\(\int \frac {-972-231 x+3 x^2+e^{3 e^3} (-3 x+3 x^2)+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} (-72 x+20 x^2+2 x^3)+(-288+152 x-12 x^2-2 x^3+e^{3 e^3} (8 x-2 x^2)) \log (x)+(16-8 x+x^2) \log ^2(x)} \, dx\) [9444]

Optimal result

Integrand size = 143, antiderivative size = 33 \[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=2-\frac {3 x}{4-x-\frac {\left (5+e^{3 e^3}\right ) x}{9+x-\log (x)}} \]

[Out]

2-3*x/(4-x-x/(x+9-ln(x))*(5+exp(3*exp(3))))

Rubi [F]

\[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=\int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx \]

[In]

Int[(-972 - 231*x + 3*x^2 + E^(3*E^3)*(-3*x + 3*x^2) + (216 + 24*x)*Log[x] - 12*Log[x]^2)/(1296 - 720*x + 28*x
^2 + E^(6*E^3)*x^2 + 20*x^3 + x^4 + E^(3*E^3)*(-72*x + 20*x^2 + 2*x^3) + (-288 + 152*x - 12*x^2 - 2*x^3 + E^(3
*E^3)*(8*x - 2*x^2))*Log[x] + (16 - 8*x + x^2)*Log[x]^2),x]

[Out]

-12/(4 - x) - 12*(5 + E^(3*E^3))^2*Defer[Int][(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])^(-2),
x] - 192*(5 + E^(3*E^3))^2*Defer[Int][1/((4 - x)^2*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])
^2), x] + 96*(5 + E^(3*E^3))^2*Defer[Int][1/((4 - x)*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x]
)^2), x] - 3*(5 + E^(3*E^3))*Defer[Int][x/(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])^2, x] + 3
*(5 + E^(3*E^3))*Defer[Int][x^2/(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])^2, x] - 96*(5 + E^(
3*E^3))*Defer[Int][1/((4 - x)^2*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])), x] - 24*(5 + E^(3
*E^3))*Defer[Int][1/((-4 + x)*(36 - 10*(1 + E^(3*E^3)/10)*x - x^2 - 4*Log[x] + x*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+\left (28+e^{6 e^3}\right ) x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx \\ & = \int \frac {3 \left (-324-\left (77+e^{3 e^3}\right ) x+\left (1+e^{3 e^3}\right ) x^2+8 (9+x) \log (x)-4 \log ^2(x)\right )}{\left (36-\left (10+e^{3 e^3}\right ) x-x^2+(-4+x) \log (x)\right )^2} \, dx \\ & = 3 \int \frac {-324-\left (77+e^{3 e^3}\right ) x+\left (1+e^{3 e^3}\right ) x^2+8 (9+x) \log (x)-4 \log ^2(x)}{\left (36-\left (10+e^{3 e^3}\right ) x-x^2+(-4+x) \log (x)\right )^2} \, dx \\ & = 3 \int \left (-\frac {4}{(-4+x)^2}+\frac {\left (5+e^{3 e^3}\right ) x \left (-16+4 \left (1-e^{3 e^3}\right ) x-9 x^2+x^3\right )}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {8 \left (-5-e^{3 e^3}\right ) x}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )}\right ) \, dx \\ & = -\frac {12}{4-x}+\left (3 \left (5+e^{3 e^3}\right )\right ) \int \frac {x \left (-16+4 \left (1-e^{3 e^3}\right ) x-9 x^2+x^3\right )}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx-\left (24 \left (5+e^{3 e^3}\right )\right ) \int \frac {x}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )} \, dx \\ & = -\frac {12}{4-x}+\left (3 \left (5+e^{3 e^3}\right )\right ) \int \left (\frac {4 \left (-5-e^{3 e^3}\right )}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {64 \left (-5-e^{3 e^3}\right )}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {32 \left (5+e^{3 e^3}\right )}{(4-x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}-\frac {x}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}+\frac {x^2}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2}\right ) \, dx-\left (24 \left (5+e^{3 e^3}\right )\right ) \int \left (\frac {4}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )}+\frac {1}{(-4+x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )}\right ) \, dx \\ & = -\frac {12}{4-x}-\left (3 \left (5+e^{3 e^3}\right )\right ) \int \frac {x}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx+\left (3 \left (5+e^{3 e^3}\right )\right ) \int \frac {x^2}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx-\left (24 \left (5+e^{3 e^3}\right )\right ) \int \frac {1}{(-4+x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )} \, dx-\left (96 \left (5+e^{3 e^3}\right )\right ) \int \frac {1}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )} \, dx-\left (12 \left (5+e^{3 e^3}\right )^2\right ) \int \frac {1}{\left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx+\left (96 \left (5+e^{3 e^3}\right )^2\right ) \int \frac {1}{(4-x) \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx-\left (192 \left (5+e^{3 e^3}\right )^2\right ) \int \frac {1}{(4-x)^2 \left (36-10 \left (1+\frac {e^{3 e^3}}{10}\right ) x-x^2-4 \log (x)+x \log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=\frac {3 \left (36-\left (1+e^{3 e^3}\right ) x-4 \log (x)\right )}{-36+\left (10+e^{3 e^3}\right ) x+x^2-(-4+x) \log (x)} \]

