\(\int \frac {e^5 (54 e x+e^2 (36+6 x^3))+(-18 e x+e^2 (-12-2 x^3)) \log (\frac {e x^2+x^3}{-3+x^3})}{-3 x^2+x^5+e (-3 x+x^4)} \, dx\) [782]

Optimal result

Integrand size = 77, antiderivative size = 26 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \left (3 e^5-\log \left (\frac {e+x}{-\frac {3}{x^2}+x}\right )\right )^2 \] Output:

exp(1)*(3*exp(5)-ln((x+exp(1))/(x-3/x^2)))^2
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.75 (sec) , antiderivative size = 1186, normalized size of antiderivative = 45.62 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx =\text {Too large to display} \] Input:

Integrate[(E^5*(54*E*x + E^2*(36 + 6*x^3)) + (-18*E*x + E^2*(-12 - 2*x^3)) 
*Log[(E*x^2 + x^3)/(-3 + x^3)])/(-3*x^2 + x^5 + E*(-3*x + x^4)),x]
 

Output:

2*E*(-1/2*Log[-(-3)^(1/3) - x]^2 + (2*Log[3]*Log[3^(1/3) - x])/3 - Log[3^( 
1/3) - x]^2/2 - Log[-(-3)^(1/3) - x]*Log[(3^(1/3) - x)/((-3)^(1/3) + 3^(1/ 
3))] - Log[(-1)^(2/3)*3^(1/3) - x]^2/2 + Log[(3^(1/3) - x)/(3^(1/3) + E)]* 
Log[-E - x] + Log[((-1)^(2/3)*3^(1/3) - x)/((-1)^(2/3)*3^(1/3) + E)]*Log[- 
E - x] - Log[-E - x]^2/2 - 2*Log[-E - x]*(-1 + Log[-x]) - 6*E^5*Log[x] + ( 
2*Log[3]*Log[x])/3 - 2*Log[x]^2 + 2*Log[(-1)^(2/3)*3^(1/3) - x]*Log[-((-1/ 
3)^(1/3)*x)] + 2*Log[-(-3)^(1/3) - x]*Log[((-1)^(2/3)*x)/3^(1/3)] - Log[(- 
1)^(2/3)*3^(1/3) - x]*Log[((-I)*((-3)^(1/3) + x))/3^(5/6)] - Log[3^(1/3) - 
 x]*Log[((-3)^(1/3) + x)/((-3)^(1/3) + 3^(1/3))] + Log[-E - x]*Log[((-3)^( 
1/3) + x)/((-3)^(1/3) - E)] - 2*Log[x]*(-1 + Log[E + x]) + Log[3^(1/3) + E 
]*Log[E + x] + Log[-(-3)^(1/3) - x]*Log[-((E + x)/((-3)^(1/3) - E))] + Log 
[(-1)^(2/3)*3^(1/3) - x]*Log[(E + x)/((-1)^(2/3)*3^(1/3) + E)] + 2*Log[x]* 
Log[(3 - (-3)^(2/3)*x)/3] - Log[-(-3)^(1/3) - x]*Log[((-3)^(1/3) + (-1)^(2 
/3)*x)/((-3)^(1/3) + 3^(1/3))] + 2*Log[x]*Log[1 + (-1/3)^(1/3)*x] - Log[3^ 
(1/3) - x]*Log[(I + Sqrt[3] + ((2*I)*x)/3^(1/3))/(3*I + Sqrt[3])] - Log[(- 
1)^(2/3)*3^(1/3) - x]*Log[(((-2*I)/3)*(-3 + 3^(2/3)*x))/(3*I + Sqrt[3])] + 
 Log[-(-3)^(1/3) - x]*(3*E^5 - Log[(x^2*(E + x))/(-3 + x^3)]) + Log[3^(1/3 
) - x]*(3*E^5 - Log[(x^2*(E + x))/(-3 + x^3)]) + Log[(-1)^(2/3)*3^(1/3) - 
x]*(3*E^5 - Log[(x^2*(E + x))/(-3 + x^3)]) + 2*Log[x]*Log[(x^2*(E + x))/(- 
3 + x^3)] + Log[-E - x]*(-3*E^5 + Log[(x^2*(E + x))/(-3 + x^3)]) - Poly...
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 5.20 (sec) , antiderivative size = 1504, normalized size of antiderivative = 57.