\(\int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} (15-3 x^2)+e^{2 x} (15-6 x-3 x^2+2 x^3)+e^x (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5)+e^{e^2} (-90+24 x-18 x^2-4 x^3+4 x^4+e^x (30-6 x-6 x^2+2 x^3))}{x^6} \, dx\) [1280]

Optimal result
Mathematica [B] (verified)
Rubi [C] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [B] (verification not implemented)
Maxima [C] (verification not implemented)
Giac [B] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 139, antiderivative size = 34 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=\left (-2-\frac {3-e^{e^2}-e^x-x}{x^2}\right )^2 \left (-\frac {3}{x}+x\right ) \] Output:

(x-3/x)*(-2-(3-exp(x)-exp(exp(2))-x)/x^2)^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(34)=68\).

Time = 6.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.32 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=\frac {-27+18 x-30 x^2+6 x^3+x^4+4 x^6+e^{2 e^2} \left (-3+x^2\right )+e^{2 x} \left (-3+x^2\right )+2 e^{e^2+x} \left (-3+x^2\right )+2 e^{e^2} \left (9-3 x+3 x^2+x^3-2 x^4\right )+2 e^x \left (9-3 x+3 x^2+x^3-2 x^4\right )}{x^5} \] Input:

Integrate[(135 - 72*x + 90*x^2 - 12*x^3 - x^4 + 4*x^6 + E^(2*E^2)*(15 - 3* 
x^2) + E^(2*x)*(15 - 6*x - 3*x^2 + 2*x^3) + E^x*(-90 + 42*x - 24*x^2 + 2*x 
^3 + 6*x^4 - 4*x^5) + E^E^2*(-90 + 24*x - 18*x^2 - 4*x^3 + 4*x^4 + E^x*(30 
 - 6*x - 6*x^2 + 2*x^3)))/x^6,x]
 

Output:

(-27 + 18*x - 30*x^2 + 6*x^3 + x^4 + 4*x^6 + E^(2*E^2)*(-3 + x^2) + E^(2*x 
)*(-3 + x^2) + 2*E^(E^2 + x)*(-3 + x^2) + 2*E^E^2*(9 - 3*x + 3*x^2 + x^3 - 
 2*x^4) + 2*E^x*(9 - 3*x + 3*x^2 + x^3 - 2*x^4))/x^5
 

Rubi [C] (verified)

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

Time = 1.40 (sec) , antiderivative size = 406, normalized size of antiderivative = 11.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2010, 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.

\(\displaystyle \int \frac {4 x^6-x^4-12 x^3+90 x^2+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (2 x^3-3 x^2-6 x+15\right )+e^{e^2} \left (4 x^4-4 x^3-18 x^2+e^x \left (2 x^3-6 x^2-6 x+30\right )+24 x-90\right )+e^x \left (-4 x^5+6 x^4+2 x^3-24 x^2+42 x-90\right )-72 x+135}{x^6} \, dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (\frac {e^{2 x} \left (2 x^3-3 x^2-6 x+15\right )}{x^6}+\frac {4 x^6-\left (1-4 e^{e^2}\right ) x^4-12 \left (1+\frac {e^{e^2}}{3}\right ) x^3+90 \left (1-\frac {1}{30} e^{e^2} \left (6+e^{e^2}\right )\right ) x^2-72 \left (1-\frac {e^{e^2}}{3}\right ) x+135 \left (1+\frac {1}{9} e^{e^2} \left (e^{e^2}-6\right )\right )}{x^6}+\frac {2 e^x \left (-2 x^5+3 x^4+\left (1+e^{e^2}\right ) x^3-12 \left (1+\frac {e^{e^2}}{4}\right ) x^2+21 \left (1-\frac {e^{e^2}}{7}\right ) x-45 \left (1-\frac {e^{e^2}}{3}\right )\right )}{x^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\left (4+e^{e^2}\right ) \operatorname {ExpIntegralEi}(x)+\left (1+e^{e^2}\right ) \operatorname {ExpIntegralEi}(x)+\frac {1}{4} \left (7-e^{e^2}\right ) \operatorname {ExpIntegralEi}(x)-\frac {1}{4} \left (3-e^{e^2}\right ) \operatorname {ExpIntegralEi}(x)+2 \operatorname {ExpIntegralEi}(x)-\frac {3 e^{2 x}}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right ) e^x}{x^5}-\frac {3 \left (7-e^{e^2}\right ) e^x}{2 x^4}+\frac {3 \left (3-e^{e^2}\right ) e^x}{2 x^4}+\frac {6 \left (3-e^{e^2}\right )}{x^4}+\frac {e^{2 x}}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}+\frac {2 \left (4+e^{e^2}\right ) e^x}{x^3}-\frac {\left (7-e^{e^2}\right ) e^x}{2 x^3}+\frac {\left (3-e^{e^2}\right ) e^x}{2 x^3}+\frac {\left (4+e^{e^2}\right ) e^x}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}-\frac {\left (1+e^{e^2}\right ) e^x}{x^2}-\frac {\left (7-e^{e^2}\right ) e^x}{4 x^2}+\frac {\left (3-e^{e^2}\right ) e^x}{4 x^2}+4 x-\frac {6 e^x}{x}+\frac {\left (4+e^{e^2}\right ) e^x}{x}-\frac {\left (1+e^{e^2}\right ) e^x}{x}-\frac {\left (7-e^{e^2}\right ) e^x}{4 x}+\frac {\left (3-e^{e^2}\right ) e^x}{4 x}+\frac {1-4 e^{e^2}}{x}\)

