\(\int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{(3 x^5+x^6) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx\) [3284]

Optimal result

Integrand size = 68, antiderivative size = 17 \[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\frac {e^{\frac {4}{x^4}}}{\log (\log (6+2 x))} \]

[Out]

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

Rubi [F]

\[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx \]

[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{align*} \text {integral}& = \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{x^5 (3+x) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx \\ & = \int \left (\frac {e^{\frac {4}{x^4}} \left (-x^5-48 \log (2 (3+x)) \log (\log (2 (3+x)))-16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{3 x^5 \log (6+2 x) \log ^2(\log (6+2 x))}+\frac {e^{\frac {4}{x^4}} \left (-x^5-48 \log (2 (3+x)) \log (\log (2 (3+x)))-16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{27 x^3 \log (6+2 x) \log ^2(\log (6+2 x))}+\frac {e^{\frac {4}{x^4}} \left (-x^5-48 \log (2 (3+x)) \log (\log (2 (3+x)))-16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{243 x \log (6+2 x) \log ^2(\log (6+2 x))}+\frac {e^{\frac {4}{x^4}} \left (x^5+48 \log (2 (3+x)) \log (\log (2 (3+x)))+16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{9 x^4 \log (6+2 x) \log ^2(\log (6+2 x))}+\frac {e^{\frac {4}{x^4}} \left (x^5+48 \log (2 (3+x)) \log (\log (2 (3+x)))+16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{81 x^2 \log (6+2 x) \log ^2(\log (6+2 x))}+\frac {e^{\frac {4}{x^4}} \left (x^5+48 \log (2 (3+x)) \log (\log (2 (3+x)))+16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{243 (3+x) \log (6+2 x) \log ^2(\log (6+2 x))}\right ) \, dx \\ & = \frac {1}{243} \int \frac {e^{\frac {4}{x^4}} \left (-x^5-48 \log (2 (3+x)) \log (\log (2 (3+x)))-16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{243} \int \frac {e^{\frac {4}{x^4}} \left (x^5+48 \log (2 (3+x)) \log (\log (2 (3+x)))+16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{(3+x) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{81} \int \frac {e^{\frac {4}{x^4}} \left (x^5+48 \log (2 (3+x)) \log (\log (2 (3+x)))+16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^2 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{27} \int \frac {e^{\frac {4}{x^4}} \left (-x^5-48 \log (2 (3+x)) \log (\log (2 (3+x)))-16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^3 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{9} \int \frac {e^{\frac {4}{x^4}} \left (x^5+48 \log (2 (3+x)) \log (\log (2 (3+x)))+16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^4 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{3} \int \frac {e^{\frac {4}{x^4}} \left (-x^5-48 \log (2 (3+x)) \log (\log (2 (3+x)))-16 x \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^5 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx \\ & = \frac {1}{243} \int \frac {e^{\frac {4}{x^4}} \left (-x^5-16 (3+x) \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{243} \int \frac {e^{\frac {4}{x^4}} \left (x^5+16 (3+x) \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{(3+x) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{81} \int \frac {e^{\frac {4}{x^4}} \left (x^5+16 (3+x) \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^2 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{27} \int \frac {e^{\frac {4}{x^4}} \left (-x^5-16 (3+x) \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^3 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{9} \int \frac {e^{\frac {4}{x^4}} \left (x^5+16 (3+x) \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^4 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{3} \int \frac {e^{\frac {4}{x^4}} \left (-x^5-16 (3+x) \log (2 (3+x)) \log (\log (2 (3+x)))\right )}{x^5 \log (6+2 x) \log ^2(\log (6+2 x))} \, dx \\ & = \frac {1}{243} \int \left (-\frac {e^{\frac {4}{x^4}} x^4}{\log (6+2 x) \log ^2(\log (6+2 x))}-\frac {16 e^{\frac {4}{x^4}}}{\log (\log (6+2 x))}-\frac {48 e^{\frac {4}{x^4}}}{x \log (\log (6+2 x))}\right ) \, dx+\frac {1}{243} \int \left (\frac {e^{\frac {4}{x^4}} x^5}{(3+x) \log (6+2 x) \log ^2(\log (6+2 x))}+\frac {48 e^{\frac {4}{x^4}}}{(3+x) \log (\log (6+2 x))}+\frac {16 e^{\frac {4}{x^4}} x}{(3+x) \log (\log (6+2 x))}\right ) \, dx+\frac {1}{81} \int \left (\frac {e^{\frac {4}{x^4}} x^3}{\log (6+2 x) \log ^2(\log (6+2 x))}+\frac {48 e^{\frac {4}{x^4}}}{x^2 \log (\log (6+2 x))}+\frac {16 e^{\frac {4}{x^4}}}{x \log (\log (6+2 x))}\right ) \, dx+\frac {1}{27} \int \left (-\frac {e^{\frac {4}{x^4}} x^2}{\log (6+2 x) \log ^2(\log (6+2 x))}-\frac {48 e^{\frac {4}{x^4}}}{x^3 \log (\log (6+2 x))}-\frac {16 e^{\frac {4}{x^4}}}{x^2 \log (\log (6+2 x))}\right ) \, dx+\frac {1}{9} \int \left (\frac {e^{\frac {4}{x^4}} x}{\log (6+2 x) \log ^2(\log (6+2 x))}+\frac {48 e^{\frac {4}{x^4}}}{x^4 \log (\log (6+2 x))}+\frac {16 e^{\frac {4}{x^4}}}{x^3 \log (\log (6+2 x))}\right ) \, dx+\frac {1}{3} \int \left (-\frac {e^{\frac {4}{x^4}}}{\log (6+2 x) \log ^2(\log (6+2 x))}-\frac {48 e^{\frac {4}{x^4}}}{x^5 \log (\log (6+2 x))}-\frac {16 e^{\frac {4}{x^4}}}{x^4 \log (\log (6+2 x))}\right ) \, dx \\ & = -\left (\frac {1}{243} \int \frac {e^{\frac {4}{x^4}} x^4}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx\right )+\frac {1}{243} \int \frac {e^{\frac {4}{x^4}} x^5}{(3+x) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{81} \int \frac {e^{\frac {4}{x^4}} x^3}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx-\frac {1}{27} \int \frac {e^{\frac {4}{x^4}} x^2}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx-\frac {16}{243} \int \frac {e^{\frac {4}{x^4}}}{\log (\log (6+2 x))} \, dx+\frac {16}{243} \int \frac {e^{\frac {4}{x^4}} x}{(3+x) \log (\log (6+2 x))} \, dx+\frac {1}{9} \int \frac {e^{\frac {4}{x^4}} x}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {16}{81} \int \frac {e^{\frac {4}{x^4}}}{(3+x) \log (\log (6+2 x))} \, dx-\frac {1}{3} \int \frac {e^{\frac {4}{x^4}}}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx-16 \int \frac {e^{\frac {4}{x^4}}}{x^5 \log (\log (6+2 x))} \, dx \\ & = \frac {1}{243} \int \left (\frac {81 e^{\frac {4}{x^4}}}{\log (6+2 x) \log ^2(\log (6+2 x))}+\frac {243 e^{\frac {4}{x^4}}}{(-3-x) \log (6+2 x) \log ^2(\log (6+2 x))}-\frac {27 e^{\frac {4}{x^4}} x}{\log (6+2 x) \log ^2(\log (6+2 x))}+\frac {9 e^{\frac {4}{x^4}} x^2}{\log (6+2 x) \log ^2(\log (6+2 x))}-\frac {3 e^{\frac {4}{x^4}} x^3}{\log (6+2 x) \log ^2(\log (6+2 x))}+\frac {e^{\frac {4}{x^4}} x^4}{\log (6+2 x) \log ^2(\log (6+2 x))}\right ) \, dx-\frac {1}{243} \int \frac {e^{\frac {4}{x^4}} x^4}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {1}{81} \int \frac {e^{\frac {4}{x^4}} x^3}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx-\frac {1}{27} \int \frac {e^{\frac {4}{x^4}} x^2}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {16}{243} \int \left (\frac {e^{\frac {4}{x^4}}}{\log (\log (6+2 x))}+\frac {3 e^{\frac {4}{x^4}}}{(-3-x) \log (\log (6+2 x))}\right ) \, dx-\frac {16}{243} \int \frac {e^{\frac {4}{x^4}}}{\log (\log (6+2 x))} \, dx+\frac {1}{9} \int \frac {e^{\frac {4}{x^4}} x}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx+\frac {16}{81} \int \frac {e^{\frac {4}{x^4}}}{(3+x) \log (\log (6+2 x))} \, dx-\frac {1}{3} \int \frac {e^{\frac {4}{x^4}}}{\log (6+2 x) \log ^2(\log (6+2 x))} \, dx-16 \int \frac {e^{\frac {4}{x^4}}}{x^5 \log (\log (6+2 x))} \, dx \\ & = \frac {16}{81} \int \frac {e^{\frac {4}{x^4}}}{(-3-x) \log (\log (6+2 x))} \, dx+\frac {16}{81} \int \frac {e^{\frac {4}{x^4}}}{(3+x) \log (\log (6+2 x))} \, dx-16 \int \frac {e^{\frac {4}{x^4}}}{x^5 \log (\log (6+2 x))} \, dx+\int \frac {e^{\frac {4}{x^4}}}{(-3-x) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\frac {e^{\frac {4}{x^4}}}{\log (\log (2 (3+x)))} \]

