\(\int (1+2 e^{2 x}+(-24 e^x x+8 e^{2 x} x^2) \log (x^2)+(e^x (-12 x-6 x^2)+e^{2 x} (6 x^2+4 x^3)) \log ^2(x^2)+(72 x^3-48 e^x x^4+8 e^{2 x} x^5) \log ^3(x^2)+(36 x^3+e^x (-30 x^4-6 x^5)+e^{2 x} (6 x^5+2 x^6)) \log ^4(x^2)) \, dx\) [2608]

Optimal result

Integrand size = 145, antiderivative size = 29 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx=4+x+\left (-e^x+x^2 \left (3-e^x x\right ) \log ^2\left (x^2\right )\right )^2 \] Output:

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

Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx=e^{2 x}+x+2 e^x x^2 \left (-3+e^x x\right ) \log ^2\left (x^2\right )+x^4 \left (-3+e^x x\right )^2 \log ^4\left (x^2\right ) \] Input:

Integrate[1 + 2*E^(2*x) + (-24*E^x*x + 8*E^(2*x)*x^2)*Log[x^2] + (E^x*(-12 
*x - 6*x^2) + E^(2*x)*(6*x^2 + 4*x^3))*Log[x^2]^2 + (72*x^3 - 48*E^x*x^4 + 
 8*E^(2*x)*x^5)*Log[x^2]^3 + (36*x^3 + E^x*(-30*x^4 - 6*x^5) + E^(2*x)*(6* 
x^5 + 2*x^6))*Log[x^2]^4,x]
 

Output:

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

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.

\(\displaystyle \int \left (\left (8 e^{2 x} x^2-24 e^x x\right ) \log \left (x^2\right )+\left (e^x \left (-6 x^2-12 x\right )+e^{2 x} \left (4 x^3+6 x^2\right )\right ) \log ^2\left (x^2\right )+\left (8 e^{2 x} x^5-48 e^x x^4+72 x^3\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^{2 x} \left (2 x^6+6 x^5\right )+e^x \left (-6 x^5-30 x^4\right )\right ) \log ^4\left (x^2\right )+2 e^{2 x}+1\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -12 \int e^x x \log ^2\left (x^2\right )dx-6 \int e^x x^2 \log ^2\left (x^2\right )dx+6 \int e^{2 x} x^2 \log ^2\left (x^2\right )dx+2 \int e^{2 x} x^6 \log ^4\left (x^2\right )dx-6 \int e^x x^5 \log ^4\left (x^2\right )dx+6 \int e^{2 x} x^5 \log ^4\left (x^2\right )dx+8 \int e^{2 x} x^5 \log ^3\left (x^2\right )dx-30 \int e^x x^4 \log ^4\left (x^2\right )dx-48 \int e^x x^4 \log ^3\left (x^2\right )dx+4 \int e^{2 x} x^3 \log ^2\left (x^2\right )dx-48 \operatorname {ExpIntegralEi}(x)-4 \operatorname {ExpIntegralEi}(2 x)+24 e^x \log \left (x^2\right )+2 e^{2 x} \log \left (x^2\right )+4 e^{2 x} x^2 \log \left (x^2\right )-24 e^x x \log \left (x^2\right )-4 e^{2 x} x \log \left (x^2\right )+9 x^4 \log ^4\left (x^2\right )+48 e^x+7 e^{2 x}-4 e^{2 x} x+x\)

Input:

Int[1 + 2*E^(2*x) + (-24*E^x*x + 8*E^(2*x)*x^2)*Log[x^2] + (E^x*(-12*x - 6 
*x^2) + E^(2*x)*(6*x^2 + 4*x^3))*Log[x^2]^2 + (72*x^3 - 48*E^x*x^4 + 8*E^( 
2*x)*x^5)*Log[x^2]^3 + (36*x^3 + E^x*(-30*x^4 - 6*x^5) + E^(2*x)*(6*x^5 + 
2*x^6))*Log[x^2]^4,x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.22 (sec) , antiderivative size = 2156, normalized size of antiderivative = 74.34

\[\text {Expression too large to display}\]

Input:

int(((2*x^6+6*x^5)*exp(x)^2+(-6*x^5-30*x^4)*exp(x)+36*x^3)*ln(x^2)^4+(8*x^ 
5*exp(x)^2-48*exp(x)*x^4+72*x^3)*ln(x^2)^3+((4*x^3+6*x^2)*exp(x)^2+(-6*x^2 
-12*x)*exp(x))*ln(x^2)^2+(8*exp(x)^2*x^2-24*exp(x)*x)*ln(x^2)+2*exp(x)^2+1 
,x)
 