Input:

Int[(135 - 72*x + 90*x^2 - 12*x^3 - x^4 + 4*x^6 + E^(2*E^2)*(15 - 3*x^2) + 
 E^(2*x)*(15 - 6*x - 3*x^2 + 2*x^3) + E^x*(-90 + 42*x - 24*x^2 + 2*x^3 + 6 
*x^4 - 4*x^5) + E^E^2*(-90 + 24*x - 18*x^2 - 4*x^3 + 4*x^4 + E^x*(30 - 6*x 
 - 6*x^2 + 2*x^3)))/x^6,x]
 

Output:

(-3*E^(2*x))/x^5 + (6*E^x*(3 - E^E^2))/x^5 - (3*(3 - E^E^2)^2)/x^5 + (6*(3 
 - E^E^2))/x^4 + (3*E^x*(3 - E^E^2))/(2*x^4) - (3*E^x*(7 - E^E^2))/(2*x^4) 
 + E^(2*x)/x^3 + (E^x*(3 - E^E^2))/(2*x^3) - (E^x*(7 - E^E^2))/(2*x^3) + ( 
2*E^x*(4 + E^E^2))/x^3 - (30 - 6*E^E^2 - E^(2*E^2))/x^3 + (E^x*(3 - E^E^2) 
)/(4*x^2) - (E^x*(7 - E^E^2))/(4*x^2) - (E^x*(1 + E^E^2))/x^2 + (2*(3 + E^ 
E^2))/x^2 + (E^x*(4 + E^E^2))/x^2 - (6*E^x)/x + (1 - 4*E^E^2)/x + (E^x*(3 
- E^E^2))/(4*x) - (E^x*(7 - E^E^2))/(4*x) - (E^x*(1 + E^E^2))/x + (E^x*(4 
+ E^E^2))/x + 4*x + 2*ExpIntegralEi[x] - ((3 - E^E^2)*ExpIntegralEi[x])/4 
+ ((7 - E^E^2)*ExpIntegralEi[x])/4 + (1 + E^E^2)*ExpIntegralEi[x] - (4 + E 
^E^2)*ExpIntegralEi[x]
 

Defintions of rubi rules used

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs. \(2(31)=62\).

