\(\int \frac {-16 e^x x-16 e^x \log (x)+(e^x (-8-4 x+4 x^2)+e^x (4+4 x) \log (x)) \log (9 x^2)}{(x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)) \log ^3(9 x^2)} \, dx\) [3299]

Optimal result

Integrand size = 80, antiderivative size = 22 \[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=-2+\frac {4 e^x x}{(x+\log (x))^2 \log ^2\left (9 x^2\right )} \]

[Out]

4*exp(x)*x/ln(9*x^2)^2/(x+ln(x))^2-2

Rubi [F]

\[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=\int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx \]

[In]

Int[(-16*E^x*x - 16*E^x*Log[x] + (E^x*(-8 - 4*x + 4*x^2) + E^x*(4 + 4*x)*Log[x])*Log[9*x^2])/((x^3 + 3*x^2*Log
[x] + 3*x*Log[x]^2 + Log[x]^3)*Log[9*x^2]^3),x]

[Out]

-16*Defer[Int][E^x/((x + Log[x])^2*Log[9*x^2]^3), x] - 8*Defer[Int][E^x/((x + Log[x])^3*Log[9*x^2]^2), x] - 4*
Defer[Int][(E^x*x)/((x + Log[x])^3*Log[9*x^2]^2), x] + 4*Defer[Int][(E^x*x^2)/((x + Log[x])^3*Log[9*x^2]^2), x
] + 4*Defer[Int][(E^x*Log[x])/((x + Log[x])^3*Log[9*x^2]^2), x] + 4*Defer[Int][(E^x*x*Log[x])/((x + Log[x])^3*
Log[9*x^2]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4 e^x \left (-4 x+\left (-2-x+x^2\right ) \log \left (9 x^2\right )+\log (x) \left (-4+(1+x) \log \left (9 x^2\right )\right )\right )}{(x+\log (x))^3 \log ^3\left (9 x^2\right )} \, dx \\ & = 4 \int \frac {e^x \left (-4 x+\left (-2-x+x^2\right ) \log \left (9 x^2\right )+\log (x) \left (-4+(1+x) \log \left (9 x^2\right )\right )\right )}{(x+\log (x))^3 \log ^3\left (9 x^2\right )} \, dx \\ & = 4 \int \left (-\frac {4 e^x}{(x+\log (x))^2 \log ^3\left (9 x^2\right )}+\frac {e^x (1+x) (-2+x+\log (x))}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}\right ) \, dx \\ & = 4 \int \frac {e^x (1+x) (-2+x+\log (x))}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx-16 \int \frac {e^x}{(x+\log (x))^2 \log ^3\left (9 x^2\right )} \, dx \\ & = 4 \int \left (\frac {e^x (-2+x+\log (x))}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}+\frac {e^x x (-2+x+\log (x))}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}\right ) \, dx-16 \int \frac {e^x}{(x+\log (x))^2 \log ^3\left (9 x^2\right )} \, dx \\ & = 4 \int \frac {e^x (-2+x+\log (x))}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx+4 \int \frac {e^x x (-2+x+\log (x))}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx-16 \int \frac {e^x}{(x+\log (x))^2 \log ^3\left (9 x^2\right )} \, dx \\ & = 4 \int \left (-\frac {2 e^x}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}+\frac {e^x x}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}+\frac {e^x \log (x)}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}\right ) \, dx+4 \int \left (-\frac {2 e^x x}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}+\frac {e^x x^2}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}+\frac {e^x x \log (x)}{(x+\log (x))^3 \log ^2\left (9 x^2\right )}\right ) \, dx-16 \int \frac {e^x}{(x+\log (x))^2 \log ^3\left (9 x^2\right )} \, dx \\ & = 4 \int \frac {e^x x}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx+4 \int \frac {e^x x^2}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx+4 \int \frac {e^x \log (x)}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx+4 \int \frac {e^x x \log (x)}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx-8 \int \frac {e^x}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx-8 \int \frac {e^x x}{(x+\log (x))^3 \log ^2\left (9 x^2\right )} \, dx-16 \int \frac {e^x}{(x+\log (x))^2 \log ^3\left (9 x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=\frac {4 e^x x}{(x+\log (x))^2 \log ^2\left (9 x^2\right )} \]

[In]

Integrate[(-16*E^x*x - 16*E^x*Log[x] + (E^x*(-8 - 4*x + 4*x^2) + E^x*(4 + 4*x)*Log[x])*Log[9*x^2])/((x^3 + 3*x
^2*Log[x] + 3*x*Log[x]^2 + Log[x]^3)*Log[9*x^2]^3),x]

[Out]

(4*E^x*x)/((x + Log[x])^2*Log[9*x^2]^2)

Maple [C] (warning: unable to verify)

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

Time = 119.79 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14

\[-\frac {16 x \,{\mathrm e}^{x}}{\left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (3\right )+4 i \ln \left (x \right )\right )^{2} \left (x +\ln \left (x \right )\right )^{2}}\]

[In]

int((((4+4*x)*exp(x)*ln(x)+(4*x^2-4*x-8)*exp(x))*ln(9*x^2)-16*exp(x)*ln(x)-16*exp(x)*x)/(ln(x)^3+3*x*ln(x)^2+3
*x^2*ln(x)+x^3)/ln(9*x^2)^3,x)

[Out]

-16*x*exp(x)/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+4*I*ln(3)+4*I*ln(x))^2/
(x+ln(x))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.91 \[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=\frac {x e^{x}}{x^{2} \log \left (3\right )^{2} + 2 \, {\left (x + \log \left (3\right )\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} + {\left (x^{2} + 4 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{2} \log \left (3\right ) + x \log \left (3\right )^{2}\right )} \log \left (x\right )} \]

[In]

integrate((((4+4*x)*exp(x)*log(x)+(4*x^2-4*x-8)*exp(x))*log(9*x^2)-16*exp(x)*log(x)-16*exp(x)*x)/(log(x)^3+3*x
*log(x)^2+3*x^2*log(x)+x^3)/log(9*x^2)^3,x, algorithm="fricas")

[Out]

x*e^x/(x^2*log(3)^2 + 2*(x + log(3))*log(x)^3 + log(x)^4 + (x^2 + 4*x*log(3) + log(3)^2)*log(x)^2 + 2*(x^2*log
(3) + x*log(3)^2)*log(x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).

Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.09 \[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=\frac {x e^{x}}{x^{2} \log {\left (x \right )}^{2} + 2 x^{2} \log {\left (3 \right )} \log {\left (x \right )} + x^{2} \log {\left (3 \right )}^{2} + 2 x \log {\left (x \right )}^{3} + 4 x \log {\left (3 \right )} \log {\left (x \right )}^{2} + 2 x \log {\left (3 \right )}^{2} \log {\left (x \right )} + \log {\left (x \right )}^{4} + 2 \log {\left (3 \right )} \log {\left (x \right )}^{3} + \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2}} \]

[In]

integrate((((4+4*x)*exp(x)*ln(x)+(4*x**2-4*x-8)*exp(x))*ln(9*x**2)-16*exp(x)*ln(x)-16*exp(x)*x)/(ln(x)**3+3*x*
ln(x)**2+3*x**2*ln(x)+x**3)/ln(9*x**2)**3,x)

[Out]

x*exp(x)/(x**2*log(x)**2 + 2*x**2*log(3)*log(x) + x**2*log(3)**2 + 2*x*log(x)**3 + 4*x*log(3)*log(x)**2 + 2*x*
log(3)**2*log(x) + log(x)**4 + 2*log(3)*log(x)**3 + log(3)**2*log(x)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (21) = 42\).

Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.91 \[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=\frac {x e^{x}}{x^{2} \log \left (3\right )^{2} + 2 \, {\left (x + \log \left (3\right )\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} + {\left (x^{2} + 4 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{2} \log \left (3\right ) + x \log \left (3\right )^{2}\right )} \log \left (x\right )} \]

[In]

integrate((((4+4*x)*exp(x)*log(x)+(4*x^2-4*x-8)*exp(x))*log(9*x^2)-16*exp(x)*log(x)-16*exp(x)*x)/(log(x)^3+3*x
*log(x)^2+3*x^2*log(x)+x^3)/log(9*x^2)^3,x, algorithm="maxima")

[Out]

x*e^x/(x^2*log(3)^2 + 2*(x + log(3))*log(x)^3 + log(x)^4 + (x^2 + 4*x*log(3) + log(3)^2)*log(x)^2 + 2*(x^2*log
(3) + x*log(3)^2)*log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (21) = 42\).

Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.55 \[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=\frac {x e^{x}}{x^{2} \log \left (3\right )^{2} + 2 \, x^{2} \log \left (3\right ) \log \left (x\right ) + 2 \, x \log \left (3\right )^{2} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 4 \, x \log \left (3\right ) \log \left (x\right )^{2} + \log \left (3\right )^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x\right )^{3} + 2 \, \log \left (3\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4}} \]

[In]

integrate((((4+4*x)*exp(x)*log(x)+(4*x^2-4*x-8)*exp(x))*log(9*x^2)-16*exp(x)*log(x)-16*exp(x)*x)/(log(x)^3+3*x
*log(x)^2+3*x^2*log(x)+x^3)/log(9*x^2)^3,x, algorithm="giac")

[Out]

x*e^x/(x^2*log(3)^2 + 2*x^2*log(3)*log(x) + 2*x*log(3)^2*log(x) + x^2*log(x)^2 + 4*x*log(3)*log(x)^2 + log(3)^
2*log(x)^2 + 2*x*log(x)^3 + 2*log(3)*log(x)^3 + log(x)^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {-16 e^x x-16 e^x \log (x)+\left (e^x \left (-8-4 x+4 x^2\right )+e^x (4+4 x) \log (x)\right ) \log \left (9 x^2\right )}{\left (x^3+3 x^2 \log (x)+3 x \log ^2(x)+\log ^3(x)\right ) \log ^3\left (9 x^2\right )} \, dx=\int -\frac {16\,{\mathrm {e}}^x\,\ln \left (x\right )+\ln \left (9\,x^2\right )\,\left ({\mathrm {e}}^x\,\left (-4\,x^2+4\,x+8\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (4\,x+4\right )\right )+16\,x\,{\mathrm {e}}^x}{{\ln \left (9\,x^2\right )}^3\,\left (x^3+3\,x^2\,\ln \left (x\right )+3\,x\,{\ln \left (x\right )}^2+{\ln \left (x\right )}^3\right )} \,d x \]

[In]

int(-(16*exp(x)*log(x) + log(9*x^2)*(exp(x)*(4*x - 4*x^2 + 8) - exp(x)*log(x)*(4*x + 4)) + 16*x*exp(x))/(log(9
*x^2)^3*(3*x*log(x)^2 + 3*x^2*log(x) + log(x)^3 + x^3)),x)

[Out]

int(-(16*exp(x)*log(x) + log(9*x^2)*(exp(x)*(4*x - 4*x^2 + 8) - exp(x)*log(x)*(4*x + 4)) + 16*x*exp(x))/(log(9
*x^2)^3*(3*x*log(x)^2 + 3*x^2*log(x) + log(x)^3 + x^3)), x)