3.41.63 $\int \frac{e^{-x}(e^x(-4+8x^3)+e^{2e^{-x}(3x+6x\log (x)+3x{\log }^2(x))}(72x^2-24x^3+(96x^2-48x^3)\log (x)+(24x^2-24x^3){\log }^2(x))+e^{e^{-x}(3x+6x\log (x)+3x{\log }^2(x))}(8e^x{x}^2+72x^3-24x^4+(96x^3-48x^4)\log (x)+(24x^3-24x^4){\log }^2(x)))}{x^2}dx$

Optimal. Leaf size=29 $$-1+\frac{4}{x}+4{(e^{3e^{-x}x(1+\log (x){)}^2}+x)}^2$$

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Rubi [F]  time = 8.51, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.000, Rules used = {} $$\begin{array}{c}\int \frac{e^{-x}(e^x(-4+8x^3)+e^{2e^{-x}(3x+6x\log (x)+3x{\log }^2(x))}(72x^2-24x^3+(96x^2-48x^3)\log (x)+(24x^2-24x^3){\log }^2(x))+e^{e^{-x}(3x+6x\log (x)+3x{\log }^2(x))}(8e^x{x}^2+72x^3-24x^4+(96x^3-48x^4)\log (x)+(24x^3-24x^4){\log }^2(x)))}{x^2}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^x*(-4 + 8*x^3) + E^((2*(3*x + 6*x*Log[x] + 3*x*Log[x]^2))/E^x)*(72*x^2 - 24*x^3 + (96*x^2 - 48*x^3)*Log
[x] + (24*x^2 - 24*x^3)*Log[x]^2) + E^((3*x + 6*x*Log[x] + 3*x*Log[x]^2)/E^x)*(8*E^x*x^2 + 72*x^3 - 24*x^4 + (
96*x^3 - 48*x^4)*Log[x] + (24*x^3 - 24*x^4)*Log[x]^2))/(E^x*x^2),x]

[Out]