Time = 2.58 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.53

method result size
risch \(4 x +\frac {\left (-4 \,{\mathrm e}^{{\mathrm e}^{2}}+1\right ) x^{4}+\left (6+2 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x^{3}+\left ({\mathrm e}^{2 \,{\mathrm e}^{2}}+6 \,{\mathrm e}^{{\mathrm e}^{2}}-30\right ) x^{2}+\left (18-6 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x -3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+18 \,{\mathrm e}^{{\mathrm e}^{2}}-27}{x^{5}}+\frac {\left (x^{2}-3\right ) {\mathrm e}^{2 x}}{x^{5}}+\frac {2 \left (-2 x^{4}+x^{2} {\mathrm e}^{{\mathrm e}^{2}}+x^{3}+3 x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{2}}-3 x +9\right ) {\mathrm e}^{x}}{x^{5}}\) \(120\)
norman \(\frac {\left (6+2 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x^{3}+\left (18-6 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x +\left (18-6 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) {\mathrm e}^{x}+\left (-4 \,{\mathrm e}^{{\mathrm e}^{2}}+1\right ) x^{4}+\left ({\mathrm e}^{2 \,{\mathrm e}^{2}}+6 \,{\mathrm e}^{{\mathrm e}^{2}}-30\right ) x^{2}+{\mathrm e}^{2 x} x^{2}+\left (6+2 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x^{2} {\mathrm e}^{x}+4 x^{6}-3 \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x} x^{3}-4 \,{\mathrm e}^{x} x^{4}-3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+18 \,{\mathrm e}^{{\mathrm e}^{2}}-27}{x^{5}}\) \(127\)
parallelrisch \(\frac {4 x^{6}-4 \,{\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{{\mathrm e}^{2}} x^{4}+2 \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{2}} x^{2}+2 \,{\mathrm e}^{x} x^{3}+2 x^{3} {\mathrm e}^{{\mathrm e}^{2}}+x^{4}+6 \,{\mathrm e}^{x} x^{2}+6 x^{2} {\mathrm e}^{{\mathrm e}^{2}}+{\mathrm e}^{2 x} x^{2}+x^{2} {\mathrm e}^{2 \,{\mathrm e}^{2}}+6 x^{3}-6 \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{2}}-6 \,{\mathrm e}^{x} x -6 x \,{\mathrm e}^{{\mathrm e}^{2}}-30 x^{2}+18 \,{\mathrm e}^{x}+18 \,{\mathrm e}^{{\mathrm e}^{2}}-3 \,{\mathrm e}^{2 x}-3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+18 x -27}{x^{5}}\) \(140\)
parts \(-\frac {3 \,{\mathrm e}^{2 x}}{x^{5}}+\frac {{\mathrm e}^{2 x}}{x^{3}}+4 x +\frac {18 \,{\mathrm e}^{{\mathrm e}^{2}}+3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}-90}{3 x^{3}}+\frac {-24 \,{\mathrm e}^{{\mathrm e}^{2}}+72}{4 x^{4}}+\frac {90 \,{\mathrm e}^{{\mathrm e}^{2}}-15 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}-135}{5 x^{5}}+\frac {4 \,{\mathrm e}^{{\mathrm e}^{2}}+12}{2 x^{2}}+\frac {-4 \,{\mathrm e}^{{\mathrm e}^{2}}+1}{x}+\frac {2 \,{\mathrm e}^{x}}{x^{2}}-\frac {4 \,{\mathrm e}^{x}}{x}+2 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{2}\right )+\frac {18 \,{\mathrm e}^{x}}{x^{5}}-\frac {6 \,{\mathrm e}^{x}}{x^{4}}+\frac {6 \,{\mathrm e}^{x}}{x^{3}}+30 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{5 x^{5}}-\frac {{\mathrm e}^{x}}{20 x^{4}}-\frac {{\mathrm e}^{x}}{60 x^{3}}-\frac {{\mathrm e}^{x}}{120 x^{2}}-\frac {{\mathrm e}^{x}}{120 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{120}\right )-6 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{4 x^{4}}-\frac {{\mathrm e}^{x}}{12 x^{3}}-\frac {{\mathrm e}^{x}}{24 x^{2}}-\frac {{\mathrm e}^{x}}{24 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{24}\right )-6 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{3 x^{3}}-\frac {{\mathrm e}^{x}}{6 x^{2}}-\frac {{\mathrm e}^{x}}{6 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{6}\right )\) \(280\)
default \(4 x +\frac {2 \,{\mathrm e}^{x}}{x^{2}}+\frac {6 \,{\mathrm e}^{x}}{x^{3}}-\frac {30}{x^{3}}+\frac {18}{x^{4}}-\frac {27}{x^{5}}+\frac {6}{x^{2}}+\frac {1}{x}-\frac {4 \,{\mathrm e}^{{\mathrm e}^{2}}}{x}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{2}}}{x^{3}}-\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}}{x^{5}}-\frac {6 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{4}}+\frac {18 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{5}}+\frac {{\mathrm e}^{2 x}}{x^{3}}+\frac {18 \,{\mathrm e}^{x}}{x^{5}}+\frac {2 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{2}}+\frac {6 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{3}}-\frac {3 \,{\mathrm e}^{2 x}}{x^{5}}-\frac {6 \,{\mathrm e}^{x}}{x^{4}}-\frac {4 \,{\mathrm e}^{x}}{x}+30 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{5 x^{5}}-\frac {{\mathrm e}^{x}}{20 x^{4}}-\frac {{\mathrm e}^{x}}{60 x^{3}}-\frac {{\mathrm e}^{x}}{120 x^{2}}-\frac {{\mathrm e}^{x}}{120 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{120}\right )-6 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{4 x^{4}}-\frac {{\mathrm e}^{x}}{12 x^{3}}-\frac {{\mathrm e}^{x}}{24 x^{2}}-\frac {{\mathrm e}^{x}}{24 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{24}\right )-6 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{3 x^{3}}-\frac {{\mathrm e}^{x}}{6 x^{2}}-\frac {{\mathrm e}^{x}}{6 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{6}\right )+2 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\operatorname {expIntegral}_{1}\left (-x \right )}{2}\right )\) \(289\)