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2026, 2463, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^5 \left (e^2 \left (6 x^3+36\right )+54 e x\right )+\left (e^2 \left (-2 x^3-12\right )-18 e x\right ) \log \left (\frac {x^3+e x^2}{x^3-3}\right )}{x^5+e \left (x^4-3 x\right )-3 x^2} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^5 \left (e^2 \left (6 x^3+36\right )+54 e x\right )+\left (e^2 \left (-2 x^3-12\right )-18 e x\right ) \log \left (\frac {x^3+e x^2}{x^3-3}\right )}{x \left (x^4+e x^3-3 x-3 e\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {\left (x^2-e x+e^2\right ) \left (e^5 \left (e^2 \left (6 x^3+36\right )+54 e x\right )+\left (e^2 \left (-2 x^3-12\right )-18 e x\right ) \log \left (\frac {x^3+e x^2}{x^3-3}\right )\right )}{\left (3+e^3\right ) x \left (x^3-3\right )}-\frac {e^5 \left (e^2 \left (6 x^3+36\right )+54 e x\right )+\left (e^2 \left (-2 x^3-12\right )-18 e x\right ) \log \left (\frac {x^3+e x^2}{x^3-3}\right )}{\left (3+e^3\right ) x (x+e)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {e^2 \left (3 e^5-\log \left (-\frac {x^2 (x+e)}{3-x^3}\right )\right ) x^2}{3+e^3}-\frac {e^2 \log \left (-\frac {x^2 (x+e)}{3-x^3}\right ) x^2}{3+e^3}+\frac {3 e^7 x^2}{3+e^3}-e \log ^2\left (-x-\sqrt [3]{-3}\right )-e \log ^2\left (\sqrt [3]{3}-x\right )-e \log ^2\left ((-1)^{2/3} \sqrt [3]{3}-x\right )-\frac {4 e^4 \log ^2(x)}{3+e^3}-\frac {12 e \log ^2(x)}{3+e^3}-e \log ^2(x+e)-2 e \log \left (-x-\sqrt [3]{-3}\right ) \log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{-3}+\sqrt [3]{3}}\right )-2 e \log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3} \left (1-(-1)^{2/3}\right )}\right ) \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right )-2 e \log \left (-x-\sqrt [3]{-3}\right ) \log \left (-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{3}-x\right )}{\sqrt [3]{-3}+\sqrt [3]{3}}\right )+\frac {4}{3} e \log (3) \log (x)-\frac {12 e^9 \log (x)}{3+e^3}-4 e \log (x)+4 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (-\sqrt [3]{-\frac {1}{3}} x\right )+4 e \log \left (-x-\sqrt [3]{-3}\right ) \log \left (\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )-2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (-\frac {i \left (x+\sqrt [3]{-3}\right )}{3^{5/6}}\right )-2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (\frac {x+\sqrt [3]{-3}}{\sqrt [3]{-3}+\sqrt [3]{3}}\right )+2 e \log \left (\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right ) \log (x+e)+2 e \log \left (\frac {x+\sqrt [3]{-3}}{\sqrt [3]{-3}-e}\right ) \log (x+e)+2 e \log \left (-x-\sqrt [3]{-3}\right ) \log \left (-\frac {x+e}{\sqrt [3]{-3}-e}\right )+2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (\frac {x+e}{\sqrt [3]{3}+e}\right )+2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (\frac {x+e}{(-1)^{2/3} \sqrt [3]{3}+e}\right )-2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (-\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+\sqrt [3]{3}\right )}{\sqrt [3]{3} \left (1-(-1)^{2/3}\right )}\right )+2 e \log (x+e) \log \left (\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+\sqrt [3]{3}\right )}{(-1)^{2/3} \sqrt [3]{3}+e}\right )-\frac {4 e^4 \log (x) \log \left (\frac {x}{e}+1\right )}{3+e^3}-\frac {12 e \log (x) \log \left (\frac {x}{e}+1\right )}{3+e^3}-\frac {12 e \log (x) \left (3 e^5-\log \left (-\frac {x^2 (x+e)}{3-x^3}\right )\right )}{3+e^3}-2 e \log (x+e) \left (3 e^5-\log \left (-\frac {x^2 (x+e)}{3-x^3}\right )\right )-2 e \log \left (-x-\sqrt [3]{-3}\right ) \log \left (-\frac {x^2 (x+e)}{3-x^3}\right )-2 e \log \left (\sqrt [3]{3}-x\right ) \log \left (-\frac {x^2 (x+e)}{3-x^3}\right )-2 e \log \left ((-1)^{2/3} \sqrt [3]{3}-x\right ) \log \left (-\frac {x^2 (x+e)}{3-x^3}\right )+\frac {4 e^4 \log (x) \log \left (-\frac {x^2 (x+e)}{3-x^3}\right )}{3+e^3}+6 e^6 \log \left (3-x^3\right )+\frac {4 e^4 \log (x) \log \left (1-\frac {x^3}{3}\right )}{3+e^3}+\frac {12 e \log (x) \log \left (1-\frac {x^3}{3}\right )}{3+e^3}-2 e \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{3}-x}{\sqrt [3]{-3}+\sqrt [3]{3}}\right )-2 e \operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt [3]{3}-x\right )}{\sqrt [3]{3} \left (3-i \sqrt {3}\right )}\right )+2 e \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{3}-x}{\sqrt [3]{3}+e}\right )-2 e \operatorname {PolyLog}\left (2,\frac {i x+\sqrt [6]{-1} \sqrt [3]{3}}{3^{5/6}}\right )-2 e \operatorname {PolyLog}\left (2,\frac {2 i x+\sqrt [3]{3} \left (i+\sqrt {3}\right )}{\sqrt [3]{3} \left (3 i+\sqrt {3}\right )}\right )-4 e \operatorname {PolyLog}\left (2,\frac {x}{\sqrt [3]{3}}\right )-\frac {4 e^4 \operatorname {PolyLog}\left (2,-\frac {x}{e}\right )}{3+e^3}-\frac {12 e \operatorname {PolyLog}\left (2,-\frac {x}{e}\right )}{3+e^3}+4 e \operatorname {PolyLog}\left (2,-\frac {x}{e}\right )+\frac {4 e^4 \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )}{3 \left (3+e^3\right )}+\frac {4 e \operatorname {PolyLog}\left (2,\frac {x^3}{3}\right )}{3+e^3}-2 e \operatorname {PolyLog}\left (2,\frac {x+\sqrt [3]{-3}}{\sqrt [3]{-3}+\sqrt [3]{3}}\right )-2 e \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \left (x+\sqrt [3]{-3}\right )}{\sqrt [3]{-3}+\sqrt [3]{3}}\right )+2 e \operatorname {PolyLog}\left (2,\frac {x+\sqrt [3]{-3}}{\sqrt [3]{-3}-e}\right )+2 e \operatorname {PolyLog}\left (2,-\frac {x+e}{\sqrt [3]{-3}-e}\right )+2 e \operatorname {PolyLog}\left (2,\frac {x+e}{\sqrt [3]{3}+e}\right )+2 e \operatorname {PolyLog}\left (2,\frac {x+e}{(-1)^{2/3} \sqrt [3]{3}+e}\right )+2 e \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+\sqrt [3]{3}\right )}{(-1)^{2/3} \sqrt [3]{3}+e}\right )+4 e \operatorname {PolyLog}\left (2,\sqrt [3]{-\frac {1}{3}} x+1\right )+4 e \operatorname {PolyLog}\left (2,1-\frac {(-1)^{2/3} x}{\sqrt [3]{3}}\right )\)

