\(\int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 (-9-6 x-x^2)+(-6 e^4 x-4 e^9 x) \log (x)+(3 x^2+2 e^5 x^2) \log ^2(x)}{e^8 (9+6 x+x^2)+e^4 (-18 x-12 x^2-2 x^3) \log (x)+(9 x^2+6 x^3+x^4) \log ^2(x)} \, dx\) [4978]

Optimal result

Integrand size = 128, antiderivative size = 28 \[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {-2 e^5+x}{3+x}+\frac {x}{-e^4+x \log (x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=\int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx \]

[In]

Int[(3*E^8 + 2*E^13 - 9*x - 6*x^2 - x^3 + E^4*(-9 - 6*x - x^2) + (-6*E^4*x - 4*E^9*x)*Log[x] + (3*x^2 + 2*E^5*
x^2)*Log[x]^2)/(E^8*(9 + 6*x + x^2) + E^4*(-18*x - 12*x^2 - 2*x^3)*Log[x] + (9*x^2 + 6*x^3 + x^4)*Log[x]^2),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^8 \left (1+\frac {2 e^5}{3}\right )-e^4 (3+x)^2-x (3+x)^2-2 e^4 \left (3+2 e^5\right ) x \log (x)+\left (3+2 e^5\right ) x^2 \log ^2(x)}{(3+x)^2 \left (e^4-x \log (x)\right )^2} \, dx \\ & = \int \left (\frac {3+2 e^5}{(3+x)^2}+\frac {-e^4-x}{\left (e^4-x \log (x)\right )^2}\right ) \, dx \\ & = -\frac {3+2 e^5}{3+x}+\int \frac {-e^4-x}{\left (e^4-x \log (x)\right )^2} \, dx \\ & = -\frac {3+2 e^5}{3+x}+\int \left (-\frac {e^4}{\left (e^4-x \log (x)\right )^2}-\frac {x}{\left (e^4-x \log (x)\right )^2}\right ) \, dx \\ & = -\frac {3+2 e^5}{3+x}-e^4 \int \frac {1}{\left (e^4-x \log (x)\right )^2} \, dx-\int \frac {x}{\left (e^4-x \log (x)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {-3-2 e^5}{3+x}+\frac {x}{-e^4+x \log (x)} \]

[In]

Integrate[(3*E^8 + 2*E^13 - 9*x - 6*x^2 - x^3 + E^4*(-9 - 6*x - x^2) + (-6*E^4*x - 4*E^9*x)*Log[x] + (3*x^2 +
2*E^5*x^2)*Log[x]^2)/(E^8*(9 + 6*x + x^2) + E^4*(-18*x - 12*x^2 - 2*x^3)*Log[x] + (9*x^2 + 6*x^3 + x^4)*Log[x]
^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{5}}{3+x}-\frac {3}{3+x}-\frac {x}{-x \ln \left (x \right )+{\mathrm e}^{4}}\)	\(31\)
default	\(\frac {-3 x +\left (2 \,{\mathrm e}^{5}+3\right ) \ln \left (x \right ) x -x^{2}-2 \,{\mathrm e}^{4} {\mathrm e}^{5}-3 \,{\mathrm e}^{4}}{\left (3+x \right ) \left (-x \ln \left (x \right )+{\mathrm e}^{4}\right )}\)	\(46\)
norman	\(\frac {-3 x +\left (2 \,{\mathrm e}^{5}+3\right ) \ln \left (x \right ) x -x^{2}-2 \,{\mathrm e}^{4} {\mathrm e}^{5}-3 \,{\mathrm e}^{4}}{\left (3+x \right ) \left (-x \ln \left (x \right )+{\mathrm e}^{4}\right )}\)	\(46\)
parallelrisch	\(\frac {2 x \,{\mathrm e}^{5} \ln \left (x \right )-2 \,{\mathrm e}^{4} {\mathrm e}^{5}+3 x \ln \left (x \right )-x^{2}-3 \,{\mathrm e}^{4}-3 x}{-x^{2} \ln \left (x \right )+x \,{\mathrm e}^{4}-3 x \ln \left (x \right )+3 \,{\mathrm e}^{4}}\)	\(56\)

[In]

int(((2*x^2*exp(5)+3*x^2)*ln(x)^2+(-4*x*exp(4)*exp(5)-6*x*exp(4))*ln(x)+2*exp(4)^2*exp(5)+3*exp(4)^2+(-x^2-6*x
-9)*exp(4)-x^3-6*x^2-9*x)/((x^4+6*x^3+9*x^2)*ln(x)^2+(-2*x^3-12*x^2-18*x)*exp(4)*ln(x)+(x^2+6*x+9)*exp(4)^2),x
,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=-\frac {x^{2} - {\left (2 \, x e^{5} + 3 \, x\right )} \log \left (x\right ) + 3 \, x + 2 \, e^{9} + 3 \, e^{4}}{{\left (x + 3\right )} e^{4} - {\left (x^{2} + 3 \, x\right )} \log \left (x\right )} \]

