\(\int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e (81-2187 x^2+729 x^5-81 x^8+3 x^{11})+(729-891 x^3+171 x^6-9 x^9+e (-2187 x^2+486 x^5-27 x^8)) \log (\frac {1}{3} (3 e+x))+(243-270 x^3+27 x^6+e (-729 x^2+81 x^5)) \log ^2(\frac {1}{3} (3 e+x))+(27-81 e x^2-27 x^3) \log ^3(\frac {1}{3} (3 e+x))}{81 e+27 x} \, dx\) [1333]

Optimal result

Integrand size = 167, antiderivative size = 27 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=x+\frac {1}{4} \left (-3+\frac {x^3}{3}-\log \left (e+\frac {x}{3}\right )\right )^4 \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(27)=54\).

Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.26 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {1}{27} \left (27 x-243 x^3+\frac {81 x^6}{2}-3 x^9+\frac {x^{12}}{12}-x^3 \left (243-27 x^3+x^6\right ) \log \left (e+\frac {x}{3}\right )+\frac {9}{2} \left (-9+x^3\right )^2 \log ^2\left (e+\frac {x}{3}\right )-9 \left (-9+x^3\right ) \log ^3\left (e+\frac {x}{3}\right )+\frac {27}{4} \log ^4\left (e+\frac {x}{3}\right )+729 \log (3 e+x)\right ) \] Input:

Integrate[(729 + 27*x - 972*x^3 + 270*x^6 - 28*x^9 + x^12 + E*(81 - 2187*x 
^2 + 729*x^5 - 81*x^8 + 3*x^11) + (729 - 891*x^3 + 171*x^6 - 9*x^9 + E*(-2 
187*x^2 + 486*x^5 - 27*x^8))*Log[(3*E + x)/3] + (243 - 270*x^3 + 27*x^6 + 
E*(-729*x^2 + 81*x^5))*Log[(3*E + x)/3]^2 + (27 - 81*E*x^2 - 27*x^3)*Log[( 
3*E + x)/3]^3)/(81*E + 27*x),x]
 

Output:

(27*x - 243*x^3 + (81*x^6)/2 - 3*x^9 + x^12/12 - x^3*(243 - 27*x^3 + x^6)* 
Log[E + x/3] + (9*(-9 + x^3)^2*Log[E + x/3]^2)/2 - 9*(-9 + x^3)*Log[E + x/ 
3]^3 + (27*Log[E + x/3]^4)/4 + 729*Log[3*E + x])/27
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1209\) vs. \(2(27)=54\).

Time = 2.88 (sec) , antiderivative size = 1209, normalized size of antiderivative = 44.78, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {7293, 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.