4/x + 4*x^2 + 8*Defer[Int][E^((3*x*(1 + Log[x]^2))/E^x)*x^((6*x)/E^x), x] + 72*Defer[Int][x^((12*x)/E^x)/E^((x
*(-6 + E^x - 6*Log[x]^2))/E^x), x] + 72*Defer[Int][x^(1 + (6*x)/E^x)/E^((x*(-3 + E^x - 3*Log[x]^2))/E^x), x] -
24*Defer[Int][x^(2 + (6*x)/E^x)/E^((x*(-3 + E^x - 3*Log[x]^2))/E^x), x] - 24*Defer[Int][x^(1 + (12*x)/E^x)/E^
((x*(-6 + E^x - 6*Log[x]^2))/E^x), x] + 96*Defer[Int][(x^((12*x)/E^x)*Log[x])/E^((x*(-6 + E^x - 6*Log[x]^2))/E
^x), x] + 96*Defer[Int][(x^(1 + (6*x)/E^x)*Log[x])/E^((x*(-3 + E^x - 3*Log[x]^2))/E^x), x] - 48*Defer[Int][(x^
(2 + (6*x)/E^x)*Log[x])/E^((x*(-3 + E^x - 3*Log[x]^2))/E^x), x] - 48*Defer[Int][(x^(1 + (12*x)/E^x)*Log[x])/E^
((x*(-6 + E^x - 6*Log[x]^2))/E^x), x] + 24*Defer[Int][(x^((12*x)/E^x)*Log[x]^2)/E^((x*(-6 + E^x - 6*Log[x]^2))
/E^x), x] + 24*Defer[Int][(x^(1 + (6*x)/E^x)*Log[x]^2)/E^((x*(-3 + E^x - 3*Log[x]^2))/E^x), x] - 24*Defer[Int]
[(x^(2 + (6*x)/E^x)*Log[x]^2)/E^((x*(-3 + E^x - 3*Log[x]^2))/E^x), x] - 24*Defer[Int][(x^(1 + (12*x)/E^x)*Log[
x]^2)/E^((x*(-6 + E^x - 6*Log[x]^2))/E^x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-\frac{4}{x^2}+8x-24e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}(1+\log (x))(-3+x+(-1+x)\log (x))+8e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{6e^{-x}x}(e^x-3(-3+x)x-6(-2+x)x\log (x)-3(-1+x)x{\log }^2(x)))dx\\ & =\frac{4}{x}+4x^2+8\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{6e^{-x}x}(e^x-3(-3+x)x-6(-2+x)x\log (x)-3(-1+x)x{\log }^2(x))dx-24\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}(1+\log (x))(-3+x+(-1+x)\log (x))dx\\ & =\frac{4}{x}+4x^2+8\int (e^{x-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{6e^{-x}x}-3e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}(-3+x)x^{1+6e^{-x}x}-6e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}(-2+x)x^{1+6e^{-x}x}\log (x)-3e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}(-1+x)x^{1+6e^{-x}x}{\log }^2(x))dx-24\int (-3e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}+e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}+2e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}(-2+x)x^{12e^{-x}x}\log (x)+e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}(-1+x)x^{12e^{-x}x}{\log }^2(x))dx\\ & =\frac{4}{x}+4x^2+8\int e^{x-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{6e^{-x}x}dx-24\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}(-3+x)x^{1+6e^{-x}x}dx-24\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}dx-24\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}(-1+x)x^{12e^{-x}x}{\log }^2(x)dx-24\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}(-1+x)x^{1+6e^{-x}x}{\log }^2(x)dx-48\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}(-2+x)x^{12e^{-x}x}\log (x)dx-48\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}(-2+x)x^{1+6e^{-x}x}\log (x)dx+72\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}dx\\ & =\frac{4}{x}+4x^2+8\int e^{3e^{-x}x(1+{\log }^2(x))}{x}^{6e^{-x}x}dx-24\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}dx-24\int (-3e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{1+6e^{-x}x}+e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{2+6e^{-x}x})dx-24\int (-e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{1+6e^{-x}x}{\log }^2(x)+e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{2+6e^{-x}x}{\log }^2(x))dx-24\int (-e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}{\log }^2(x)+e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}{\log }^2(x))dx-48\int (-2e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{1+6e^{-x}x}\log (x)+e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{2+6e^{-x}x}\log (x))dx-48\int (-2e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}\log (x)+e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}\log (x))dx+72\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}dx\\ & =\frac{4}{x}+4x^2+8\int e^{3e^{-x}x(1+{\log }^2(x))}{x}^{6e^{-x}x}dx-24\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{2+6e^{-x}x}dx-24\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}dx+24\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}{\log }^2(x)dx+24\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{1+6e^{-x}x}{\log }^2(x)dx-24\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{2+6e^{-x}x}{\log }^2(x)dx-24\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}{\log }^2(x)dx-48\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{2+6e^{-x}x}\log (x)dx-48\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{1+12e^{-x}x}\log (x)dx+72\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}dx+72\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{1+6e^{-x}x}dx+96\int e^{-e^{-x}x(-6+e^x-6{\log }^2(x))}{x}^{12e^{-x}x}\log (x)dx+96\int e^{-e^{-x}x(-3+e^x-3{\log }^2(x))}{x}^{1+6e^{-x}x}\log (x)dx\end{array}\end{array}$$

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Mathematica [B]  time = 4.97, size = 83, normalized size = 2.86 $$\begin{array}{c}\frac{4}{x}+4x^2+4e^{6e^{-x}x+6e^{-x}x{\log }^2(x)}{x}^{12e^{-x}x}+8e^{3e^{-x}x+3e^{-x}x{\log }^2(x)}{x}^{1+6e^{-x}x}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-4 + 8*x^3) + E^((2*(3*x + 6*x*Log[x] + 3*x*Log[x]^2))/E^x)*(72*x^2 - 24*x^3 + (96*x^2 - 48*x^
3)*Log[x] + (24*x^2 - 24*x^3)*Log[x]^2) + E^((3*x + 6*x*Log[x] + 3*x*Log[x]^2)/E^x)*(8*E^x*x^2 + 72*x^3 - 24*x
^4 + (96*x^3 - 48*x^4)*Log[x] + (24*x^3 - 24*x^4)*Log[x]^2))/(E^x*x^2),x]