[In]

integrate(((2*x^2*exp(5)+3*x^2)*log(x)^2+(-4*x*exp(4)*exp(5)-6*x*exp(4))*log(x)+2*exp(4)^2*exp(5)+3*exp(4)^2+(
-x^2-6*x-9)*exp(4)-x^3-6*x^2-9*x)/((x^4+6*x^3+9*x^2)*log(x)^2+(-2*x^3-12*x^2-18*x)*exp(4)*log(x)+(x^2+6*x+9)*e
xp(4)^2),x, algorithm="fricas")

[Out]

-(x^2 - (2*x*e^5 + 3*x)*log(x) + 3*x + 2*e^9 + 3*e^4)/((x + 3)*e^4 - (x^2 + 3*x)*log(x))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x}{x \log {\left (x \right )} - e^{4}} - \frac {3 + 2 e^{5}}{x + 3} \]

[In]

integrate(((2*x**2*exp(5)+3*x**2)*ln(x)**2+(-4*x*exp(4)*exp(5)-6*x*exp(4))*ln(x)+2*exp(4)**2*exp(5)+3*exp(4)**
2+(-x**2-6*x-9)*exp(4)-x**3-6*x**2-9*x)/((x**4+6*x**3+9*x**2)*ln(x)**2+(-2*x**3-12*x**2-18*x)*exp(4)*ln(x)+(x*
*2+6*x+9)*exp(4)**2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {x {\left (2 \, e^{5} + 3\right )} \log \left (x\right ) - x^{2} - 3 \, x - 2 \, e^{9} - 3 \, e^{4}}{x e^{4} - {\left (x^{2} + 3 \, x\right )} \log \left (x\right ) + 3 \, e^{4}} \]

[In]

integrate(((2*x^2*exp(5)+3*x^2)*log(x)^2+(-4*x*exp(4)*exp(5)-6*x*exp(4))*log(x)+2*exp(4)^2*exp(5)+3*exp(4)^2+(
-x^2-6*x-9)*exp(4)-x^3-6*x^2-9*x)/((x^4+6*x^3+9*x^2)*log(x)^2+(-2*x^3-12*x^2-18*x)*exp(4)*log(x)+(x^2+6*x+9)*e
xp(4)^2),x, algorithm="maxima")

[Out]

(x*(2*e^5 + 3)*log(x) - x^2 - 3*x - 2*e^9 - 3*e^4)/(x*e^4 - (x^2 + 3*x)*log(x) + 3*e^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).

Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=-\frac {2 \, x e^{5} \log \left (x\right ) - x^{2} + 3 \, x \log \left (x\right ) - 3 \, x - 2 \, e^{9} - 3 \, e^{4}}{x^{2} \log \left (x\right ) - x e^{4} + 3 \, x \log \left (x\right ) - 3 \, e^{4}} \]

[In]

integrate(((2*x^2*exp(5)+3*x^2)*log(x)^2+(-4*x*exp(4)*exp(5)-6*x*exp(4))*log(x)+2*exp(4)^2*exp(5)+3*exp(4)^2+(
-x^2-6*x-9)*exp(4)-x^3-6*x^2-9*x)/((x^4+6*x^3+9*x^2)*log(x)^2+(-2*x^3-12*x^2-18*x)*exp(4)*log(x)+(x^2+6*x+9)*e
xp(4)^2),x, algorithm="giac")

[Out]

-(2*x*e^5*log(x) - x^2 + 3*x*log(x) - 3*x - 2*e^9 - 3*e^4)/(x^2*log(x) - x*e^4 + 3*x*log(x) - 3*e^4)

Mupad [B] (verification not implemented)

Time = 11.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {3 e^8+2 e^{13}-9 x-6 x^2-x^3+e^4 \left (-9-6 x-x^2\right )+\left (-6 e^4 x-4 e^9 x\right ) \log (x)+\left (3 x^2+2 e^5 x^2\right ) \log ^2(x)}{e^8 \left (9+6 x+x^2\right )+e^4 \left (-18 x-12 x^2-2 x^3\right ) \log (x)+\left (9 x^2+6 x^3+x^4\right ) \log ^2(x)} \, dx=\frac {1}{\ln \left (x\right )-\frac {{\mathrm {e}}^4}{x}}-\frac {2\,{\mathrm {e}}^5+3}{x+3} \]

[In]

int(-(9*x - 3*exp(8) - 2*exp(13) + log(x)*(6*x*exp(4) + 4*x*exp(9)) + exp(4)*(6*x + x^2 + 9) - log(x)^2*(2*x^2
*exp(5) + 3*x^2) + 6*x^2 + x^3)/(log(x)^2*(9*x^2 + 6*x^3 + x^4) + exp(8)*(6*x + x^2 + 9) - exp(4)*log(x)*(18*x
+ 12*x^2 + 2*x^3)),x)

[Out]

1/(log(x) - exp(4)/x) - (2*exp(5) + 3)/(x + 3)