Input:

int(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4*x^3-18 
*x^2+24*x-90)*exp(exp(2))+(2*x^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^4+2*x^ 
3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x,method=_R 
ETURNVERBOSE)
 

Output:

4*x+((-4*exp(exp(2))+1)*x^4+(6+2*exp(exp(2)))*x^3+(exp(exp(2))^2+6*exp(exp 
(2))-30)*x^2+(18-6*exp(exp(2)))*x-3*exp(exp(2))^2+18*exp(exp(2))-27)/x^5+( 
x^2-3)/x^5*exp(x)^2+2*(-2*x^4+x^2*exp(exp(2))+x^3+3*x^2-3*exp(exp(2))-3*x+ 
9)/x^5*exp(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (24) = 48\).

Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.12 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=\frac {4 \, x^{6} + x^{4} + 6 \, x^{3} - 30 \, x^{2} + {\left (x^{2} - 3\right )} e^{\left (2 \, x\right )} - 2 \, {\left (2 \, x^{4} - x^{3} - 3 \, x^{2} + 3 \, x - 9\right )} e^{x} + {\left (x^{2} - 3\right )} e^{\left (2 \, e^{2}\right )} - 2 \, {\left (2 \, x^{4} - x^{3} - 3 \, x^{2} - {\left (x^{2} - 3\right )} e^{x} + 3 \, x - 9\right )} e^{\left (e^{2}\right )} + 18 \, x - 27}{x^{5}} \] Input:

integrate(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4* 
x^3-18*x^2+24*x-90)*exp(exp(2))+(2*x^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^ 
4+2*x^3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x, al 
gorithm="fricas")
 

Output:

(4*x^6 + x^4 + 6*x^3 - 30*x^2 + (x^2 - 3)*e^(2*x) - 2*(2*x^4 - x^3 - 3*x^2 
 + 3*x - 9)*e^x + (x^2 - 3)*e^(2*e^2) - 2*(2*x^4 - x^3 - 3*x^2 - (x^2 - 3) 
*e^x + 3*x - 9)*e^(e^2) + 18*x - 27)/x^5
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (24) = 48\).

Time = 1.59 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.15 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=4 x + \frac {x^{4} \cdot \left (1 - 4 e^{e^{2}}\right ) + x^{3} \cdot \left (6 + 2 e^{e^{2}}\right ) + x^{2} \left (-30 + 6 e^{e^{2}} + e^{2 e^{2}}\right ) + x \left (18 - 6 e^{e^{2}}\right ) - 3 e^{2 e^{2}} - 27 + 18 e^{e^{2}}}{x^{5}} + \frac {\left (x^{7} - 3 x^{5}\right ) e^{2 x} + \left (- 4 x^{9} + 2 x^{8} + 6 x^{7} + 2 x^{7} e^{e^{2}} - 6 x^{6} - 6 x^{5} e^{e^{2}} + 18 x^{5}\right ) e^{x}}{x^{10}} \] Input:

integrate(((-3*x**2+15)*exp(exp(2))**2+((2*x**3-6*x**2-6*x+30)*exp(x)+4*x* 
*4-4*x**3-18*x**2+24*x-90)*exp(exp(2))+(2*x**3-3*x**2-6*x+15)*exp(x)**2+(- 
4*x**5+6*x**4+2*x**3-24*x**2+42*x-90)*exp(x)+4*x**6-x**4-12*x**3+90*x**2-7 
2*x+135)/x**6,x)
 

Output:

4*x + (x**4*(1 - 4*exp(exp(2))) + x**3*(6 + 2*exp(exp(2))) + x**2*(-30 + 6 
*exp(exp(2)) + exp(2*exp(2))) + x*(18 - 6*exp(exp(2))) - 3*exp(2*exp(2)) - 
 27 + 18*exp(exp(2)))/x**5 + ((x**7 - 3*x**5)*exp(2*x) + (-4*x**9 + 2*x**8 
 + 6*x**7 + 2*x**7*exp(exp(2)) - 6*x**6 - 6*x**5*exp(exp(2)) + 18*x**5)*ex 
p(x))/x**10
 

Maxima [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 5.68 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=-2 \, e^{\left (e^{2}\right )} \Gamma \left (-2, -x\right ) - 6 \, e^{\left (e^{2}\right )} \Gamma \left (-3, -x\right ) + 6 \, e^{\left (e^{2}\right )} \Gamma \left (-4, -x\right ) + 30 \, e^{\left (e^{2}\right )} \Gamma \left (-5, -x\right ) + 4 \, x - \frac {4 \, e^{\left (e^{2}\right )}}{x} + \frac {1}{x} + \frac {2 \, e^{\left (e^{2}\right )}}{x^{2}} + \frac {6}{x^{2}} + \frac {e^{\left (2 \, e^{2}\right )}}{x^{3}} + \frac {6 \, e^{\left (e^{2}\right )}}{x^{3}} - \frac {30}{x^{3}} - \frac {6 \, e^{\left (e^{2}\right )}}{x^{4}} + \frac {18}{x^{4}} - \frac {3 \, e^{\left (2 \, e^{2}\right )}}{x^{5}} + \frac {18 \, e^{\left (e^{2}\right )}}{x^{5}} - \frac {27}{x^{5}} - 4 \, {\rm Ei}\left (x\right ) + 6 \, \Gamma \left (-1, -x\right ) - 2 \, \Gamma \left (-2, -x\right ) - 8 \, \Gamma \left (-2, -2 \, x\right ) - 24 \, \Gamma \left (-3, -x\right ) - 24 \, \Gamma \left (-3, -2 \, x\right ) - 42 \, \Gamma \left (-4, -x\right ) + 96 \, \Gamma \left (-4, -2 \, x\right ) - 90 \, \Gamma \left (-5, -x\right ) + 480 \, \Gamma \left (-5, -2 \, x\right ) \] Input:

integrate(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4* 
x^3-18*x^2+24*x-90)*exp(exp(2))+(2*x^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^ 
4+2*x^3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x, al 
gorithm="maxima")
 

Output:

-2*e^(e^2)*gamma(-2, -x) - 6*e^(e^2)*gamma(-3, -x) + 6*e^(e^2)*gamma(-4, - 
x) + 30*e^(e^2)*gamma(-5, -x) + 4*x - 4*e^(e^2)/x + 1/x + 2*e^(e^2)/x^2 + 
6/x^2 + e^(2*e^2)/x^3 + 6*e^(e^2)/x^3 - 30/x^3 - 6*e^(e^2)/x^4 + 18/x^4 - 
3*e^(2*e^2)/x^5 + 18*e^(e^2)/x^5 - 27/x^5 - 4*Ei(x) + 6*gamma(-1, -x) - 2* 
gamma(-2, -x) - 8*gamma(-2, -2*x) - 24*gamma(-3, -x) - 24*gamma(-3, -2*x) 
- 42*gamma(-4, -x) + 96*gamma(-4, -2*x) - 90*gamma(-5, -x) + 480*gamma(-5, 
 -2*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (24) = 48\).

Time = 0.12 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.09 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=\frac {4 \, x^{6} - 4 \, x^{4} e^{x} - 4 \, x^{4} e^{\left (e^{2}\right )} + x^{4} + 2 \, x^{3} e^{x} + 2 \, x^{3} e^{\left (e^{2}\right )} + 6 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{\left (x + e^{2}\right )} + 6 \, x^{2} e^{x} + x^{2} e^{\left (2 \, e^{2}\right )} + 6 \, x^{2} e^{\left (e^{2}\right )} - 30 \, x^{2} - 6 \, x e^{x} - 6 \, x e^{\left (e^{2}\right )} + 18 \, x - 3 \, e^{\left (2 \, x\right )} - 6 \, e^{\left (x + e^{2}\right )} + 18 \, e^{x} - 3 \, e^{\left (2 \, e^{2}\right )} + 18 \, e^{\left (e^{2}\right )} - 27}{x^{5}} \] Input:

integrate(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4* 
x^3-18*x^2+24*x-90)*exp(exp(2))+(2*x^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^ 
4+2*x^3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x, al 
gorithm="giac")
 