\(\displaystyle \int \frac {x^{12}-28 x^9+270 x^6-972 x^3+\left (-27 x^3-81 e x^2+27\right ) \log ^3\left (\frac {1}{3} (x+3 e)\right )+e \left (3 x^{11}-81 x^8+729 x^5-2187 x^2+81\right )+\left (27 x^6-270 x^3+e \left (81 x^5-729 x^2\right )+243\right ) \log ^2\left (\frac {1}{3} (x+3 e)\right )+\left (-9 x^9+171 x^6-891 x^3+e \left (-27 x^8+486 x^5-2187 x^2\right )+729\right ) \log \left (\frac {1}{3} (x+3 e)\right )+27 x+729}{27 x+81 e} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^{12}}{27 (x+3 e)}-\frac {28 x^9}{27 (x+3 e)}+\frac {10 x^6}{x+3 e}-\frac {36 x^3}{x+3 e}-\frac {\left (x^3+3 e x^2-1\right ) \log ^3\left (\frac {x}{3}+e\right )}{x+3 e}+\frac {\left (x^3-9\right ) \left (x^3+3 e x^2-1\right ) \log ^2\left (\frac {x}{3}+e\right )}{x+3 e}-\frac {\left (x^3-9\right )^2 \left (x^3+3 e x^2-1\right ) \log \left (\frac {x}{3}+e\right )}{3 (x+3 e)}+\frac {e \left (x^{11}-27 x^8+243 x^5-729 x^2+27\right )}{9 (x+3 e)}+\frac {x}{x+3 e}+\frac {27}{x+3 e}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{12}}{324}-\frac {1}{27} \log \left (\frac {x}{3}+e\right ) x^9-\frac {x^9}{9}-\frac {3}{8} e \left (1+e^3\right ) x^8+\frac {3 e^4 x^8}{8}+\frac {3 e x^8}{8}+\frac {9}{7} e^2 \left (1+e^3\right ) x^7-\frac {9 e^5 x^7}{7}-\frac {9 e^2 x^7}{7}+\frac {1}{6} \log ^2\left (\frac {x}{3}+e\right ) x^6+\frac {19}{18} \log \left (\frac {x}{3}+e\right ) x^6-\frac {9}{2} e^3 \left (1+e^3\right ) x^6+\frac {9 e^6 x^6}{2}+\frac {9 e^3 x^6}{2}+\frac {161 x^6}{108}-\frac {1}{5} e \log \left (\frac {x}{3}+e\right ) x^5+\frac {27}{5} e \left (1+3 e^3+3 e^6\right ) x^5-\frac {81 e^7 x^5}{5}-\frac {81 e^4 x^5}{5}-\frac {799 e x^5}{150}+\frac {3}{4} e^2 \log \left (\frac {x}{3}+e\right ) x^4-\frac {81}{4} e^2 \left (1+3 e^3+3 e^6\right ) x^4+\frac {243 e^8 x^4}{4}+\frac {243 e^5 x^4}{4}+\frac {1583 e^2 x^4}{80}-\frac {10}{3} \log ^2\left (\frac {x}{3}+e\right ) x^3-\left (11+3 e^3\right ) \log \left (\frac {x}{3}+e\right ) x^3+81 e^3 \left (1+3 e^3+3 e^6\right ) x^3+\frac {1}{3} \left (11+3 e^3\right ) x^3-243 e^9 x^3-243 e^6 x^3-\frac {1583 e^3 x^3}{20}-12 x^3+\frac {9}{2} e \left (2+3 e^3\right ) \log \left (\frac {x}{3}+e\right ) x^2-\frac {3}{2} e \left (11+3 e^3\right ) x^2-\frac {9}{4} e \left (2+3 e^3\right ) x^2-\frac {81}{2} e \left (1+3 e^3\right )^3 x^2+\frac {2187 e^{10} x^2}{2}+\frac {2187 e^7 x^2}{2}+\frac {14247 e^4 x^2}{40}+54 e x^2+9 e^2 \left (11+3 e^3\right ) x+\frac {81}{2} e^2 \left (2+3 e^3\right ) x+243 e^2 \left (1+3 e^3\right )^3 x-6561 e^{11} x-6561 e^8 x-\frac {52461 e^5 x}{20}-486 e^2 x+x+\frac {1}{108} (x+3 e)^6-\frac {6}{25} e (x+3 e)^5+\frac {45}{16} e^2 (x+3 e)^4+\frac {1}{4} \log ^4\left (\frac {x}{3}+e\right )-20 e^3 (x+3 e)^3-\frac {2}{3} (x+3 e)^3-\frac {1}{3} (x+3 e)^3 \log ^3\left (\frac {x}{3}+e\right )+3 e (x+3 e)^2 \log ^3\left (\frac {x}{3}+e\right )-9 e^2 (x+3 e) \log ^3\left (\frac {x}{3}+e\right )+3 \left (1+3 e^3\right ) \log ^3\left (\frac {x}{3}+e\right )+\frac {405}{4} e^4 (x+3 e)^2+\frac {27}{2} e (x+3 e)^2+\frac {1}{3} (x+3 e)^3 \log ^2\left (\frac {x}{3}+e\right )-3 e (x+3 e)^2 \log ^2\left (\frac {x}{3}+e\right )+9 e^2 (x+3 e) \log ^2\left (\frac {x}{3}+e\right )+\frac {27}{2} \left (1+3 e^3\right )^2 \log ^2\left (\frac {x}{3}+e\right )-\frac {243}{2} e^6 \log ^2\left (\frac {x}{3}+e\right )-90 e^3 \log ^2\left (\frac {x}{3}+e\right )-\frac {1}{18} (x+3 e)^6 \log \left (\frac {x}{3}+e\right )+\frac {6}{5} e (x+3 e)^5 \log \left (\frac {x}{3}+e\right )-\frac {45}{4} e^2 (x+3 e)^4 \log \left (\frac {x}{3}+e\right )+60 e^3 (x+3 e)^3 \log \left (\frac {x}{3}+e\right )+2 (x+3 e)^3 \log \left (\frac {x}{3}+e\right )-\frac {405}{2} e^4 (x+3 e)^2 \log \left (\frac {x}{3}+e\right )-27 e (x+3 e)^2 \log \left (\frac {x}{3}+e\right )-27 e^2 \left (2+3 e^3\right ) (x+3 e) \log \left (\frac {x}{3}+e\right )+486 e^5 (x+3 e) \log \left (\frac {x}{3}+e\right )+162 e^2 (x+3 e) \log \left (\frac {x}{3}+e\right )+3 e \left (1-243 e^2-2187 e^5-6561 e^8-6561 e^{11}\right ) \log (x+3 e)-27 e^3 \left (11+3 e^3\right ) \log (x+3 e)-\frac {81}{2} e^3 \left (2+3 e^3\right ) \log (x+3 e)+19683 e^{12} \log (x+3 e)+19683 e^9 \log (x+3 e)+\frac {128223}{20} e^6 \log (x+3 e)+972 e^3 \log (x+3 e)-3 e \log (x+3 e)+27 \log (x+3 e)\)

