3.14.8 \(\int \frac {e^{-x} (24-8 e+8 x+e^x (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e (192+16 x-16 x^2))+(12-12 x-4 x^2+e (-4+4 x)+e^x (-144-16 e^2-96 x-16 x^2+e (96+32 x))) \log (x^2))}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx\) [1308]

3.14.8.1 Optimal result

Integrand size = 122, antiderivative size = 32 \begin {dmath*} \int \frac {e^{-x} \left (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx=4 x \left (\frac {5}{x}+x-\left (4+\frac {e^{-x}}{-3+e-x}\right ) \log \left (x^2\right )\right ) \end {dmath*}

output

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

3.14.8.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \begin {dmath*} \int \frac {e^{-x} \left (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx=4 x^2-\frac {4 x \left (12-4 e-e^{-x}+4 x\right ) \log \left (x^2\right )}{3-e+x} \end {dmath*}

input

Integrate[(24 - 8*E + 8*x + E^x*(-288 - 120*x + 16*x^2 + 8*x^3 + E^2*(-32 
+ 8*x) + E*(192 + 16*x - 16*x^2)) + (12 - 12*x - 4*x^2 + E*(-4 + 4*x) + E^ 
x*(-144 - 16*E^2 - 96*x - 16*x^2 + E*(96 + 32*x)))*Log[x^2])/(E^x*(9 + E^2 
 + E*(-6 - 2*x) + 6*x + x^2)),x]
 

output

4*x^2 - (4*x*(12 - 4*E - E^(-x) + 4*x)*Log[x^2])/(3 - E + x)
 

3.14.8.3 Rubi [F]

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.

\(\displaystyle \int \frac {e^{-x} \left (\left (-4 x^2+e^x \left (-16 x^2-96 x+e (32 x+96)-16 e^2-144\right )-12 x+e (4 x-4)+12\right ) \log \left (x^2\right )+e^x \left (8 x^3+16 x^2+e \left (-16 x^2+16 x+192\right )-120 x+e^2 (8 x-32)-288\right )+8 x-8 e+24\right )}{x^2+6 x+e (-2 x-6)+e^2+9} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{-x} \left (\left (-4 x^2+e^x \left (-16 x^2-96 x+e (32 x+96)-16 e^2-144\right )-12 x+e (4 x-4)+12\right ) \log \left (x^2\right )+e^x \left (8 x^3+16 x^2+e \left (-16 x^2+16 x+192\right )-120 x+e^2 (8 x-32)-288\right )+8 x+24 \left (1-\frac {e}{3}\right )\right )}{x^2+2 (3-e) x+(e-3)^2}dx\)

\(\Big \downarrow \) 7277

\(\displaystyle 4 \int \frac {e^{-x} \left (2 x-2 e^x \left (-x^3-2 x^2+15 x+e^2 (4-x)-2 e \left (-x^2+x+12\right )+36\right )+\left (-x^2-3 x-e (1-x)-4 e^x \left (x^2+6 x-2 e (x+3)+e^2+9\right )+3\right ) \log \left (x^2\right )+2 (3-e)\right )}{(x-e+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 4 \int \left (-\frac {e^{-x} \log \left (x^2\right ) x^2}{(-x+e-3)^2}-\frac {3 e^{-x} \log \left (x^2\right ) x}{(-x+e-3)^2}+\frac {2 e^{-x} x}{(-x+e-3)^2}+2 \left (x-2 \log \left (x^2\right )-4\right )+\frac {e^{1-x} (x-1) \log \left (x^2\right )}{(-x+e-3)^2}+\frac {3 e^{-x} \log \left (x^2\right )}{(-x+e-3)^2}-\frac {2 (-3+e) e^{-x}}{(-x+e-3)^2}\right )dx\)

\(\Big \downarrow \) 2009

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

input

Int[(24 - 8*E + 8*x + E^x*(-288 - 120*x + 16*x^2 + 8*x^3 + E^2*(-32 + 8*x) 
 + E*(192 + 16*x - 16*x^2)) + (12 - 12*x - 4*x^2 + E*(-4 + 4*x) + E^x*(-14 
4 - 16*E^2 - 96*x - 16*x^2 + E*(96 + 32*x)))*Log[x^2])/(E^x*(9 + E^2 + E*( 
-6 - 2*x) + 6*x + x^2)),x]
 

output

$Aborted
 

3.14.8.3.1 Defintions of rubi rules used

rule 2009

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

rule 7277

Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> 
 Simp[1/(4^p*c^p)   Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} 
, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] &&  !AlgebraicFu 
nctionQ[u, x]
 

rule 7292

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

rule 7293

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

3.14.8.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(32)=64\).

Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09

method	result	size
norman	\(\frac {\left (\left (4 \,{\mathrm e}-12\right ) x^{2} {\mathrm e}^{x}+\left (-16 \,{\mathrm e}+48\right ) x \,{\mathrm e}^{x} \ln \left (x^{2}\right )-4 x \ln \left (x^{2}\right )-4 \,{\mathrm e}^{x} x^{3}+16 x^{2} {\mathrm e}^{x} \ln \left (x^{2}\right )\right ) {\mathrm e}^{-x}}{{\mathrm e}-3-x}\)	\(67\)
parallelrisch	\(-\frac {\left (-8 x^{2} {\mathrm e} \,{\mathrm e}^{x}+32 \,{\mathrm e} \,{\mathrm e}^{x} \ln \left (x^{2}\right ) x +8 \,{\mathrm e}^{x} x^{3}-32 x^{2} {\mathrm e}^{x} \ln \left (x^{2}\right )+24 \,{\mathrm e}^{x} x^{2}-96 x \,{\mathrm e}^{x} \ln \left (x^{2}\right )+8 x \ln \left (x^{2}\right )\right ) {\mathrm e}^{-x}}{2 \left ({\mathrm e}-3-x \right )}\)	\(78\)
default	\(\frac {4 \left (x^{2} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )+\left (-{\mathrm e} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )+3 \ln \left (x^{2}\right )-6 \ln \left (x \right )\right ) x +\left (-2 \,{\mathrm e}+6\right ) x \ln \left (x \right )+2 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{-x}}{\left ({\mathrm e}-3-x \right )^{2}}+4 x^{2}-16 x \ln \left (x^{2}\right )\)	\(86\)
parts	\(\frac {4 \left (x^{2} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )+\left (-{\mathrm e} \left (\ln \left (x^{2}\right )-2 \ln \left (x \right )\right )+3 \ln \left (x^{2}\right )-6 \ln \left (x \right )\right ) x +\left (-2 \,{\mathrm e}+6\right ) x \ln \left (x \right )+2 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{-x}}{\left ({\mathrm e}-3-x \right )^{2}}+4 x^{2}-16 x \ln \left (x^{2}\right )\)	\(86\)

input

int((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(-4+4*x)*exp( 
1)-4*x^2-12*x+12)*ln(x^2)+((8*x-32)*exp(1)^2+(-16*x^2+16*x+192)*exp(1)+8*x 
^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*exp(1)+x^2 
+6*x+9)/exp(x),x,method=_RETURNVERBOSE)
 

output

((4*exp(1)-12)*x^2*exp(x)+(-16*exp(1)+48)*x*exp(x)*ln(x^2)-4*x*ln(x^2)-4*e 
xp(x)*x^3+16*x^2*exp(x)*ln(x^2))/(exp(1)-3-x)/exp(x)
 

3.14.8.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \begin {dmath*} \int \frac {e^{-x} \left (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx=\frac {4 \, {\left ({\left (x^{3} - x^{2} e + 3 \, x^{2}\right )} e^{x} - {\left (4 \, {\left (x^{2} - x e + 3 \, x\right )} e^{x} - x\right )} \log \left (x^{2}\right )\right )} e^{\left (-x\right )}}{x - e + 3} \end {dmath*}

input

integrate((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(-4+4*x 
)*exp(1)-4*x^2-12*x+12)*log(x^2)+((8*x-32)*exp(1)^2+(-16*x^2+16*x+192)*exp 
(1)+8*x^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*exp 
(1)+x^2+6*x+9)/exp(x),x, algorithm=\
 

output

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

3.14.8.6 Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {e^{-x} \left (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx=4 x^{2} - 16 x \log {\left (x^{2} \right )} + \frac {4 x e^{- x} \log {\left (x^{2} \right )}}{x - e + 3} \end {dmath*}

input

integrate((((-16*exp(1)**2+(32*x+96)*exp(1)-16*x**2-96*x-144)*exp(x)+(-4+4 
*x)*exp(1)-4*x**2-12*x+12)*ln(x**2)+((8*x-32)*exp(1)**2+(-16*x**2+16*x+192 
)*exp(1)+8*x**3+16*x**2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)**2+(-2* 
x-6)*exp(1)+x**2+6*x+9)/exp(x),x)
 

output

4*x**2 - 16*x*log(x**2) + 4*x*exp(-x)*log(x**2)/(x - E + 3)
 