[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)]]

Maple [A] (verified)

Time = 49.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {4}{x^{4}}}}{\ln \left (\ln \left (2 x +6\right )\right )}\)	\(17\)
parallelrisch	\(\frac {{\mathrm e}^{\frac {4}{x^{4}}}}{\ln \left (\ln \left (2 x +6\right )\right )}\)	\(19\)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\frac {e^{\left (\frac {4}{x^{4}}\right )}}{\log \left (\log \left (2 \, x + 6\right )\right )} \]

[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))

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\frac {e^{\frac {4}{x^{4}}}}{\log {\left (\log {\left (2 x + 6 \right )} \right )}} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\frac {e^{\left (\frac {4}{x^{4}}\right )}}{\log \left (\log \left (2\right ) + \log \left (x + 3\right )\right )} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\frac {e^{\left (\frac {4}{x^{4}}\right )}}{\log \left (\log \left (2 \, x + 6\right )\right )} \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {-e^{\frac {4}{x^4}} x^5+e^{\frac {4}{x^4}} (-48-16 x) \log (6+2 x) \log (\log (6+2 x))}{\left (3 x^5+x^6\right ) \log (6+2 x) \log ^2(\log (6+2 x))} \, dx=\int -\frac {x^5\,{\mathrm {e}}^{\frac {4}{x^4}}+\ln \left (\ln \left (2\,x+6\right )\right )\,{\mathrm {e}}^{\frac {4}{x^4}}\,\ln \left (2\,x+6\right )\,\left (16\,x+48\right )}{{\ln \left (\ln \left (2\,x+6\right )\right )}^2\,\ln \left (2\,x+6\right )\,\left (x^6+3\,x^5\right )} \,d x \]

[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)