\(\int \frac {-16-27 x-2 x^2+(3 x+x^2) \log (e^{4 x} x^4)+(5 x-2 x^2+(-4 x-x^2) \log (e^{4 x} x^4)+(-5+2 x+(4+x) \log (e^{4 x} x^4)) \log (-5+2 x+(4+x) \log (e^{4 x} x^4))) \log (x-\log (-5+2 x+(4+x) \log (e^{4 x} x^4)))}{(5 x-2 x^2+(-4 x-x^2) \log (e^{4 x} x^4)+(-5+2 x+(4+x) \log (e^{4 x} x^4)) \log (-5+2 x+(4+x) \log (e^{4 x} x^4))) \log ^2(x-\log (-5+2 x+(4+x) \log (e^{4 x} x^4)))} \, dx\) [8484]

Optimal result

Integrand size = 222, antiderivative size = 31 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=-5+\frac {x}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \]

[Out]

x/ln(-ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+x)-5

Rubi [F]

\[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx \]

[In]

Int[(-16 - 27*x - 2*x^2 + (3*x + x^2)*Log[E^(4*x)*x^4] + (5*x - 2*x^2 + (-4*x - x^2)*Log[E^(4*x)*x^4] + (-5 +
2*x + (4 + x)*Log[E^(4*x)*x^4])*Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*Log[E
^(4*x)*x^4]]])/((5*x - 2*x^2 + (-4*x - x^2)*Log[E^(4*x)*x^4] + (-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*Log[-5 +
2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2),x]

[Out]

16*Defer[Int][1/((-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*(x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x -
Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2), x] + 27*Defer[Int][x/((-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*(x -
Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2), x] + 2*Defer[
Int][x^2/((-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*(x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5
+ 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2), x] - 3*Defer[Int][(x*Log[E^(4*x)*x^4])/((-5 + 2*x + (4 + x)*Log[E^(4*x)
*x^4])*(x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2), x]
- Defer[Int][(x^2*Log[E^(4*x)*x^4])/((-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*(x - Log[-5 + 2*x + (4 + x)*Log[E^
(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2), x] + Defer[Int][Log[x - Log[-5 + 2*x + (4 +
x)*Log[E^(4*x)*x^4]]]^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-16-27 x-2 x^2+x (3+x) \log \left (e^{4 x} x^4\right )-\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5-2 x-(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx \\ & = \int \left (\frac {16+27 x+2 x^2-3 x \log \left (e^{4 x} x^4\right )-x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}+\frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}\right ) \, dx \\ & = \int \frac {16+27 x+2 x^2-3 x \log \left (e^{4 x} x^4\right )-x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx \\ & = \int \frac {-16-27 x-2 x^2+x (3+x) \log \left (e^{4 x} x^4\right )}{\left (5-2 x-(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx \\ & = \int \left (\frac {16}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}+\frac {27 x}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}+\frac {2 x^2}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}-\frac {3 x \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}-\frac {x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx \\ & = 2 \int \frac {x^2}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-3 \int \frac {x \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+16 \int \frac {1}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+27 \int \frac {x}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-\int \frac {x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+4 \log \left (e^{4 x} x^4\right )+x \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx \\ & = 2 \int \frac {x^2}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-3 \int \frac {x \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+16 \int \frac {1}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+27 \int \frac {x}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx-\int \frac {x^2 \log \left (e^{4 x} x^4\right )}{\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx+\int \frac {1}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \]

[In]

Integrate[(-16 - 27*x - 2*x^2 + (3*x + x^2)*Log[E^(4*x)*x^4] + (5*x - 2*x^2 + (-4*x - x^2)*Log[E^(4*x)*x^4] +
(-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)
*Log[E^(4*x)*x^4]]])/((5*x - 2*x^2 + (-4*x - x^2)*Log[E^(4*x)*x^4] + (-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*Log
[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2),x]

[Out]

x/Log[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 29.17 (sec) , antiderivative size = 329, normalized size of antiderivative = 10.61

\[\frac {x}{\ln \left (-\ln \left (\left (4+x \right ) \left (4 \ln \left (x \right )+4 \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) {\left (-\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{4}\right ) \left (-\operatorname {csgn}\left (i x^{4}\right )+\operatorname {csgn}\left (i x^{3}\right )\right ) \left (-\operatorname {csgn}\left (i x^{4}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right ) \left (-\operatorname {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i x^{4}\right )\right ) \left (-\operatorname {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right )\right )}{2}\right )+2 x -5\right )+x \right )}\]

[In]

int(((((4+x)*ln(x^4*exp(x)^4)+2*x-5)*ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*ln(x^4*exp(x)^4)-2*x^2+5*x)*l
n(-ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+x)+(x^2+3*x)*ln(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*ln(x^4*exp(x)^4)+2*x-
5)*ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*ln(x^4*exp(x)^4)-2*x^2+5*x)/ln(-ln((4+x)*ln(x^4*exp(x)^4)+2*x-5
)+x)^2,x)

[Out]

x/ln(-ln((4+x)*(4*ln(x)+4*ln(exp(x))-1/2*I*Pi*csgn(I*exp(2*x))*(-csgn(I*exp(2*x))+csgn(I*exp(x)))^2-1/2*I*Pi*c
sgn(I*exp(3*x))*(-csgn(I*exp(3*x))+csgn(I*exp(2*x)))*(-csgn(I*exp(3*x))+csgn(I*exp(x)))-1/2*I*Pi*csgn(I*exp(4*
x))*(-csgn(I*exp(4*x))+csgn(I*exp(3*x)))*(-csgn(I*exp(4*x))+csgn(I*exp(x)))-1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)
+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-
csgn(I*x^4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4*exp(4*x))*(-csgn(I*x^4*exp(4*x))+csgn(I*
x^4))*(-csgn(I*x^4*exp(4*x))+csgn(I*exp(4*x))))+2*x-5)+x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x - \log \left ({\left (x + 4\right )} \log \left (x^{4} e^{\left (4 \, x\right )}\right ) + 2 \, x - 5\right )\right )} \]

