3.33.84 $\int \frac{-e^{\frac{4}{x^4}}{x}^5+e^{\frac{4}{x^4}}(-48-16x)\log (6+2x)\log (\log (6+2x))}{(3x^5+x^6)\log (6+2x){\log }^2(\log (6+2x))}dx$

Optimal. Leaf size=17 $$\frac{e^{\frac{4}{x^4}}}{\log (\log (6+2x))}$$

________________________________________________________________________________________

Rubi [F]  time = 12.18, 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 = {} $$\begin{array}{c}\int \frac{-e^{\frac{4}{x^4}}{x}^5+e^{\frac{4}{x^4}}(-48-16x)\log (6+2x)\log (\log (6+2x))}{(3x^5+x^6)\log (6+2x){\log }^2(\log (6+2x))}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-(E^(4/x^4)*x^5) + E^(4/x^4)*(-48 - 16*x)*Log[6 + 2*x]*Log[Log[6 + 2*x]])/((3*x^5 + x^6)*Log[6 + 2*x]*Log
[Log[6 + 2*x]]^2),x]

[Out]

Defer[Int][E^(4/x^4)/((-3 - x)*Log[6 + 2*x]*Log[Log[6 + 2*x]]^2), x] + (16*Defer[Int][E^(4/x^4)/((-3 - x)*Log[
Log[6 + 2*x]]), x])/81 - 16*Defer[Int][E^(4/x^4)/(x^5*Log[Log[6 + 2*x]]), x] + (16*Defer[Int][E^(4/x^4)/((3 +
x)*Log[Log[6 + 2*x]]), x])/81