Input:

Int[(729 + 27*x - 972*x^3 + 270*x^6 - 28*x^9 + x^12 + E*(81 - 2187*x^2 + 7 
29*x^5 - 81*x^8 + 3*x^11) + (729 - 891*x^3 + 171*x^6 - 9*x^9 + E*(-2187*x^ 
2 + 486*x^5 - 27*x^8))*Log[(3*E + x)/3] + (243 - 270*x^3 + 27*x^6 + E*(-72 
9*x^2 + 81*x^5))*Log[(3*E + x)/3]^2 + (27 - 81*E*x^2 - 27*x^3)*Log[(3*E + 
x)/3]^3)/(81*E + 27*x),x]
 

Output:

x - 486*E^2*x - (52461*E^5*x)/20 - 6561*E^8*x - 6561*E^11*x + 243*E^2*(1 + 
 3*E^3)^3*x + (81*E^2*(2 + 3*E^3)*x)/2 + 9*E^2*(11 + 3*E^3)*x + 54*E*x^2 + 
 (14247*E^4*x^2)/40 + (2187*E^7*x^2)/2 + (2187*E^10*x^2)/2 - (81*E*(1 + 3* 
E^3)^3*x^2)/2 - (9*E*(2 + 3*E^3)*x^2)/4 - (3*E*(11 + 3*E^3)*x^2)/2 - 12*x^ 
3 - (1583*E^3*x^3)/20 - 243*E^6*x^3 - 243*E^9*x^3 + ((11 + 3*E^3)*x^3)/3 + 
 81*E^3*(1 + 3*E^3 + 3*E^6)*x^3 + (1583*E^2*x^4)/80 + (243*E^5*x^4)/4 + (2 
43*E^8*x^4)/4 - (81*E^2*(1 + 3*E^3 + 3*E^6)*x^4)/4 - (799*E*x^5)/150 - (81 
*E^4*x^5)/5 - (81*E^7*x^5)/5 + (27*E*(1 + 3*E^3 + 3*E^6)*x^5)/5 + (161*x^6 
)/108 + (9*E^3*x^6)/2 + (9*E^6*x^6)/2 - (9*E^3*(1 + E^3)*x^6)/2 - (9*E^2*x 
^7)/7 - (9*E^5*x^7)/7 + (9*E^2*(1 + E^3)*x^7)/7 + (3*E*x^8)/8 + (3*E^4*x^8 
)/8 - (3*E*(1 + E^3)*x^8)/8 - x^9/9 + x^12/324 + (27*E*(3*E + x)^2)/2 + (4 
05*E^4*(3*E + x)^2)/4 - (2*(3*E + x)^3)/3 - 20*E^3*(3*E + x)^3 + (45*E^2*( 
3*E + x)^4)/16 - (6*E*(3*E + x)^5)/25 + (3*E + x)^6/108 + (9*E*(2 + 3*E^3) 
*x^2*Log[E + x/3])/2 - (11 + 3*E^3)*x^3*Log[E + x/3] + (3*E^2*x^4*Log[E + 
x/3])/4 - (E*x^5*Log[E + x/3])/5 + (19*x^6*Log[E + x/3])/18 - (x^9*Log[E + 
 x/3])/27 + 162*E^2*(3*E + x)*Log[E + x/3] + 486*E^5*(3*E + x)*Log[E + x/3 
] - 27*E^2*(2 + 3*E^3)*(3*E + x)*Log[E + x/3] - 27*E*(3*E + x)^2*Log[E + x 
/3] - (405*E^4*(3*E + x)^2*Log[E + x/3])/2 + 2*(3*E + x)^3*Log[E + x/3] + 
60*E^3*(3*E + x)^3*Log[E + x/3] - (45*E^2*(3*E + x)^4*Log[E + x/3])/4 + (6 
*E*(3*E + x)^5*Log[E + x/3])/5 - ((3*E + x)^6*Log[E + x/3])/18 - 90*E^3...
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(661\) vs. \(2(22)=44\).