Input:

Int[(E^5*(54*E*x + E^2*(36 + 6*x^3)) + (-18*E*x + E^2*(-12 - 2*x^3))*Log[( 
E*x^2 + x^3)/(-3 + x^3)])/(-3*x^2 + x^5 + E*(-3*x + x^4)),x]
 

Output:

(3*E^7*x^2)/(3 + E^3) - E*Log[-(-3)^(1/3) - x]^2 - E*Log[3^(1/3) - x]^2 - 
2*E*Log[-(-3)^(1/3) - x]*Log[(3^(1/3) - x)/((-3)^(1/3) + 3^(1/3))] - 2*E*L 
og[(3^(1/3) - x)/(3^(1/3)*(1 - (-1)^(2/3)))]*Log[(-1)^(2/3)*3^(1/3) - x] - 
 E*Log[(-1)^(2/3)*3^(1/3) - x]^2 - 2*E*Log[-(-3)^(1/3) - x]*Log[-(((-1)^(2 
/3)*((-1)^(2/3)*3^(1/3) - x))/((-3)^(1/3) + 3^(1/3)))] - 4*E*Log[x] - (12* 
E^9*Log[x])/(3 + E^3) + (4*E*Log[3]*Log[x])/3 - (12*E*Log[x]^2)/(3 + E^3) 
- (4*E^4*Log[x]^2)/(3 + E^3) + 4*E*Log[(-1)^(2/3)*3^(1/3) - x]*Log[-((-1/3 
)^(1/3)*x)] + 4*E*Log[-(-3)^(1/3) - x]*Log[((-1)^(2/3)*x)/3^(1/3)] - 2*E*L 
og[(-1)^(2/3)*3^(1/3) - x]*Log[((-I)*((-3)^(1/3) + x))/3^(5/6)] - 2*E*Log[ 
3^(1/3) - x]*Log[((-3)^(1/3) + x)/((-3)^(1/3) + 3^(1/3))] + 2*E*Log[(3^(1/ 
3) - x)/(3^(1/3) + E)]*Log[E + x] + 2*E*Log[((-3)^(1/3) + x)/((-3)^(1/3) - 
 E)]*Log[E + x] - E*Log[E + x]^2 + 2*E*Log[-(-3)^(1/3) - x]*Log[-((E + x)/ 
((-3)^(1/3) - E))] + 2*E*Log[3^(1/3) - x]*Log[(E + x)/(3^(1/3) + E)] + 2*E 
*Log[(-1)^(2/3)*3^(1/3) - x]*Log[(E + x)/((-1)^(2/3)*3^(1/3) + E)] - 2*E*L 
og[3^(1/3) - x]*Log[-(((-1)^(2/3)*(3^(1/3) + (-1)^(1/3)*x))/(3^(1/3)*(1 - 
(-1)^(2/3))))] + 2*E*Log[E + x]*Log[((-1)^(2/3)*(3^(1/3) + (-1)^(1/3)*x))/ 
((-1)^(2/3)*3^(1/3) + E)] - (12*E*Log[x]*Log[1 + x/E])/(3 + E^3) - (4*E^4* 
Log[x]*Log[1 + x/E])/(3 + E^3) - (E^2*x^2*(3*E^5 - Log[-((x^2*(E + x))/(3 
- x^3))]))/(3 + E^3) - (12*E*Log[x]*(3*E^5 - Log[-((x^2*(E + x))/(3 - x^3) 
)]))/(3 + E^3) - 2*E*Log[E + x]*(3*E^5 - Log[-((x^2*(E + x))/(3 - x^3))...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96

method	result	size
norman	\(-6 \,{\mathrm e} \,{\mathrm e}^{5} \ln \left (\frac {x^{2} {\mathrm e}+x^{3}}{x^{3}-3}\right )+{\mathrm e} \ln \left (\frac {x^{2} {\mathrm e}+x^{3}}{x^{3}-3}\right )^{2}\)	\(51\)
risch	\(\text {Expression too large to display}\)	\(1571\)
default	\(\text {Expression too large to display}\)	\(1579\)
parts	\(\text {Expression too large to display}\)	\(1579\)

Input:

int((((-2*x^3-12)*exp(1)^2-18*x*exp(1))*ln((x^2*exp(1)+x^3)/(x^3-3))+((6*x 
^3+36)*exp(1)^2+54*x*exp(1))*exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x,method 
=_RETURNVERBOSE)
 