Output:

9/16*Pi^4*x^4*csgn(I*x^2)^12+x+exp(x)^2+9/16*Pi^4*x^4*csgn(I*x)^8*csgn(I*x 
^2)^4-9/2*Pi^4*x^4*csgn(I*x)^7*csgn(I*x^2)^5+63/4*Pi^4*x^4*csgn(I*x)^6*csg 
n(I*x^2)^6-63/2*Pi^4*x^4*csgn(I*x)^5*csgn(I*x^2)^7+315/8*Pi^4*x^4*csgn(I*x 
)^4*csgn(I*x^2)^8-63/2*Pi^4*x^4*csgn(I*x)^3*csgn(I*x^2)^9+63/4*Pi^4*x^4*cs 
gn(I*x)^2*csgn(I*x^2)^10-9/2*Pi^4*x^4*csgn(I*x)*csgn(I*x^2)^11+1/16*Pi^4*x 
^6*csgn(I*x^2)^12*exp(x)^2-3/8*Pi^4*x^5*csgn(I*x^2)^12*exp(x)+1/16*Pi^4*x^ 
6*csgn(I*x)^8*csgn(I*x^2)^4*exp(x)^2-3/8*Pi^4*x^5*csgn(I*x)^8*csgn(I*x^2)^ 
4*exp(x)-1/2*Pi^4*x^6*csgn(I*x)^7*csgn(I*x^2)^5*exp(x)^2+3*Pi^4*x^5*csgn(I 
*x)^7*csgn(I*x^2)^5*exp(x)+7/4*Pi^4*x^6*csgn(I*x)^6*csgn(I*x^2)^6*exp(x)^2 
-21/2*Pi^4*x^5*csgn(I*x)^6*csgn(I*x^2)^6*exp(x)-7/2*Pi^4*x^6*csgn(I*x)^5*c 
sgn(I*x^2)^7*exp(x)^2+21*Pi^4*x^5*csgn(I*x)^5*csgn(I*x^2)^7*exp(x)+35/8*Pi 
^4*x^6*csgn(I*x)^4*csgn(I*x^2)^8*exp(x)^2-105/4*Pi^4*x^5*csgn(I*x)^4*csgn( 
I*x^2)^8*exp(x)-7/2*Pi^4*x^6*csgn(I*x)^3*csgn(I*x^2)^9*exp(x)^2+21*Pi^4*x^ 
5*csgn(I*x)^3*csgn(I*x^2)^9*exp(x)+7/4*Pi^4*x^6*csgn(I*x)^2*csgn(I*x^2)^10 
*exp(x)^2-21/2*Pi^4*x^5*csgn(I*x)^2*csgn(I*x^2)^10*exp(x)-1/2*Pi^4*x^6*csg 
n(I*x)*csgn(I*x^2)^11*exp(x)^2+3*Pi^4*x^5*csgn(I*x)*csgn(I*x^2)^11*exp(x)+ 
((8*x^2-8*x+4)*exp(x)^2+(48-48*x)*exp(x))*ln(x)+(16*exp(x)^2*x^6-96*x^5*ex 
p(x)+144*x^4)*ln(x)^4+I*Pi^3*x^4*csgn(I*x^2)^3*(x^2*csgn(I*x)^6*exp(x)^2-6 
*x^2*csgn(I*x)^5*csgn(I*x^2)*exp(x)^2+15*x^2*csgn(I*x)^4*csgn(I*x^2)^2*exp 
(x)^2-20*x^2*csgn(I*x)^3*csgn(I*x^2)^3*exp(x)^2+15*x^2*csgn(I*x)^2*csgn...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx={\left (x^{6} e^{\left (2 \, x\right )} - 6 \, x^{5} e^{x} + 9 \, x^{4}\right )} \log \left (x^{2}\right )^{4} + 2 \, {\left (x^{3} e^{\left (2 \, x\right )} - 3 \, x^{2} e^{x}\right )} \log \left (x^{2}\right )^{2} + x + e^{\left (2 \, x\right )} \] Input:

integrate(((2*x^6+6*x^5)*exp(x)^2+(-6*x^5-30*x^4)*exp(x)+36*x^3)*log(x^2)^ 
4+(8*x^5*exp(x)^2-48*exp(x)*x^4+72*x^3)*log(x^2)^3+((4*x^3+6*x^2)*exp(x)^2 
+(-6*x^2-12*x)*exp(x))*log(x^2)^2+(8*exp(x)^2*x^2-24*exp(x)*x)*log(x^2)+2* 
exp(x)^2+1,x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx=9 x^{4} \log {\left (x^{2} \right )}^{4} + x + \left (- 6 x^{5} \log {\left (x^{2} \right )}^{4} - 6 x^{2} \log {\left (x^{2} \right )}^{2}\right ) e^{x} + \left (x^{6} \log {\left (x^{2} \right )}^{4} + 2 x^{3} \log {\left (x^{2} \right )}^{2} + 1\right ) e^{2 x} \] Input:

integrate(((2*x**6+6*x**5)*exp(x)**2+(-6*x**5-30*x**4)*exp(x)+36*x**3)*ln( 
x**2)**4+(8*x**5*exp(x)**2-48*exp(x)*x**4+72*x**3)*ln(x**2)**3+((4*x**3+6* 
x**2)*exp(x)**2+(-6*x**2-12*x)*exp(x))*ln(x**2)**2+(8*exp(x)**2*x**2-24*ex 
p(x)*x)*ln(x**2)+2*exp(x)**2+1,x)
 

Output:

9*x**4*log(x**2)**4 + x + (-6*x**5*log(x**2)**4 - 6*x**2*log(x**2)**2)*exp 
(x) + (x**6*log(x**2)**4 + 2*x**3*log(x**2)**2 + 1)*exp(2*x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx=144 \, x^{4} \log \left (x\right )^{4} + 8 \, {\left (2 \, x^{6} \log \left (x\right )^{4} + x^{3} \log \left (x\right )^{2}\right )} e^{\left (2 \, x\right )} - 24 \, {\left (4 \, x^{5} \log \left (x\right )^{4} + x^{2} \log \left (x\right )^{2}\right )} e^{x} + x + e^{\left (2 \, x\right )} \] Input:

integrate(((2*x^6+6*x^5)*exp(x)^2+(-6*x^5-30*x^4)*exp(x)+36*x^3)*log(x^2)^ 
4+(8*x^5*exp(x)^2-48*exp(x)*x^4+72*x^3)*log(x^2)^3+((4*x^3+6*x^2)*exp(x)^2 
+(-6*x^2-12*x)*exp(x))*log(x^2)^2+(8*exp(x)^2*x^2-24*exp(x)*x)*log(x^2)+2* 
exp(x)^2+1,x, algorithm="maxima")
 

Output:

144*x^4*log(x)^4 + 8*(2*x^6*log(x)^4 + x^3*log(x)^2)*e^(2*x) - 24*(4*x^5*l 
og(x)^4 + x^2*log(x)^2)*e^x + x + e^(2*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (24) = 48\).

Time = 0.47 (sec) , antiderivative size = 259, normalized size of antiderivative = 8.93 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx=-18 \, x^{4} \log \left (x^{2}\right )^{3} - {\left (4 \, x^{5} - 10 \, x^{4} + 20 \, x^{3} - 30 \, x^{2} + 30 \, x - 15\right )} e^{\left (2 \, x\right )} \log \left (x^{2}\right )^{3} + 48 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} \log \left (x^{2}\right )^{3} + {\left (x^{6} e^{\left (2 \, x\right )} - 6 \, x^{5} e^{x} + 9 \, x^{4}\right )} \log \left (x^{2}\right )^{4} + {\left (18 \, x^{4} + {\left (4 \, x^{5} - 10 \, x^{4} + 20 \, x^{3} - 30 \, x^{2} + 30 \, x - 15\right )} e^{\left (2 \, x\right )} - 48 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x}\right )} \log \left (x^{2}\right )^{3} - 2 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} \log \left (x^{2}\right ) + 24 \, {\left (x - 1\right )} e^{x} \log \left (x^{2}\right ) + 2 \, {\left (x^{3} e^{\left (2 \, x\right )} - 3 \, x^{2} e^{x}\right )} \log \left (x^{2}\right )^{2} + 2 \, {\left ({\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - 12 \, {\left (x - 1\right )} e^{x}\right )} \log \left (x^{2}\right ) + x + e^{\left (2 \, x\right )} \] Input:

integrate(((2*x^6+6*x^5)*exp(x)^2+(-6*x^5-30*x^4)*exp(x)+36*x^3)*log(x^2)^ 
4+(8*x^5*exp(x)^2-48*exp(x)*x^4+72*x^3)*log(x^2)^3+((4*x^3+6*x^2)*exp(x)^2 
+(-6*x^2-12*x)*exp(x))*log(x^2)^2+(8*exp(x)^2*x^2-24*exp(x)*x)*log(x^2)+2* 
exp(x)^2+1,x, algorithm="giac")
 