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-e^{\frac{4}{x^4}}{x}^5+e^{\frac{4}{x^4}}(-48-16x)\log (6+2x)\log (\log (6+2x))}{x^5(3+x)\log (6+2x){\log }^2(\log (6+2x))}dx\\ & =\int (\frac{e^{\frac{4}{x^4}}(-x^5-48\log (2(3+x))\log (\log (2(3+x)))-16x\log (2(3+x))\log (\log (2(3+x))))}{3x^5\log (6+2x){\log }^2(\log (6+2x))}+\frac{e^{\frac{4}{x^4}}(-x^5-48\log (2(3+x))\log (\log (2(3+x)))-16x\log (2(3+x))\log (\log (2(3+x))))}{27x^3\log (6+2x){\log }^2(\log (6+2x))}+\frac{e^{\frac{4}{x^4}}(-x^5-48\log (2(3+x))\log (\log (2(3+x)))-16x\log (2(3+x))\log (\log (2(3+x))))}{243x\log (6+2x){\log }^2(\log (6+2x))}+\frac{e^{\frac{4}{x^4}}(x^5+48\log (2(3+x))\log (\log (2(3+x)))+16x\log (2(3+x))\log (\log (2(3+x))))}{9x^4\log (6+2x){\log }^2(\log (6+2x))}+\frac{e^{\frac{4}{x^4}}(x^5+48\log (2(3+x))\log (\log (2(3+x)))+16x\log (2(3+x))\log (\log (2(3+x))))}{81x^2\log (6+2x){\log }^2(\log (6+2x))}+\frac{e^{\frac{4}{x^4}}(x^5+48\log (2(3+x))\log (\log (2(3+x)))+16x\log (2(3+x))\log (\log (2(3+x))))}{243(3+x)\log (6+2x){\log }^2(\log (6+2x))})dx\\ & =\frac{1}{243}\int \frac{e^{\frac{4}{x^4}}(-x^5-48\log (2(3+x))\log (\log (2(3+x)))-16x\log (2(3+x))\log (\log (2(3+x))))}{x\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{243}\int \frac{e^{\frac{4}{x^4}}(x^5+48\log (2(3+x))\log (\log (2(3+x)))+16x\log (2(3+x))\log (\log (2(3+x))))}{(3+x)\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{81}\int \frac{e^{\frac{4}{x^4}}(x^5+48\log (2(3+x))\log (\log (2(3+x)))+16x\log (2(3+x))\log (\log (2(3+x))))}{x^2\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{27}\int \frac{e^{\frac{4}{x^4}}(-x^5-48\log (2(3+x))\log (\log (2(3+x)))-16x\log (2(3+x))\log (\log (2(3+x))))}{x^3\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{9}\int \frac{e^{\frac{4}{x^4}}(x^5+48\log (2(3+x))\log (\log (2(3+x)))+16x\log (2(3+x))\log (\log (2(3+x))))}{x^4\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{3}\int \frac{e^{\frac{4}{x^4}}(-x^5-48\log (2(3+x))\log (\log (2(3+x)))-16x\log (2(3+x))\log (\log (2(3+x))))}{x^5\log (6+2x){\log }^2(\log (6+2x))}dx\\ & =\frac{1}{243}\int \frac{e^{\frac{4}{x^4}}(-x^5-16(3+x)\log (2(3+x))\log (\log (2(3+x))))}{x\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{243}\int \frac{e^{\frac{4}{x^4}}(x^5+16(3+x)\log (2(3+x))\log (\log (2(3+x))))}{(3+x)\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{81}\int \frac{e^{\frac{4}{x^4}}(x^5+16(3+x)\log (2(3+x))\log (\log (2(3+x))))}{x^2\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{27}\int \frac{e^{\frac{4}{x^4}}(-x^5-16(3+x)\log (2(3+x))\log (\log (2(3+x))))}{x^3\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{9}\int \frac{e^{\frac{4}{x^4}}(x^5+16(3+x)\log (2(3+x))\log (\log (2(3+x))))}{x^4\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{3}\int \frac{e^{\frac{4}{x^4}}(-x^5-16(3+x)\log (2(3+x))\log (\log (2(3+x))))}{x^5\log (6+2x){\log }^2(\log (6+2x))}dx\\ & =\frac{1}{243}\int (-\frac{e^{\frac{4}{x^4}}{x}^4}{\log (6+2x){\log }^2(\log (6+2x))}-\frac{16e^{\frac{4}{x^4}}}{\log (\log (6+2x))}-\frac{48e^{\frac{4}{x^4}}}{x\log (\log (6+2x))})dx+\frac{1}{243}\int (\frac{e^{\frac{4}{x^4}}{x}^5}{(3+x)\log (6+2x){\log }^2(\log (6+2x))}+\frac{48e^{\frac{4}{x^4}}}{(3+x)\log (\log (6+2x))}+\frac{16e^{\frac{4}{x^4}}x}{(3+x)\log (\log (6+2x))})dx+\frac{1}{81}\int (\frac{e^{\frac{4}{x^4}}{x}^3}{\log (6+2x){\log }^2(\log (6+2x))}+\frac{48e^{\frac{4}{x^4}}}{x^2\log (\log (6+2x))}+\frac{16e^{\frac{4}{x^4}}}{x\log (\log (6+2x))})dx+\frac{1}{27}\int (-\frac{e^{\frac{4}{x^4}}{x}^2}{\log (6+2x){\log }^2(\log (6+2x))}-\frac{48e^{\frac{4}{x^4}}}{x^3\log (\log (6+2x))}-\frac{16e^{\frac{4}{x^4}}}{x^2\log (\log (6+2x))})dx+\frac{1}{9}\int (\frac{e^{\frac{4}{x^4}}x}{\log (6+2x){\log }^2(\log (6+2x))}+\frac{48e^{\frac{4}{x^4}}}{x^4\log (\log (6+2x))}+\frac{16e^{\frac{4}{x^4}}}{x^3\log (\log (6+2x))})dx+\frac{1}{3}\int (-\frac{e^{\frac{4}{x^4}}}{\log (6+2x){\log }^2(\log (6+2x))}-\frac{48e^{\frac{4}{x^4}}}{x^5\log (\log (6+2x))}-\frac{16e^{\frac{4}{x^4}}}{x^4\log (\log (6+2x))})dx\\ & =-(\frac{1}{243}\int \frac{e^{\frac{4}{x^4}}{x}^4}{\log (6+2x){\log }^2(\log (6+2x))}dx)+\frac{1}{243}\int \frac{e^{\frac{4}{x^4}}{x}^5}{(3+x)\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{81}\int \frac{e^{\frac{4}{x^4}}{x}^3}{\log (6+2x){\log }^2(\log (6+2x))}dx-\frac{1}{27}\int \frac{e^{\frac{4}{x^4}}{x}^2}{\log (6+2x){\log }^2(\log (6+2x))}dx-\frac{16}{243}\int \frac{e^{\frac{4}{x^4}}}{\log (\log (6+2x))}dx+\frac{16}{243}\int \frac{e^{\frac{4}{x^4}}x}{(3+x)\log (\log (6+2x))}dx+\frac{1}{9}\int \frac{e^{\frac{4}{x^4}}x}{\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{16}{81}\int \frac{e^{\frac{4}{x^4}}}{(3+x)\log (\log (6+2x))}dx-\frac{1}{3}\int \frac{e^{\frac{4}{x^4}}}{\log (6+2x){\log }^2(\log (6+2x))}dx-16\int \frac{e^{\frac{4}{x^4}}}{x^5\log (\log (6+2x))}dx\\ & =\frac{1}{243}\int (\frac{81e^{\frac{4}{x^4}}}{\log (6+2x){\log }^2(\log (6+2x))}+\frac{243e^{\frac{4}{x^4}}}{(-3-x)\log (6+2x){\log }^2(\log (6+2x))}-\frac{27e^{\frac{4}{x^4}}x}{\log (6+2x){\log }^2(\log (6+2x))}+\frac{9e^{\frac{4}{x^4}}{x}^2}{\log (6+2x){\log }^2(\log (6+2x))}-\frac{3e^{\frac{4}{x^4}}{x}^3}{\log (6+2x){\log }^2(\log (6+2x))}+\frac{e^{\frac{4}{x^4}}{x}^4}{\log (6+2x){\log }^2(\log (6+2x))})dx-\frac{1}{243}\int \frac{e^{\frac{4}{x^4}}{x}^4}{\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{1}{81}\int \frac{e^{\frac{4}{x^4}}{x}^3}{\log (6+2x){\log }^2(\log (6+2x))}dx-\frac{1}{27}\int \frac{e^{\frac{4}{x^4}}{x}^2}{\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{16}{243}\int (\frac{e^{\frac{4}{x^4}}}{\log (\log (6+2x))}+\frac{3e^{\frac{4}{x^4}}}{(-3-x)\log (\log (6+2x))})dx-\frac{16}{243}\int \frac{e^{\frac{4}{x^4}}}{\log (\log (6+2x))}dx+\frac{1}{9}\int \frac{e^{\frac{4}{x^4}}x}{\log (6+2x){\log }^2(\log (6+2x))}dx+\frac{16}{81}\int \frac{e^{\frac{4}{x^4}}}{(3+x)\log (\log (6+2x))}dx-\frac{1}{3}\int \frac{e^{\frac{4}{x^4}}}{\log (6+2x){\log }^2(\log (6+2x))}dx-16\int \frac{e^{\frac{4}{x^4}}}{x^5\log (\log (6+2x))}dx\\ & =\frac{16}{81}\int \frac{e^{\frac{4}{x^4}}}{(-3-x)\log (\log (6+2x))}dx+\frac{16}{81}\int \frac{e^{\frac{4}{x^4}}}{(3+x)\log (\log (6+2x))}dx-16\int \frac{e^{\frac{4}{x^4}}}{x^5\log (\log (6+2x))}dx+\int \frac{e^{\frac{4}{x^4}}}{(-3-x)\log (6+2x){\log }^2(\log (6+2x))}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 17, normalized size = 1.00 $$\begin{array}{c}\frac{e^{\frac{4}{x^4}}}{\log (\log (2(3+x)))}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-(E^(4/x^4)*x^5) + E^(4/x^4)*(-48 - 16*x)*Log[6 + 2*x]*Log[Log[6 + 2*x]])/((3*x^5 + x^6)*Log[6 + 2*
x]*Log[Log[6 + 2*x]]^2),x]