Output:

(4*x^6 - 4*x^4*e^x - 4*x^4*e^(e^2) + x^4 + 2*x^3*e^x + 2*x^3*e^(e^2) + 6*x 
^3 + x^2*e^(2*x) + 2*x^2*e^(x + e^2) + 6*x^2*e^x + x^2*e^(2*e^2) + 6*x^2*e 
^(e^2) - 30*x^2 - 6*x*e^x - 6*x*e^(e^2) + 18*x - 3*e^(2*x) - 6*e^(x + e^2) 
 + 18*e^x - 3*e^(2*e^2) + 18*e^(e^2) - 27)/x^5
 

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.82 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=4\,x+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^2}+{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{x+{\mathrm {e}}^2}+6\,{\mathrm {e}}^{{\mathrm {e}}^2}+6\,{\mathrm {e}}^x-30}{x^3}-\frac {3\,{\left ({\mathrm {e}}^{{\mathrm {e}}^2}+{\mathrm {e}}^x-3\right )}^2}{x^5}-\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^2}+4\,{\mathrm {e}}^x-1}{x}+\frac {2\,{\mathrm {e}}^{{\mathrm {e}}^2}+2\,{\mathrm {e}}^x+6}{x^2}-\frac {6\,{\mathrm {e}}^{{\mathrm {e}}^2}+6\,{\mathrm {e}}^x-18}{x^4} \] Input:

int(-(72*x + exp(2*exp(2))*(3*x^2 - 15) + exp(2*x)*(6*x + 3*x^2 - 2*x^3 - 
15) - exp(x)*(42*x - 24*x^2 + 2*x^3 + 6*x^4 - 4*x^5 - 90) - 90*x^2 + 12*x^ 
3 + x^4 - 4*x^6 + exp(exp(2))*(18*x^2 - 24*x + 4*x^3 - 4*x^4 + exp(x)*(6*x 
 + 6*x^2 - 2*x^3 - 30) + 90) - 135)/x^6,x)
 

Output:

4*x + (exp(2*exp(2)) + exp(2*x) + 2*exp(x + exp(2)) + 6*exp(exp(2)) + 6*ex 
p(x) - 30)/x^3 - (3*(exp(exp(2)) + exp(x) - 3)^2)/x^5 - (4*exp(exp(2)) + 4 
*exp(x) - 1)/x + (2*exp(exp(2)) + 2*exp(x) + 6)/x^2 - (6*exp(exp(2)) + 6*e 
xp(x) - 18)/x^4
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.82 \[ \int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} \left (15-3 x^2\right )+e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )+e^x \left (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5\right )+e^{e^2} \left (-90+24 x-18 x^2-4 x^3+4 x^4+e^x \left (30-6 x-6 x^2+2 x^3\right )\right )}{x^6} \, dx=\frac {e^{2 e^{2}} x^{2}-3 e^{2 e^{2}}+2 e^{e^{2}+x} x^{2}-6 e^{e^{2}+x}-4 e^{e^{2}} x^{4}+2 e^{e^{2}} x^{3}+6 e^{e^{2}} x^{2}-6 e^{e^{2}} x +18 e^{e^{2}}+e^{2 x} x^{2}-3 e^{2 x}-4 e^{x} x^{4}+2 e^{x} x^{3}+6 e^{x} x^{2}-6 e^{x} x +18 e^{x}+4 x^{6}+x^{4}+6 x^{3}-30 x^{2}+18 x -27}{x^{5}} \] Input:

int(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4*x^3-18 
*x^2+24*x-90)*exp(exp(2))+(2*x^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^4+2*x^ 
3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x)
 

Output:

(e**(2*e**2)*x**2 - 3*e**(2*e**2) + 2*e**(e**2 + x)*x**2 - 6*e**(e**2 + x) 
 - 4*e**(e**2)*x**4 + 2*e**(e**2)*x**3 + 6*e**(e**2)*x**2 - 6*e**(e**2)*x 
+ 18*e**(e**2) + e**(2*x)*x**2 - 3*e**(2*x) - 4*e**x*x**4 + 2*e**x*x**3 + 
6*e**x*x**2 - 6*e**x*x + 18*e**x + 4*x**6 + x**4 + 6*x**3 - 30*x**2 + 18*x 
 - 27)/x**5