[Out]

4/x + 4*x^2 + 4*E^((6*x)/E^x + (6*x*Log[x]^2)/E^x)*x^((12*x)/E^x) + 8*E^((3*x)/E^x + (3*x*Log[x]^2)/E^x)*x^(1
+ (6*x)/E^x)

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fricas [B]  time = 0.65, size = 57, normalized size = 1.97 $$\begin{array}{c}\frac{4(x^3+2x^2{e}^{\left(3(x\log (x{)}^2+2x\log (x)+x)e^{(-x)}\right)}+x{e}^{\left(6(x\log (x{)}^2+2x\log (x)+x)e^{(-x)}\right)}+1)}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x^3+24*x^2)*log(x)^2+(-48*x^3+96*x^2)*log(x)-24*x^3+72*x^2)*exp((3*x*log(x)^2+6*x*log(x)+3*x)
/exp(x))^2+((-24*x^4+24*x^3)*log(x)^2+(-48*x^4+96*x^3)*log(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*log(x)^2+6*
x*log(x)+3*x)/exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x, algorithm="fricas")

[Out]

4*(x^3 + 2*x^2*e^(3*(x*log(x)^2 + 2*x*log(x) + x)*e^(-x)) + x*e^(6*(x*log(x)^2 + 2*x*log(x) + x)*e^(-x)) + 1)/
x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{4(6(x^3+(x^3-x^2)\log (x{)}^2-3x^2+2(x^3-2x^2)\log (x))e^{\left(6(x\log (x{)}^2+2x\log (x)+x)e^{(-x)}\right)}+2(3x^4-9x^3-x^2{e}^x+3(x^4-x^3)\log (x{)}^2+6(x^4-2x^3)\log (x))e^{\left(3(x\log (x{)}^2+2x\log (x)+x)e^{(-x)}\right)}-(2x^3-1)e^x)e^{(-x)}}{x^2}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x^3+24*x^2)*log(x)^2+(-48*x^3+96*x^2)*log(x)-24*x^3+72*x^2)*exp((3*x*log(x)^2+6*x*log(x)+3*x)
/exp(x))^2+((-24*x^4+24*x^3)*log(x)^2+(-48*x^4+96*x^3)*log(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*log(x)^2+6*
x*log(x)+3*x)/exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x, algorithm="giac")

[Out]

integrate(-4*(6*(x^3 + (x^3 - x^2)*log(x)^2 - 3*x^2 + 2*(x^3 - 2*x^2)*log(x))*e^(6*(x*log(x)^2 + 2*x*log(x) +
x)*e^(-x)) + 2*(3*x^4 - 9*x^3 - x^2*e^x + 3*(x^4 - x^3)*log(x)^2 + 6*(x^4 - 2*x^3)*log(x))*e^(3*(x*log(x)^2 +
2*x*log(x) + x)*e^(-x)) - (2*x^3 - 1)*e^x)*e^(-x)/x^2, x)

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maple [A]  time = 0.12, size = 45, normalized size = 1.55

method	result	size
risch	$4x^2+\frac{4}{x}+4{\mathrm{e}}^{6x{(\ln (x)+1)}^2{\mathrm{e}}^{-x}}+8x{\mathrm{e}}^{3x{(\ln (x)+1)}^2{\mathrm{e}}^{-x}}$	$45$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-24*x^3+24*x^2)*ln(x)^2+(-48*x^3+96*x^2)*ln(x)-24*x^3+72*x^2)*exp((3*x*ln(x)^2+6*x*ln(x)+3*x)/exp(x))^2
+((-24*x^4+24*x^3)*ln(x)^2+(-48*x^4+96*x^3)*ln(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*ln(x)^2+6*x*ln(x)+3*x)/
exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x,method=_RETURNVERBOSE)

[Out]