Time = 0.02 (sec) , antiderivative size = 662, normalized size of antiderivative = 24.52

\[x +3 \,{\mathrm e}+\frac {\ln \left ({\mathrm e}+\frac {x}{3}\right )^{4}}{4}-\frac {6561 \,{\mathrm e}^{12}}{4}+27 \ln \left ({\mathrm e}+\frac {x}{3}\right )-2187 \,{\mathrm e}^{9}-\frac {2187 \,{\mathrm e}^{6}}{2}+\frac {x^{12}}{324}-\frac {x^{9}}{9}-243 \,{\mathrm e}^{3}-9 x^{3}+\frac {3 x^{6}}{2}+3 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3}+\frac {27 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}}{2}-729 \ln \left ({\mathrm e}+\frac {x}{3}\right ) \left ({\mathrm e}+\frac {x}{3}\right )^{9}-9 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3} \left ({\mathrm e}+\frac {x}{3}\right )^{3}-81 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} \left ({\mathrm e}+\frac {x}{3}\right )^{3}-243 \ln \left ({\mathrm e}+\frac {x}{3}\right ) \left ({\mathrm e}+\frac {x}{3}\right )^{3}+\frac {243 \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} \left ({\mathrm e}+\frac {x}{3}\right )^{6}}{2}+729 \ln \left ({\mathrm e}+\frac {x}{3}\right ) \left ({\mathrm e}+\frac {x}{3}\right )^{6}+81 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{2}+27 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{2}-3 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{5}-18 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{5}+3 \,{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3} x^{2}+{\mathrm e} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{8}+243 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x -135 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{4}+9 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3} x +81 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x -\frac {45 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{4}}{2}+12 \,{\mathrm e}^{2} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{7}-243 \,{\mathrm e}^{5} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x -1458 \,{\mathrm e}^{5} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x -\frac {405 \,{\mathrm e}^{4} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{2}}{2}-1215 \,{\mathrm e}^{4} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{2}-90 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2} x^{3}-540 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{3}+1134 \,{\mathrm e}^{5} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{4}+2187 \,{\mathrm e}^{8} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x +2916 \,{\mathrm e}^{7} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{2}+2268 \,{\mathrm e}^{6} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{3}+378 \,{\mathrm e}^{4} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{5}+84 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right ) x^{6}+729 \,{\mathrm e}^{9} \ln \left ({\mathrm e}+\frac {x}{3}\right )-\frac {243 \,{\mathrm e}^{6} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}}{2}+9 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{3}+81 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right )^{2}+243 \,{\mathrm e}^{3} \ln \left ({\mathrm e}+\frac {x}{3}\right )-729 \,{\mathrm e}^{6} \ln \left ({\mathrm e}+\frac {x}{3}\right )\]

Input:

int(((-81*x^2*exp(1)-27*x^3+27)*ln(exp(1)+1/3*x)^3+((81*x^5-729*x^2)*exp(1 
)+27*x^6-270*x^3+243)*ln(exp(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^2)*exp(1 
)-9*x^9+171*x^6-891*x^3+729)*ln(exp(1)+1/3*x)+(3*x^11-81*x^8+729*x^5-2187* 
x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+27*x),x)
 

Output:

x+3*exp(1)+1/4*ln(exp(1)+1/3*x)^4+27*ln(exp(1)+1/3*x)+1/324*x^12-243*exp(1 
)^3-1/9*x^9-9*x^3+3/2*x^6-2187*exp(1)^9+3*ln(exp(1)+1/3*x)^3+27/2*ln(exp(1 
)+1/3*x)^2-729*ln(exp(1)+1/3*x)*(exp(1)+1/3*x)^9+729*exp(1)^9*ln(exp(1)+1/ 
3*x)-243/2*exp(1)^6*ln(exp(1)+1/3*x)^2+9*exp(1)^3*ln(exp(1)+1/3*x)^3-9*ln( 
exp(1)+1/3*x)^3*(exp(1)+1/3*x)^3+81*exp(1)^3*ln(exp(1)+1/3*x)^2-81*ln(exp( 
1)+1/3*x)^2*(exp(1)+1/3*x)^3+243*exp(1)^3*ln(exp(1)+1/3*x)-243*ln(exp(1)+1 
/3*x)*(exp(1)+1/3*x)^3+243/2*ln(exp(1)+1/3*x)^2*(exp(1)+1/3*x)^6-729*exp(1 
)^6*ln(exp(1)+1/3*x)+729*ln(exp(1)+1/3*x)*(exp(1)+1/3*x)^6-6561/4*exp(1)^1 
2-2187/2*exp(1)^6+81*exp(1)*ln(exp(1)+1/3*x)*x^2+27*exp(1)*ln(exp(1)+1/3*x 
)^2*x^2+243*exp(1)^2*ln(exp(1)+1/3*x)*x-3*exp(1)*ln(exp(1)+1/3*x)^2*x^5-13 
5*exp(1)^2*ln(exp(1)+1/3*x)*x^4-18*exp(1)*ln(exp(1)+1/3*x)*x^5+9*exp(1)^2* 
ln(exp(1)+1/3*x)^3*x+3*exp(1)*ln(exp(1)+1/3*x)^3*x^2+81*exp(1)^2*ln(exp(1) 
+1/3*x)^2*x+exp(1)*ln(exp(1)+1/3*x)*x^8-45/2*exp(1)^2*ln(exp(1)+1/3*x)^2*x 
^4+12*exp(1)^2*ln(exp(1)+1/3*x)*x^7-243*exp(1)^5*ln(exp(1)+1/3*x)^2*x-1458 
*exp(1)^5*ln(exp(1)+1/3*x)*x-405/2*exp(1)^4*ln(exp(1)+1/3*x)^2*x^2-1215*ex 
p(1)^4*ln(exp(1)+1/3*x)*x^2-90*exp(1)^3*ln(exp(1)+1/3*x)^2*x^3-540*exp(1)^ 
3*ln(exp(1)+1/3*x)*x^3+1134*exp(1)^5*ln(exp(1)+1/3*x)*x^4+2187*exp(1)^8*ln 
(exp(1)+1/3*x)*x+2916*exp(1)^7*ln(exp(1)+1/3*x)*x^2+2268*exp(1)^6*ln(exp(1 
)+1/3*x)*x^3+378*exp(1)^4*ln(exp(1)+1/3*x)*x^5+84*exp(1)^3*ln(exp(1)+1/3*x 
)*x^6
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (20) = 40\).

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {1}{324} \, x^{12} - \frac {1}{9} \, x^{9} + \frac {3}{2} \, x^{6} - \frac {1}{3} \, {\left (x^{3} - 9\right )} \log \left (\frac {1}{3} \, x + e\right )^{3} + \frac {1}{4} \, \log \left (\frac {1}{3} \, x + e\right )^{4} - 9 \, x^{3} + \frac {1}{6} \, {\left (x^{6} - 18 \, x^{3} + 81\right )} \log \left (\frac {1}{3} \, x + e\right )^{2} - \frac {1}{27} \, {\left (x^{9} - 27 \, x^{6} + 243 \, x^{3} - 729\right )} \log \left (\frac {1}{3} \, x + e\right ) + x \] Input:

integrate(((-81*x^2*exp(1)-27*x^3+27)*log(exp(1)+1/3*x)^3+((81*x^5-729*x^2 
)*exp(1)+27*x^6-270*x^3+243)*log(exp(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^ 
2)*exp(1)-9*x^9+171*x^6-891*x^3+729)*log(exp(1)+1/3*x)+(3*x^11-81*x^8+729* 
x^5-2187*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+2 
7*x),x, algorithm="fricas")
 

Output:

1/324*x^12 - 1/9*x^9 + 3/2*x^6 - 1/3*(x^3 - 9)*log(1/3*x + e)^3 + 1/4*log( 
1/3*x + e)^4 - 9*x^3 + 1/6*(x^6 - 18*x^3 + 81)*log(1/3*x + e)^2 - 1/27*(x^ 
9 - 27*x^6 + 243*x^3 - 729)*log(1/3*x + e) + x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (19) = 38\).