Output:

-6*exp(1)*exp(5)*ln((x^2*exp(1)+x^3)/(x^3-3))+exp(1)*ln((x^2*exp(1)+x^3)/( 
x^3-3))^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right )^{2} - 6 \, e^{6} \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right ) \] Input:

integrate((((-2*x^3-12)*exp(1)^2-18*exp(1)*x)*log((x^2*exp(1)+x^3)/(x^3-3) 
)+((6*x^3+36)*exp(1)^2+54*exp(1)*x)*exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x 
, algorithm="fricas")
 

Output:

e*log((x^3 + x^2*e)/(x^3 - 3))^2 - 6*e^6*log((x^3 + x^2*e)/(x^3 - 3))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 1.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=- 12 e^{6} \log {\left (x \right )} + e \log {\left (\frac {x^{3} + e x^{2}}{x^{3} - 3} \right )}^{2} - 6 e^{6} \log {\left (x + e \right )} + 6 e^{6} \log {\left (x^{3} - 3 \right )} \] Input:

integrate((((-2*x**3-12)*exp(1)**2-18*exp(1)*x)*ln((x**2*exp(1)+x**3)/(x** 
3-3))+((6*x**3+36)*exp(1)**2+54*exp(1)*x)*exp(5))/((x**4-3*x)*exp(1)+x**5- 
3*x**2),x)
 

Output:

-12*exp(6)*log(x) + E*log((x**3 + E*x**2)/(x**3 - 3))**2 - 6*exp(6)*log(x 
+ E) + 6*exp(6)*log(x**3 - 3)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (27) = 54\).

Time = 2.96 (sec) , antiderivative size = 513, normalized size of antiderivative = 19.73 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx =\text {Too large to display} \] Input:

integrate((((-2*x^3-12)*exp(1)^2-18*exp(1)*x)*log((x^2*exp(1)+x^3)/(x^3-3) 
)+((6*x^3+36)*exp(1)^2+54*exp(1)*x)*exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x 
, algorithm="maxima")
 

Output:

e*log(x^3 - 3)^2 + e*log(x + e)^2 + 4*e*log(x + e)*log(x) + 4*e*log(x)^2 - 
 2*(6*e^(-1)*log(x) - 6*3^(1/6)*(3^(1/3)*e + 3^(2/3))*arctan(1/3*3^(1/6)*( 
2*x + 3^(1/3)))/(3^(2/3)*e^3 + 3*3^(2/3)) - (2*3^(2/3)*e^2 - 3*3^(1/3) + 3 
*e)*log(x^2 + 3^(1/3)*x + 3^(2/3))/(3^(2/3)*e^3 + 3*3^(2/3)) - 2*(3^(2/3)* 
e^2 + 3*3^(1/3) - 3*e)*log(x - 3^(1/3))/(3^(2/3)*e^3 + 3*3^(2/3)) - 18*log 
(x + e)/(e^4 + 3*e))*e^7 + (6*3^(1/6)*(3^(1/3)*e + 3^(2/3))*arctan(1/3*3^( 
1/6)*(2*x + 3^(1/3)))/(3^(2/3)*e^3 + 3*3^(2/3)) + (2*3^(2/3)*e^2 - 3*3^(1/ 
3) + 3*e)*log(x^2 + 3^(1/3)*x + 3^(2/3))/(3^(2/3)*e^3 + 3*3^(2/3)) + 2*(3^ 
(2/3)*e^2 + 3*3^(1/3) - 3*e)*log(x - 3^(1/3))/(3^(2/3)*e^3 + 3*3^(2/3)) - 
6*e^2*log(x + e)/(e^3 + 3))*e^7 - 9*(2*3^(1/6)*(3^(2/3)*e + 3^(1/3)*e^2)*a 
rctan(1/3*3^(1/6)*(2*x + 3^(1/3)))/(3^(2/3)*e^3 + 3*3^(2/3)) - (3^(1/3)*e 
+ 2*3^(2/3) - e^2)*log(x^2 + 3^(1/3)*x + 3^(2/3))/(3^(2/3)*e^3 + 3*3^(2/3) 
) + 2*(3^(1/3)*e - 3^(2/3) - e^2)*log(x - 3^(1/3))/(3^(2/3)*e^3 + 3*3^(2/3 
)) + 6*log(x + e)/(e^3 + 3))*e^6 - 2*(e*log(x + e) + 2*e*log(x))*log(x^3 - 
 3)
 