3.14.8.7 Maxima [F]

\begin {dmath*} \int \frac {e^{-x} \left (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx=\int { \frac {4 \, {\left (2 \, {\left (x^{3} + 2 \, x^{2} + {\left (x - 4\right )} e^{2} - 2 \, {\left (x^{2} - x - 12\right )} e - 15 \, x - 36\right )} e^{x} - {\left (x^{2} - {\left (x - 1\right )} e + 4 \, {\left (x^{2} - 2 \, {\left (x + 3\right )} e + 6 \, x + e^{2} + 9\right )} e^{x} + 3 \, x - 3\right )} \log \left (x^{2}\right ) + 2 \, x - 2 \, e + 6\right )} e^{\left (-x\right )}}{x^{2} - 2 \, {\left (x + 3\right )} e + 6 \, x + e^{2} + 9} \,d x } \end {dmath*}

input

integrate((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(-4+4*x 
)*exp(1)-4*x^2-12*x+12)*log(x^2)+((8*x-32)*exp(1)^2+(-16*x^2+16*x+192)*exp 
(1)+8*x^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*exp 
(1)+x^2+6*x+9)/exp(x),x, algorithm=\
 

output

8*(e - 3)*integrate(e^(-x)/(x^2 - 2*x*(e - 3) + e^2 - 6*e + 9), x) + 8*e^( 
-e + 4)*exp_integral_e(2, x - e + 3)/(x - e + 3) - 24*e^(-e + 3)*exp_integ 
ral_e(2, x - e + 3)/(x - e + 3) + 4*(x^3 - x^2*(e - 3) + 2*x*e^(-x)*log(x) 
 - 8*(x^2 - x*(e - 3))*log(x))/(x - e + 3)
 

3.14.8.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94 \begin {dmath*} \int \frac {e^{-x} \left (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx=\frac {4 \, {\left (x^{3} - x^{2} e - 4 \, x^{2} \log \left (x^{2}\right ) + 4 \, x e \log \left (x^{2}\right ) + x e^{\left (-x\right )} \log \left (x^{2}\right ) + 3 \, x^{2} - 12 \, x \log \left (x^{2}\right )\right )}}{x - e + 3} \end {dmath*}

input

integrate((((-16*exp(1)^2+(32*x+96)*exp(1)-16*x^2-96*x-144)*exp(x)+(-4+4*x 
)*exp(1)-4*x^2-12*x+12)*log(x^2)+((8*x-32)*exp(1)^2+(-16*x^2+16*x+192)*exp 
(1)+8*x^3+16*x^2-120*x-288)*exp(x)-8*exp(1)+8*x+24)/(exp(1)^2+(-2*x-6)*exp 
(1)+x^2+6*x+9)/exp(x),x, algorithm=\
 

output

4*(x^3 - x^2*e - 4*x^2*log(x^2) + 4*x*e*log(x^2) + x*e^(-x)*log(x^2) + 3*x 
^2 - 12*x*log(x^2))/(x - e + 3)
 

3.14.8.9 Mupad [F(-1)]

Timed out. \begin {dmath*} \int \frac {e^{-x} \left (24-8 e+8 x+e^x \left (-288-120 x+16 x^2+8 x^3+e^2 (-32+8 x)+e \left (192+16 x-16 x^2\right )\right )+\left (12-12 x-4 x^2+e (-4+4 x)+e^x \left (-144-16 e^2-96 x-16 x^2+e (96+32 x)\right )\right ) \log \left (x^2\right )\right )}{9+e^2+e (-6-2 x)+6 x+x^2} \, dx=\int \frac {{\mathrm {e}}^{-x}\,\left (8\,x-8\,\mathrm {e}+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (-16\,x^2+16\,x+192\right )-120\,x+16\,x^2+8\,x^3+{\mathrm {e}}^2\,\left (8\,x-32\right )-288\right )-\ln \left (x^2\right )\,\left (12\,x+{\mathrm {e}}^x\,\left (96\,x+16\,{\mathrm {e}}^2+16\,x^2-\mathrm {e}\,\left (32\,x+96\right )+144\right )+4\,x^2-\mathrm {e}\,\left (4\,x-4\right )-12\right )+24\right )}{6\,x+{\mathrm {e}}^2+x^2-\mathrm {e}\,\left (2\,x+6\right )+9} \,d x \end {dmath*}

input

int((exp(-x)*(8*x - 8*exp(1) + exp(x)*(exp(1)*(16*x - 16*x^2 + 192) - 120* 
x + 16*x^2 + 8*x^3 + exp(2)*(8*x - 32) - 288) - log(x^2)*(12*x + exp(x)*(9 
6*x + 16*exp(2) + 16*x^2 - exp(1)*(32*x + 96) + 144) + 4*x^2 - exp(1)*(4*x 
 - 4) - 12) + 24))/(6*x + exp(2) + x^2 - exp(1)*(2*x + 6) + 9),x)
 

output

int((exp(-x)*(8*x - 8*exp(1) + exp(x)*(exp(1)*(16*x - 16*x^2 + 192) - 120* 
x + 16*x^2 + 8*x^3 + exp(2)*(8*x - 32) - 288) - log(x^2)*(12*x + exp(x)*(9 
6*x + 16*exp(2) + 16*x^2 - exp(1)*(32*x + 96) + 144) + 4*x^2 - exp(1)*(4*x 
 - 4) - 12) + 24))/(6*x + exp(2) + x^2 - exp(1)*(2*x + 6) + 9), x)