[In]

integrate(((((4+x)*log(x^4*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*
x^2+5*x)*log(-log((4+x)*log(x^4*exp(x)^4)+2*x-5)+x)+(x^2+3*x)*log(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*log(x^4
*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*x^2+5*x)/log(-log((4+x)*lo
g(x^4*exp(x)^4)+2*x-5)+x)^2,x, algorithm="fricas")

[Out]

x/log(x - log((x + 4)*log(x^4*e^(4*x)) + 2*x - 5))

Sympy [A] (verification not implemented)

Time = 117.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log {\left (x - \log {\left (2 x + \left (x + 4\right ) \log {\left (x^{4} e^{4 x} \right )} - 5 \right )} \right )}} \]

[In]

integrate(((((4+x)*ln(x**4*exp(x)**4)+2*x-5)*ln((4+x)*ln(x**4*exp(x)**4)+2*x-5)+(-x**2-4*x)*ln(x**4*exp(x)**4)
-2*x**2+5*x)*ln(-ln((4+x)*ln(x**4*exp(x)**4)+2*x-5)+x)+(x**2+3*x)*ln(x**4*exp(x)**4)-2*x**2-27*x-16)/(((4+x)*l
n(x**4*exp(x)**4)+2*x-5)*ln((4+x)*ln(x**4*exp(x)**4)+2*x-5)+(-x**2-4*x)*ln(x**4*exp(x)**4)-2*x**2+5*x)/ln(-ln(
(4+x)*ln(x**4*exp(x)**4)+2*x-5)+x)**2,x)

[Out]

x/log(x - log(2*x + (x + 4)*log(x**4*exp(4*x)) - 5))

Maxima [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x - \log \left (4 \, x^{2} + 4 \, {\left (x + 4\right )} \log \left (x\right ) + 18 \, x - 5\right )\right )} \]

[In]

integrate(((((4+x)*log(x^4*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*
x^2+5*x)*log(-log((4+x)*log(x^4*exp(x)^4)+2*x-5)+x)+(x^2+3*x)*log(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*log(x^4
*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*x^2+5*x)/log(-log((4+x)*lo
g(x^4*exp(x)^4)+2*x-5)+x)^2,x, algorithm="maxima")

[Out]

x/log(x - log(4*x^2 + 4*(x + 4)*log(x) + 18*x - 5))

Giac [A] (verification not implemented)

none

Time = 1.94 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x - \log \left (4 \, x^{2} + x \log \left (x^{4}\right ) + 18 \, x + 4 \, \log \left (x^{4}\right ) - 5\right )\right )} \]

[In]

integrate(((((4+x)*log(x^4*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*
x^2+5*x)*log(-log((4+x)*log(x^4*exp(x)^4)+2*x-5)+x)+(x^2+3*x)*log(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*log(x^4
*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*x^2+5*x)/log(-log((4+x)*lo
g(x^4*exp(x)^4)+2*x-5)+x)^2,x, algorithm="giac")

[Out]

x/log(x - log(4*x^2 + x*log(x^4) + 18*x + 4*log(x^4) - 5))

Mupad [F(-1)]

Timed out. \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\int -\frac {27\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+3\,x\right )+2\,x^2-\ln \left (x-\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\right )\,\left (5\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+4\,x\right )+\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\,\left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )-2\,x^2\right )+16}{{\ln \left (x-\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\right )}^2\,\left (5\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+4\,x\right )+\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\,\left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )-2\,x^2\right )} \,d x \]

[In]

int(-(27*x - log(x^4*exp(4*x))*(3*x + x^2) + 2*x^2 - log(x - log(2*x + log(x^4*exp(4*x))*(x + 4) - 5))*(5*x -
log(x^4*exp(4*x))*(4*x + x^2) + log(2*x + log(x^4*exp(4*x))*(x + 4) - 5)*(2*x + log(x^4*exp(4*x))*(x + 4) - 5)
- 2*x^2) + 16)/(log(x - log(2*x + log(x^4*exp(4*x))*(x + 4) - 5))^2*(5*x - log(x^4*exp(4*x))*(4*x + x^2) + lo
g(2*x + log(x^4*exp(4*x))*(x + 4) - 5)*(2*x + log(x^4*exp(4*x))*(x + 4) - 5) - 2*x^2)),x)

[Out]

int(-(27*x - log(x^4*exp(4*x))*(3*x + x^2) + 2*x^2 - log(x - log(2*x + log(x^4*exp(4*x))*(x + 4) - 5))*(5*x -
log(x^4*exp(4*x))*(4*x + x^2) + log(2*x + log(x^4*exp(4*x))*(x + 4) - 5)*(2*x + log(x^4*exp(4*x))*(x + 4) - 5)
- 2*x^2) + 16)/(log(x - log(2*x + log(x^4*exp(4*x))*(x + 4) - 5))^2*(5*x - log(x^4*exp(4*x))*(4*x + x^2) + lo
g(2*x + log(x^4*exp(4*x))*(x + 4) - 5)*(2*x + log(x^4*exp(4*x))*(x + 4) - 5) - 2*x^2)), x)