[Out]

E^(4/x^4)/Log[Log[2*(3 + x)]]

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 16, normalized size = 0.94 $$\begin{array}{c}\frac{e^{\left(\frac{4}{x^4}\right)}}{\log (\log (2x+6))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x-48)*exp(2/x^4)^2*log(2*x+6)*log(log(2*x+6))-x^5*exp(2/x^4)^2)/(x^6+3*x^5)/log(2*x+6)/log(log
(2*x+6))^2,x, algorithm="fricas")

[Out]

e^(4/x^4)/log(log(2*x + 6))

________________________________________________________________________________________

giac [A]  time = 0.26, size = 16, normalized size = 0.94 $$\begin{array}{c}\frac{e^{\left(\frac{4}{x^4}\right)}}{\log (\log (2x+6))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x-48)*exp(2/x^4)^2*log(2*x+6)*log(log(2*x+6))-x^5*exp(2/x^4)^2)/(x^6+3*x^5)/log(2*x+6)/log(log
(2*x+6))^2,x, algorithm="giac")

[Out]

e^(4/x^4)/log(log(2*x + 6))

________________________________________________________________________________________

maple [A]  time = 0.45, size = 17, normalized size = 1.00

method	result	size
risch	$\frac{{\mathrm{e}}^{\frac{4}{x^4}}}{\ln (\ln (2x+6))}$	$17$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x-48)*exp(2/x^4)^2*ln(2*x+6)*ln(ln(2*x+6))-x^5*exp(2/x^4)^2)/(x^6+3*x^5)/ln(2*x+6)/ln(ln(2*x+6))^2,x
,method=_RETURNVERBOSE)

[Out]

exp(4/x^4)/ln(ln(2*x+6))

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 17, normalized size = 1.00 $$\begin{array}{c}\frac{e^{\left(\frac{4}{x^4}\right)}}{\log (\log (2)+\log (x+3))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x-48)*exp(2/x^4)^2*log(2*x+6)*log(log(2*x+6))-x^5*exp(2/x^4)^2)/(x^6+3*x^5)/log(2*x+6)/log(log
(2*x+6))^2,x, algorithm="maxima")

[Out]

e^(4/x^4)/log(log(2) + log(x + 3))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.06 $$\begin{array}{c}\int -\frac{x^5{\mathrm{e}}^{\frac{4}{x^4}}+\ln (\ln (2x+6)){\mathrm{e}}^{\frac{4}{x^4}}\ln (2x+6)(16x+48)}{{\ln (\ln (2x+6))}^2\ln (2x+6)(x^6+3x^5)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^5*exp(4/x^4) + log(log(2*x + 6))*exp(4/x^4)*log(2*x + 6)*(16*x + 48))/(log(log(2*x + 6))^2*log(2*x + 6
)*(3*x^5 + x^6)),x)

[Out]

int(-(x^5*exp(4/x^4) + log(log(2*x + 6))*exp(4/x^4)*log(2*x + 6)*(16*x + 48))/(log(log(2*x + 6))^2*log(2*x + 6
)*(3*x^5 + x^6)), x)

________________________________________________________________________________________

sympy [A]  time = 0.53, size = 14, normalized size = 0.82 $$\begin{array}{c}\frac{e^{\frac{4}{x^4}}}{\log (\log (2x+6))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x-48)*exp(2/x**4)**2*ln(2*x+6)*ln(ln(2*x+6))-x**5*exp(2/x**4)**2)/(x**6+3*x**5)/ln(2*x+6)/ln(l
n(2*x+6))**2,x)

[Out]

exp(4/x**4)/log(log(2*x + 6))

________________________________________________________________________________________