Time = 0.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {x^{12}}{324} - \frac {x^{9}}{9} + \frac {3 x^{6}}{2} - 9 x^{3} + x + \left (3 - \frac {x^{3}}{3}\right ) \log {\left (\frac {x}{3} + e \right )}^{3} + \left (\frac {x^{6}}{6} - 3 x^{3} + \frac {27}{2}\right ) \log {\left (\frac {x}{3} + e \right )}^{2} + \left (- \frac {x^{9}}{27} + x^{6} - 9 x^{3}\right ) \log {\left (\frac {x}{3} + e \right )} + \frac {\log {\left (\frac {x}{3} + e \right )}^{4}}{4} + 27 \log {\left (x + 3 e \right )} \] Input:

integrate(((-81*x**2*exp(1)-27*x**3+27)*ln(exp(1)+1/3*x)**3+((81*x**5-729* 
x**2)*exp(1)+27*x**6-270*x**3+243)*ln(exp(1)+1/3*x)**2+((-27*x**8+486*x**5 
-2187*x**2)*exp(1)-9*x**9+171*x**6-891*x**3+729)*ln(exp(1)+1/3*x)+(3*x**11 
-81*x**8+729*x**5-2187*x**2+81)*exp(1)+x**12-28*x**9+270*x**6-972*x**3+27* 
x+729)/(81*exp(1)+27*x),x)
 

Output:

x**12/324 - x**9/9 + 3*x**6/2 - 9*x**3 + x + (3 - x**3/3)*log(x/3 + E)**3 
+ (x**6/6 - 3*x**3 + 27/2)*log(x/3 + E)**2 + (-x**9/27 + x**6 - 9*x**3)*lo 
g(x/3 + E) + log(x/3 + E)**4/4 + 27*log(x + 3*E)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1976 vs. \(2 (20) = 40\).

Time = 0.15 (sec) , antiderivative size = 1976, normalized size of antiderivative = 73.19 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\text {Too large to display} \] Input:

integrate(((-81*x^2*exp(1)-27*x^3+27)*log(exp(1)+1/3*x)^3+((81*x^5-729*x^2 
)*exp(1)+27*x^6-270*x^3+243)*log(exp(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^ 
2)*exp(1)-9*x^9+171*x^6-891*x^3+729)*log(exp(1)+1/3*x)+(3*x^11-81*x^8+729* 
x^5-2187*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+2 
7*x),x, algorithm="maxima")
 

Output:

1/324*x^12 - 1/99*x^11*e + 1/30*x^10*e^2 - 1/9*x^9*e^3 - 1/9*x^9 + 3/8*x^8 
*e^4 + 23/64*x^8*e - 9/7*x^7*e^5 - 459/392*x^7*e^2 + 1/108*(18*log(1/3*x + 
 e)^2 - 6*log(1/3*x + e) + 1)*(x + 3*e)^6 + 9/2*x^6*e^6 + 431/112*x^6*e^3 
- 18/125*(25*e*log(1/3*x + e)^2 - 10*e*log(1/3*x + e) + 2*e)*(x + 3*e)^5 + 
 161/108*x^6 - 81/5*x^5*e^7 - 17883/1400*x^5*e^4 - 691/150*x^5*e + 135/32* 
(8*e^2*log(1/3*x + e)^2 - 4*e^2*log(1/3*x + e) + e^2)*(x + 3*e)^4 + 243/4* 
x^4*e^8 + 47979/1120*x^4*e^5 + 1097/80*x^4*e^2 + 27/4*e^3*log(1/3*x + e)^4 
 - 20*(9*e^3*log(1/3*x + e)^2 - 6*e^3*log(1/3*x + e) + 2*e^3)*(x + 3*e)^3 
- 1/27*(9*log(1/3*x + e)^3 - 9*log(1/3*x + e)^2 + 6*log(1/3*x + e) - 2)*(x 
 + 3*e)^3 - 10/27*(9*log(1/3*x + e)^2 - 6*log(1/3*x + e) + 2)*(x + 3*e)^3 
- 243*x^3*e^9 - 40419/280*x^3*e^6 - 717/20*x^3*e^3 + 243*e^6*log(1/3*x + e 
)^3 + 90*e^3*log(1/3*x + e)^3 + 1/4*log(1/3*x + e)^4 + 9/8*(4*e*log(1/3*x 
+ e)^3 - 6*e*log(1/3*x + e)^2 + 6*e*log(1/3*x + e) - 3*e)*(x + 3*e)^2 + 12 
15/4*(2*e^4*log(1/3*x + e)^2 - 2*e^4*log(1/3*x + e) + e^4)*(x + 3*e)^2 + 4 
5/2*(2*e*log(1/3*x + e)^2 - 2*e*log(1/3*x + e) + e)*(x + 3*e)^2 - 25/3*x^3 
 + 2187/2*x^2*e^10 + 261711/560*x^2*e^7 + 1323/40*x^2*e^4 + 51/4*x^2*e - 6 
561/2*e^9*log(x + 3*e)^2 - 4617/2*e^6*log(x + 3*e)^2 - 891/2*e^3*log(x + 3 
*e)^2 - 1/280*(35*x^8 - 120*x^7*e + 420*x^6*e^2 - 1512*x^5*e^3 + 5670*x^4* 
e^4 - 22680*x^3*e^5 + 102060*x^2*e^6 - 612360*x*e^7 + 1837080*e^8*log(x + 
3*e))*e*log(1/3*x + e) + 9/10*(4*x^5 - 15*x^4*e + 60*x^3*e^2 - 270*x^2*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (20) = 40\).

