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

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

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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 {gather*} \int \frac {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{x^2}+8 x-24 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} (1+\log (x)) (-3+x+(-1+x) \log (x))+8 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x} \left (e^x-3 (-3+x) x-6 (-2+x) x \log (x)-3 (-1+x) x \log ^2(x)\right )\right ) \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x} \left (e^x-3 (-3+x) x-6 (-2+x) x \log (x)-3 (-1+x) x \log ^2(x)\right ) \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} (1+\log (x)) (-3+x+(-1+x) \log (x)) \, dx\\ &=\frac {4}{x}+4 x^2+8 \int \left (e^{x-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x}-3 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-3+x) x^{1+6 e^{-x} x}-6 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-2+x) x^{1+6 e^{-x} x} \log (x)-3 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-1+x) x^{1+6 e^{-x} x} \log ^2(x)\right ) \, dx-24 \int \left (-3 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x}+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x}+2 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-2+x) x^{12 e^{-x} x} \log (x)+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-1+x) x^{12 e^{-x} x} \log ^2(x)\right ) \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{x-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-3+x) x^{1+6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-1+x) x^{12 e^{-x} x} \log ^2(x) \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-1+x) x^{1+6 e^{-x} x} \log ^2(x) \, dx-48 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-2+x) x^{12 e^{-x} x} \log (x) \, dx-48 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-2+x) x^{1+6 e^{-x} x} \log (x) \, dx+72 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{3 e^{-x} x \left (1+\log ^2(x)\right )} x^{6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \, dx-24 \int \left (-3 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x}+e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x}\right ) \, dx-24 \int \left (-e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log ^2(x)+e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log ^2(x)\right ) \, dx-24 \int \left (-e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log ^2(x)+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log ^2(x)\right ) \, dx-48 \int \left (-2 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log (x)+e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log (x)\right ) \, dx-48 \int \left (-2 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log (x)+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log (x)\right ) \, dx+72 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{3 e^{-x} x \left (1+\log ^2(x)\right )} x^{6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \, dx+24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log ^2(x) \, dx+24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log ^2(x) \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log ^2(x) \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log ^2(x) \, dx-48 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log (x) \, dx-48 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log (x) \, dx+72 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \, dx+72 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \, dx+96 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log (x) \, dx+96 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log (x) \, dx\\ \end {aligned} \end {gather*}

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

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 {gather*} \frac {4 \, {\left (x^{3} + 2 \, x^{2} e^{\left (3 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} + x e^{\left (6 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} + 1\right )}}{x} \end {gather*}

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 {gather*} \int -\frac {4 \, {\left (6 \, {\left (x^{3} + {\left (x^{3} - x^{2}\right )} \log \relax (x)^{2} - 3 \, x^{2} + 2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x)\right )} e^{\left (6 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} + 2 \, {\left (3 \, x^{4} - 9 \, x^{3} - x^{2} e^{x} + 3 \, {\left (x^{4} - x^{3}\right )} \log \relax (x)^{2} + 6 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (x)\right )} e^{\left (3 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} - {\left (2 \, x^{3} - 1\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{2}}\,{d x} \end {gather*}

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	\(4 x^{2}+\frac {4}{x}+4 \,{\mathrm e}^{6 x \left (\ln \relax (x )+1\right )^{2} {\mathrm e}^{-x}}+8 x \,{\mathrm e}^{3 x \left (\ln \relax (x )+1\right )^{2} {\mathrm e}^{-x}}\)	\(45\)

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 {gather*} 4 \, x^{2} + \frac {4}{x} + 4 \, e^{\left (6 \, x e^{\left (-x\right )} \log \relax (x)^{2} + 12 \, x e^{\left (-x\right )} \log \relax (x) + 6 \, x e^{\left (-x\right )}\right )} + 4 \, \int -2 \, {\left (3 \, {\left (x^{2} - x\right )} \log \relax (x)^{2} + 3 \, x^{2} + 6 \, {\left (x^{2} - 2 \, x\right )} \log \relax (x) - 9 \, x - e^{x}\right )} e^{\left (3 \, x e^{\left (-x\right )} \log \relax (x)^{2} + 6 \, x e^{\left (-x\right )} \log \relax (x) + 3 \, x e^{\left (-x\right )} - x\right )}\,{d x} \end {gather*}

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 {gather*} 4\,x^{12\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{6\,x\,{\mathrm {e}}^{-x}\,{\ln \relax (x)}^2+6\,x\,{\mathrm {e}}^{-x}}+\frac {4}{x}+4\,x^2+8\,x\,x^{6\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^{-x}\,{\ln \relax (x)}^2+3\,x\,{\mathrm {e}}^{-x}} \end {gather*}

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 {gather*} 4 x^{2} + 8 x e^{\left (3 x \log {\relax (x )}^{2} + 6 x \log {\relax (x )} + 3 x\right ) e^{- x}} + 4 e^{2 \left (3 x \log {\relax (x )}^{2} + 6 x \log {\relax (x )} + 3 x\right ) e^{- x}} + \frac {4}{x} \end {gather*}

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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