[In]

Integrate[(-972 - 231*x + 3*x^2 + E^(3*E^3)*(-3*x + 3*x^2) + (216 + 24*x)*Log[x] - 12*Log[x]^2)/(1296 - 720*x
+ 28*x^2 + E^(6*E^3)*x^2 + 20*x^3 + x^4 + E^(3*E^3)*(-72*x + 20*x^2 + 2*x^3) + (-288 + 152*x - 12*x^2 - 2*x^3
+ E^(3*E^3)*(8*x - 2*x^2))*Log[x] + (16 - 8*x + x^2)*Log[x]^2),x]

[Out]

(3*(36 - (1 + E^(3*E^3))*x - 4*Log[x]))/(-36 + (10 + E^(3*E^3))*x + x^2 - (-4 + x)*Log[x])

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {3 \left (4 \ln \left (x \right )+\left ({\mathrm e}^{3 \,{\mathrm e}^{3}}+1\right ) x -36\right )}{{\mathrm e}^{3 \,{\mathrm e}^{3}} x +x^{2}-x \ln \left (x \right )+10 x +4 \ln \left (x \right )-36}\)	\(44\)
norman	\(\frac {-12 \ln \left (x \right )+\left (-3 \,{\mathrm e}^{3 \,{\mathrm e}^{3}}-3\right ) x +108}{{\mathrm e}^{3 \,{\mathrm e}^{3}} x +x^{2}-x \ln \left (x \right )+10 x +4 \ln \left (x \right )-36}\)	\(45\)
parallelrisch	\(\frac {108-3 \,{\mathrm e}^{3 \,{\mathrm e}^{3}} x -3 x -12 \ln \left (x \right )}{{\mathrm e}^{3 \,{\mathrm e}^{3}} x +x^{2}-x \ln \left (x \right )+10 x +4 \ln \left (x \right )-36}\)	\(45\)
risch	\(\frac {12}{x -4}-\frac {3 \left (5+{\mathrm e}^{3 \,{\mathrm e}^{3}}\right ) x^{2}}{\left (x -4\right ) \left ({\mathrm e}^{3 \,{\mathrm e}^{3}} x +x^{2}-x \ln \left (x \right )+10 x +4 \ln \left (x \right )-36\right )}\)	\(52\)

[In]

int((-12*ln(x)^2+(24*x+216)*ln(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*ln(x)^2+((-2*x^2+8*
x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*ln(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))+x^4+20*x^
3+28*x^2-720*x+1296),x,method=_RETURNVERBOSE)

[Out]

-3*(4*ln(x)+(exp(exp(3))^3+1)*x-36)/(exp(exp(3))^3*x+x^2-x*ln(x)+10*x+4*ln(x)-36)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x e^{\left (3 \, e^{3}\right )} + x + 4 \, \log \left (x\right ) - 36\right )}}{x^{2} + x e^{\left (3 \, e^{3}\right )} - {\left (x - 4\right )} \log \left (x\right ) + 10 \, x - 36} \]

[In]

integrate((-12*log(x)^2+(24*x+216)*log(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*log(x)^2+((
-2*x^2+8*x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*log(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))
+x^4+20*x^3+28*x^2-720*x+1296),x, algorithm="fricas")

[Out]

-3*(x*e^(3*e^3) + x + 4*log(x) - 36)/(x^2 + x*e^(3*e^3) - (x - 4)*log(x) + 10*x - 36)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).

Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=\frac {15 x^{2} + 3 x^{2} e^{3 e^{3}}}{- x^{3} - x^{2} e^{3 e^{3}} - 6 x^{2} + 76 x + 4 x e^{3 e^{3}} + \left (x^{2} - 8 x + 16\right ) \log {\left (x \right )} - 144} + \frac {12}{x - 4} \]

[In]

integrate((-12*ln(x)**2+(24*x+216)*ln(x)+(3*x**2-3*x)*exp(3*exp(3))+3*x**2-231*x-972)/((x**2-8*x+16)*ln(x)**2+
((-2*x**2+8*x)*exp(3*exp(3))-2*x**3-12*x**2+152*x-288)*ln(x)+x**2*exp(3*exp(3))**2+(2*x**3+20*x**2-72*x)*exp(3
*exp(3))+x**4+20*x**3+28*x**2-720*x+1296),x)

[Out]

(15*x**2 + 3*x**2*exp(3*exp(3)))/(-x**3 - x**2*exp(3*exp(3)) - 6*x**2 + 76*x + 4*x*exp(3*exp(3)) + (x**2 - 8*x
+ 16)*log(x) - 144) + 12/(x - 4)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x {\left (e^{\left (3 \, e^{3}\right )} + 1\right )} + 4 \, \log \left (x\right ) - 36\right )}}{x^{2} + x {\left (e^{\left (3 \, e^{3}\right )} + 10\right )} - {\left (x - 4\right )} \log \left (x\right ) - 36} \]

[In]

integrate((-12*log(x)^2+(24*x+216)*log(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*log(x)^2+((
-2*x^2+8*x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*log(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))
+x^4+20*x^3+28*x^2-720*x+1296),x, algorithm="maxima")

[Out]

-3*(x*(e^(3*e^3) + 1) + 4*log(x) - 36)/(x^2 + x*(e^(3*e^3) + 10) - (x - 4)*log(x) - 36)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x e^{\left (3 \, e^{3}\right )} + x + 4 \, \log \left (x\right ) - 36\right )}}{x^{2} + x e^{\left (3 \, e^{3}\right )} - x \log \left (x\right ) + 10 \, x + 4 \, \log \left (x\right ) - 36} \]

[In]

integrate((-12*log(x)^2+(24*x+216)*log(x)+(3*x^2-3*x)*exp(3*exp(3))+3*x^2-231*x-972)/((x^2-8*x+16)*log(x)^2+((
-2*x^2+8*x)*exp(3*exp(3))-2*x^3-12*x^2+152*x-288)*log(x)+x^2*exp(3*exp(3))^2+(2*x^3+20*x^2-72*x)*exp(3*exp(3))
+x^4+20*x^3+28*x^2-720*x+1296),x, algorithm="giac")

[Out]

-3*(x*e^(3*e^3) + x + 4*log(x) - 36)/(x^2 + x*e^(3*e^3) - x*log(x) + 10*x + 4*log(x) - 36)

Mupad [B] (verification not implemented)

Time = 15.87 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {-972-231 x+3 x^2+e^{3 e^3} \left (-3 x+3 x^2\right )+(216+24 x) \log (x)-12 \log ^2(x)}{1296-720 x+28 x^2+e^{6 e^3} x^2+20 x^3+x^4+e^{3 e^3} \left (-72 x+20 x^2+2 x^3\right )+\left (-288+152 x-12 x^2-2 x^3+e^{3 e^3} \left (8 x-2 x^2\right )\right ) \log (x)+\left (16-8 x+x^2\right ) \log ^2(x)} \, dx=-\frac {12\,\ln \left (x\right )+x\,\left (3\,{\mathrm {e}}^{3\,{\mathrm {e}}^3}+3\right )-108}{10\,x+4\,\ln \left (x\right )+x\,{\mathrm {e}}^{3\,{\mathrm {e}}^3}-x\,\ln \left (x\right )+x^2-36} \]

[In]

int(-(231*x + 12*log(x)^2 - log(x)*(24*x + 216) - 3*x^2 + exp(3*exp(3))*(3*x - 3*x^2) + 972)/(exp(3*exp(3))*(2
0*x^2 - 72*x + 2*x^3) - 720*x + x^2*exp(6*exp(3)) + log(x)^2*(x^2 - 8*x + 16) + 28*x^2 + 20*x^3 + x^4 - log(x)
*(12*x^2 - 152*x + 2*x^3 - exp(3*exp(3))*(8*x - 2*x^2) + 288) + 1296),x)

[Out]

-(12*log(x) + x*(3*exp(3*exp(3)) + 3) - 108)/(10*x + 4*log(x) + x*exp(3*exp(3)) - x*log(x) + x^2 - 36)