Output:

-18*x^4*log(x^2)^3 - (4*x^5 - 10*x^4 + 20*x^3 - 30*x^2 + 30*x - 15)*e^(2*x 
)*log(x^2)^3 + 48*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x*log(x^2)^3 + (x^6 
*e^(2*x) - 6*x^5*e^x + 9*x^4)*log(x^2)^4 + (18*x^4 + (4*x^5 - 10*x^4 + 20* 
x^3 - 30*x^2 + 30*x - 15)*e^(2*x) - 48*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)* 
e^x)*log(x^2)^3 - 2*(2*x^2 - 2*x + 1)*e^(2*x)*log(x^2) + 24*(x - 1)*e^x*lo 
g(x^2) + 2*(x^3*e^(2*x) - 3*x^2*e^x)*log(x^2)^2 + 2*((2*x^2 - 2*x + 1)*e^( 
2*x) - 12*(x - 1)*e^x)*log(x^2) + x + e^(2*x)
 

Mupad [B] (verification not implemented)

Time = 2.00 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx=\left (x^6\,{\mathrm {e}}^{2\,x}-6\,x^5\,{\mathrm {e}}^x+9\,x^4\right )\,{\ln \left (x^2\right )}^4+\left (2\,x^3\,{\mathrm {e}}^{2\,x}-6\,x^2\,{\mathrm {e}}^x\right )\,{\ln \left (x^2\right )}^2+x+{\mathrm {e}}^{2\,x} \] Input:

int(2*exp(2*x) + log(x^2)^4*(exp(2*x)*(6*x^5 + 2*x^6) - exp(x)*(30*x^4 + 6 
*x^5) + 36*x^3) + log(x^2)^3*(8*x^5*exp(2*x) - 48*x^4*exp(x) + 72*x^3) + l 
og(x^2)*(8*x^2*exp(2*x) - 24*x*exp(x)) + log(x^2)^2*(exp(2*x)*(6*x^2 + 4*x 
^3) - exp(x)*(12*x + 6*x^2)) + 1,x)
 

Output:

x + exp(2*x) + log(x^2)^4*(x^6*exp(2*x) - 6*x^5*exp(x) + 9*x^4) - log(x^2) 
^2*(6*x^2*exp(x) - 2*x^3*exp(2*x))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66 \[ \int \left (1+2 e^{2 x}+\left (-24 e^x x+8 e^{2 x} x^2\right ) \log \left (x^2\right )+\left (e^x \left (-12 x-6 x^2\right )+e^{2 x} \left (6 x^2+4 x^3\right )\right ) \log ^2\left (x^2\right )+\left (72 x^3-48 e^x x^4+8 e^{2 x} x^5\right ) \log ^3\left (x^2\right )+\left (36 x^3+e^x \left (-30 x^4-6 x^5\right )+e^{2 x} \left (6 x^5+2 x^6\right )\right ) \log ^4\left (x^2\right )\right ) \, dx=e^{2 x} \mathrm {log}\left (x^{2}\right )^{4} x^{6}+2 e^{2 x} \mathrm {log}\left (x^{2}\right )^{2} x^{3}+e^{2 x}-6 e^{x} \mathrm {log}\left (x^{2}\right )^{4} x^{5}-6 e^{x} \mathrm {log}\left (x^{2}\right )^{2} x^{2}+9 \mathrm {log}\left (x^{2}\right )^{4} x^{4}+x \] Input:

int(((2*x^6+6*x^5)*exp(x)^2+(-6*x^5-30*x^4)*exp(x)+36*x^3)*log(x^2)^4+(8*x 
^5*exp(x)^2-48*exp(x)*x^4+72*x^3)*log(x^2)^3+((4*x^3+6*x^2)*exp(x)^2+(-6*x 
^2-12*x)*exp(x))*log(x^2)^2+(8*exp(x)^2*x^2-24*exp(x)*x)*log(x^2)+2*exp(x) 
^2+1,x)
 

Output:

e**(2*x)*log(x**2)**4*x**6 + 2*e**(2*x)*log(x**2)**2*x**3 + e**(2*x) - 6*e 
**x*log(x**2)**4*x**5 - 6*e**x*log(x**2)**2*x**2 + 9*log(x**2)**4*x**4 + x