Time = 0.15 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.22 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=\frac {1}{324} \, x^{12} - \frac {1}{27} \, x^{9} \log \left (\frac {1}{3} \, x + e\right ) - \frac {1}{9} \, x^{9} + \frac {1}{6} \, x^{6} \log \left (\frac {1}{3} \, x + e\right )^{2} + x^{6} \log \left (\frac {1}{3} \, x + e\right ) + \frac {3}{2} \, x^{6} - \frac {1}{3} \, x^{3} \log \left (\frac {1}{3} \, x + e\right )^{3} - 3 \, x^{3} \log \left (\frac {1}{3} \, x + e\right )^{2} - 9 \, x^{3} \log \left (\frac {1}{3} \, x + e\right ) + \frac {1}{4} \, \log \left (\frac {1}{3} \, x + e\right )^{4} - 9 \, x^{3} + 3 \, \log \left (\frac {1}{3} \, x + e\right )^{3} + \frac {27}{2} \, \log \left (\frac {1}{3} \, x + e\right )^{2} + x + 27 \, \log \left (\frac {1}{3} \, x + e\right ) \] Input:

integrate(((-81*x^2*exp(1)-27*x^3+27)*log(exp(1)+1/3*x)^3+((81*x^5-729*x^2 
)*exp(1)+27*x^6-270*x^3+243)*log(exp(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^ 
2)*exp(1)-9*x^9+171*x^6-891*x^3+729)*log(exp(1)+1/3*x)+(3*x^11-81*x^8+729* 
x^5-2187*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+2 
7*x),x, algorithm="giac")
 

Output:

1/324*x^12 - 1/27*x^9*log(1/3*x + e) - 1/9*x^9 + 1/6*x^6*log(1/3*x + e)^2 
+ x^6*log(1/3*x + e) + 3/2*x^6 - 1/3*x^3*log(1/3*x + e)^3 - 3*x^3*log(1/3* 
x + e)^2 - 9*x^3*log(1/3*x + e) + 1/4*log(1/3*x + e)^4 - 9*x^3 + 3*log(1/3 
*x + e)^3 + 27/2*log(1/3*x + e)^2 + x + 27*log(1/3*x + e)
 

Mupad [B] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=x+27\,\ln \left (x+3\,\mathrm {e}\right )-{\ln \left (\frac {x}{3}+\mathrm {e}\right )}^3\,\left (\frac {x^3}{3}-3\right )+\frac {{\ln \left (\frac {x}{3}+\mathrm {e}\right )}^4}{4}-\ln \left (\frac {x}{3}+\mathrm {e}\right )\,\left (\frac {x^9}{27}-x^6+9\,x^3\right )+{\ln \left (\frac {x}{3}+\mathrm {e}\right )}^2\,\left (\frac {x^6}{6}-3\,x^3+\frac {27}{2}\right )-9\,x^3+\frac {3\,x^6}{2}-\frac {x^9}{9}+\frac {x^{12}}{324} \] Input:

int((27*x - log(x/3 + exp(1))^2*(exp(1)*(729*x^2 - 81*x^5) + 270*x^3 - 27* 
x^6 - 243) - log(x/3 + exp(1))*(exp(1)*(2187*x^2 - 486*x^5 + 27*x^8) + 891 
*x^3 - 171*x^6 + 9*x^9 - 729) - 972*x^3 + 270*x^6 - 28*x^9 + x^12 - log(x/ 
3 + exp(1))^3*(81*x^2*exp(1) + 27*x^3 - 27) + exp(1)*(729*x^5 - 2187*x^2 - 
 81*x^8 + 3*x^11 + 81) + 729)/(27*x + 81*exp(1)),x)
 