4*x^2+4/x+4*exp(6*x*(ln(x)+1)^2*exp(-x))+8*x*exp(3*x*(ln(x)+1)^2*exp(-x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}4x^2+\frac{4}{x}+4e^{(6x{e}^{(-x)}\log (x{)}^2+12x{e}^{(-x)}\log (x)+6x{e}^{(-x)})}+4\int -2(3(x^2-x)\log (x{)}^2+3x^2+6(x^2-2x)\log (x)-9x-e^x)e^{(3x{e}^{(-x)}\log (x{)}^2+6x{e}^{(-x)}\log (x)+3x{e}^{(-x)}-x)}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x^3+24*x^2)*log(x)^2+(-48*x^3+96*x^2)*log(x)-24*x^3+72*x^2)*exp((3*x*log(x)^2+6*x*log(x)+3*x)
/exp(x))^2+((-24*x^4+24*x^3)*log(x)^2+(-48*x^4+96*x^3)*log(x)+8*exp(x)*x^2-24*x^4+72*x^3)*exp((3*x*log(x)^2+6*
x*log(x)+3*x)/exp(x))+(8*x^3-4)*exp(x))/exp(x)/x^2,x, algorithm="maxima")

[Out]

4*x^2 + 4/x + 4*e^(6*x*e^(-x)*log(x)^2 + 12*x*e^(-x)*log(x) + 6*x*e^(-x)) + 4*integrate(-2*(3*(x^2 - x)*log(x)
^2 + 3*x^2 + 6*(x^2 - 2*x)*log(x) - 9*x - e^x)*e^(3*x*e^(-x)*log(x)^2 + 6*x*e^(-x)*log(x) + 3*x*e^(-x) - x), x
)

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mupad [B]  time = 3.22, size = 74, normalized size = 2.55 $$\begin{array}{c}4x^{12x{\mathrm{e}}^{-x}}{\mathrm{e}}^{6x{\mathrm{e}}^{-x}{\ln (x)}^2+6x{\mathrm{e}}^{-x}}+\frac{4}{x}+4x^2+8x{x}^{6x{\mathrm{e}}^{-x}}{\mathrm{e}}^{3x{\mathrm{e}}^{-x}{\ln (x)}^2+3x{\mathrm{e}}^{-x}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(exp(2*exp(-x)*(3*x + 3*x*log(x)^2 + 6*x*log(x)))*(log(x)*(96*x^2 - 48*x^3) + log(x)^2*(24*x^2 -
24*x^3) + 72*x^2 - 24*x^3) + exp(exp(-x)*(3*x + 3*x*log(x)^2 + 6*x*log(x)))*(log(x)*(96*x^3 - 48*x^4) + 8*x^2*
exp(x) + log(x)^2*(24*x^3 - 24*x^4) + 72*x^3 - 24*x^4) + exp(x)*(8*x^3 - 4)))/x^2,x)

[Out]

4*x^(12*x*exp(-x))*exp(6*x*exp(-x) + 6*x*exp(-x)*log(x)^2) + 4/x + 4*x^2 + 8*x*x^(6*x*exp(-x))*exp(3*x*exp(-x)
+ 3*x*exp(-x)*log(x)^2)

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sympy [B]  time = 176.54, size = 60, normalized size = 2.07 $$\begin{array}{c}4x^2+8x{e}^{(3x\log {(x)}^2+6x\log (x)+3x)e^{-x}}+4e^{2(3x\log {(x)}^2+6x\log (x)+3x)e^{-x}}+\frac{4}{x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x**3+24*x**2)*ln(x)**2+(-48*x**3+96*x**2)*ln(x)-24*x**3+72*x**2)*exp((3*x*ln(x)**2+6*x*ln(x)+
3*x)/exp(x))**2+((-24*x**4+24*x**3)*ln(x)**2+(-48*x**4+96*x**3)*ln(x)+8*exp(x)*x**2-24*x**4+72*x**3)*exp((3*x*
ln(x)**2+6*x*ln(x)+3*x)/exp(x))+(8*x**3-4)*exp(x))/exp(x)/x**2,x)

[Out]

4*x**2 + 8*x*exp((3*x*log(x)**2 + 6*x*log(x) + 3*x)*exp(-x)) + 4*exp(2*(3*x*log(x)**2 + 6*x*log(x) + 3*x)*exp(
-x)) + 4/x

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