, normalized size = 0.79 \begin {gather*} {\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{-3\,x}\,{\mathrm {e}}^{-3}}{\ln \left (4\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(x) - log(log(4*x)) - 3*x - 3)*exp(exp(log(x) - log(log(4*x)) - 3*x - 3))*(log(4*x)*(3*x - 1) + 1
))/(x*log(4*x)),x)

[Out]

exp((x*exp(-3*x)*exp(-3))/log(4*x))

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sympy [A]  time = 0.51, size = 15, normalized size = 0.79 \begin {gather*} e^{\frac {x e^{- 3 x - 3}}{\log {\left (4 x \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+1)*ln(4*x)-1)*exp(-ln(ln(4*x))+ln(x)-3*x-3)*exp(exp(-ln(ln(4*x))+ln(x)-3*x-3))/x/ln(4*x),x)

[Out]

exp(x*exp(-3*x - 3)/log(4*x))

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