Output:

x + 27*log(x + 3*exp(1)) - log(x/3 + exp(1))^3*(x^3/3 - 3) + log(x/3 + exp 
(1))^4/4 - log(x/3 + exp(1))*(9*x^3 - x^6 + x^9/27) + log(x/3 + exp(1))^2* 
(x^6/6 - 3*x^3 + 27/2) - 9*x^3 + (3*x^6)/2 - x^9/9 + x^12/324
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.30 \[ \int \frac {729+27 x-972 x^3+270 x^6-28 x^9+x^{12}+e \left (81-2187 x^2+729 x^5-81 x^8+3 x^{11}\right )+\left (729-891 x^3+171 x^6-9 x^9+e \left (-2187 x^2+486 x^5-27 x^8\right )\right ) \log \left (\frac {1}{3} (3 e+x)\right )+\left (243-270 x^3+27 x^6+e \left (-729 x^2+81 x^5\right )\right ) \log ^2\left (\frac {1}{3} (3 e+x)\right )+\left (27-81 e x^2-27 x^3\right ) \log ^3\left (\frac {1}{3} (3 e+x)\right )}{81 e+27 x} \, dx=729 \,\mathrm {log}\left (3 e +x \right ) e^{9}+729 \,\mathrm {log}\left (3 e +x \right ) e^{6}+243 \,\mathrm {log}\left (3 e +x \right ) e^{3}+27 \,\mathrm {log}\left (3 e +x \right )+\frac {\mathrm {log}\left (e +\frac {x}{3}\right )^{4}}{4}-\frac {\mathrm {log}\left (e +\frac {x}{3}\right )^{3} x^{3}}{3}+3 \mathrm {log}\left (e +\frac {x}{3}\right )^{3}+\frac {\mathrm {log}\left (e +\frac {x}{3}\right )^{2} x^{6}}{6}-3 \mathrm {log}\left (e +\frac {x}{3}\right )^{2} x^{3}+\frac {27 \mathrm {log}\left (e +\frac {x}{3}\right )^{2}}{2}-729 \,\mathrm {log}\left (e +\frac {x}{3}\right ) e^{9}-729 \,\mathrm {log}\left (e +\frac {x}{3}\right ) e^{6}-243 \,\mathrm {log}\left (e +\frac {x}{3}\right ) e^{3}-\frac {\mathrm {log}\left (e +\frac {x}{3}\right ) x^{9}}{27}+\mathrm {log}\left (e +\frac {x}{3}\right ) x^{6}-9 \,\mathrm {log}\left (e +\frac {x}{3}\right ) x^{3}+\frac {x^{12}}{324}-\frac {x^{9}}{9}+\frac {3 x^{6}}{2}-9 x^{3}+x \] Input:

int(((-81*x^2*exp(1)-27*x^3+27)*log(exp(1)+1/3*x)^3+((81*x^5-729*x^2)*exp( 
1)+27*x^6-270*x^3+243)*log(exp(1)+1/3*x)^2+((-27*x^8+486*x^5-2187*x^2)*exp 
(1)-9*x^9+171*x^6-891*x^3+729)*log(exp(1)+1/3*x)+(3*x^11-81*x^8+729*x^5-21 
87*x^2+81)*exp(1)+x^12-28*x^9+270*x^6-972*x^3+27*x+729)/(81*exp(1)+27*x),x 
)
 

Output:

(236196*log(3*e + x)*e**9 + 236196*log(3*e + x)*e**6 + 78732*log(3*e + x)* 
e**3 + 8748*log(3*e + x) + 81*log((3*e + x)/3)**4 - 108*log((3*e + x)/3)** 
3*x**3 + 972*log((3*e + x)/3)**3 + 54*log((3*e + x)/3)**2*x**6 - 972*log(( 
3*e + x)/3)**2*x**3 + 4374*log((3*e + x)/3)**2 - 236196*log((3*e + x)/3)*e 
**9 - 236196*log((3*e + x)/3)*e**6 - 78732*log((3*e + x)/3)*e**3 - 12*log( 
(3*e + x)/3)*x**9 + 324*log((3*e + x)/3)*x**6 - 2916*log((3*e + x)/3)*x**3 
 + x**12 - 36*x**9 + 486*x**6 - 2916*x**3 + 324*x)/324