Giac [F]

\[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=\int { \frac {2 \, {\left (3 \, {\left ({\left (x^{3} + 6\right )} e^{2} + 9 \, x e\right )} e^{5} - {\left ({\left (x^{3} + 6\right )} e^{2} + 9 \, x e\right )} \log \left (\frac {x^{3} + x^{2} e}{x^{3} - 3}\right )\right )}}{x^{5} - 3 \, x^{2} + {\left (x^{4} - 3 \, x\right )} e} \,d x } \] Input:

integrate((((-2*x^3-12)*exp(1)^2-18*exp(1)*x)*log((x^2*exp(1)+x^3)/(x^3-3) 
)+((6*x^3+36)*exp(1)^2+54*exp(1)*x)*exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x 
, algorithm="giac")
 

Output:

integrate(2*(3*((x^3 + 6)*e^2 + 9*x*e)*e^5 - ((x^3 + 6)*e^2 + 9*x*e)*log(( 
x^3 + x^2*e)/(x^3 - 3)))/(x^5 - 3*x^2 + (x^4 - 3*x)*e), x)
 

Mupad [B] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=\mathrm {e}\,{\ln \left (\frac {x^3+\mathrm {e}\,x^2}{x^3-3}\right )}^2+6\,{\mathrm {e}}^6\,\ln \left (x^3-3\right )-6\,{\mathrm {e}}^6\,\ln \left (x+\mathrm {e}\right )-12\,{\mathrm {e}}^6\,\ln \left (x\right ) \] Input:

int((log((x^2*exp(1) + x^3)/(x^3 - 3))*(18*x*exp(1) + exp(2)*(2*x^3 + 12)) 
 - exp(5)*(54*x*exp(1) + exp(2)*(6*x^3 + 36)))/(exp(1)*(3*x - x^4) + 3*x^2 
 - x^5),x)
 

Output:

log((x^2*exp(1) + x^3)/(x^3 - 3))^2*exp(1) + 6*exp(6)*log(x^3 - 3) - 6*exp 
(6)*log(x + exp(1)) - 12*exp(6)*log(x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {e^5 \left (54 e x+e^2 \left (36+6 x^3\right )\right )+\left (-18 e x+e^2 \left (-12-2 x^3\right )\right ) \log \left (\frac {e x^2+x^3}{-3+x^3}\right )}{-3 x^2+x^5+e \left (-3 x+x^4\right )} \, dx=e \left (6 \,\mathrm {log}\left (3^{\frac {2}{3}}+3^{\frac {1}{3}} x +x^{2}\right ) e^{5}+6 \,\mathrm {log}\left (-3^{\frac {1}{3}}+x \right ) e^{5}-6 \,\mathrm {log}\left (e +x \right ) e^{5}+\mathrm {log}\left (\frac {e \,x^{2}+x^{3}}{x^{3}-3}\right )^{2}-12 \,\mathrm {log}\left (x \right ) e^{5}\right ) \] Input:

int((((-2*x^3-12)*exp(1)^2-18*exp(1)*x)*log((x^2*exp(1)+x^3)/(x^3-3))+((6* 
x^3+36)*exp(1)^2+54*exp(1)*x)*exp(5))/((x^4-3*x)*exp(1)+x^5-3*x^2),x)
 

Output:

e*(6*log(3**(2/3) + 3**(1/3)*x + x**2)*e**5 + 6*log( - 3**(1/3) + x)*e**5 
- 6*log(e + x)*e**5 + log((e*x**2 + x**3)/(x**3 - 3))**2 - 12*log(x)*e**5)