3.18.77 \(\int \frac {e^{-x} (3 e^{1+x}+x^{\frac {1}{3} e^{-x} (-2 x+2 e^x x)} (-6 x+e^x (3+6 x)+(-8 x+8 e^x x+6 x^2) \log (x)+(-2 x+2 e^x x+2 x^2) \log ^2(x)))}{3 x} \, dx\)

Optimal. Leaf size=25 \[ \left (e+x^{\frac {1}{3} \left (2 x-2 e^{-x} x\right )}\right ) (3+\log (x)) \]

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Rubi [F]  time = 18.44, 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 (3 e^{1+x}+x^{\frac {1}{3} e^{-x} \left (-2 x+2 e^x x\right )} \left (-6 x+e^x (3+6 x)+\left (-8 x+8 e^x x+6 x^2\right ) \log (x)+\left (-2 x+2 e^x x+2 x^2\right ) \log ^2(x)\right )\right )}{3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*E^(1 + x) + x^((-2*x + 2*E^x*x)/(3*E^x))*(-6*x + E^x*(3 + 6*x) + (-8*x + 8*E^x*x + 6*x^2)*Log[x] + (-2*
x + 2*E^x*x + 2*x^2)*Log[x]^2))/(3*E^x*x),x]

[Out]

(3*x^((2*(1 - E^(-x))*x)/3)*(1 + 2*x))/(3 + 2*(1 - E^(-x))*x) + E*Log[x] + (8*Log[x]*Defer[Int][x^((-2*(-1 + E
^(-x))*x)/3), x])/3 - 2*Defer[Int][1/(E^x*x^((2*(-1 + E^(-x))*x)/3)), x] - (8*Log[x]*Defer[Int][1/(E^x*x^((2*(
-1 + E^(-x))*x)/3)), x])/3 + (3*Defer[Int][x^(-1 - (2*(-1 + E^(-x))*x)/3), x])/(3 + 2*(1 - E^(-x))*x) + 2*Log[
x]*Defer[Int][x^(1 - (2*(-1 + E^(-x))*x)/3)/E^x, x] + (2*Defer[Int][Log[x]^2/x^((2*(-1 + E^(-x))*x)/3), x])/3
- (2*Defer[Int][Log[x]^2/(E^x*x^((2*(-1 + E^(-x))*x)/3)), x])/3 + (2*Defer[Int][(x^(1 - (2*(-1 + E^(-x))*x)/3)
*Log[x]^2)/E^x, x])/3 - (8*Defer[Int][Defer[Int][x^((-2*(-1 + E^(-x))*x)/3), x]/x, x])/3 + (8*Defer[Int][Defer
[Int][1/(E^x*x^((2*(-1 + E^(-x))*x)/3)), x]/x, x])/3 - 2*Defer[Int][Defer[Int][x^(1 - (2*(-1 + E^(-x))*x)/3)/E
^x, x]/x, x]

Rubi steps

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

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Mathematica [A]  time = 5.21, size = 31, normalized size = 1.24 \begin {gather*} \frac {1}{3} \left (3 e \log (x)+x^{-\frac {2}{3} \left (-1+e^{-x}\right ) x} (9+3 \log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*E^(1 + x) + x^((-2*x + 2*E^x*x)/(3*E^x))*(-6*x + E^x*(3 + 6*x) + (-8*x + 8*E^x*x + 6*x^2)*Log[x]
+ (-2*x + 2*E^x*x + 2*x^2)*Log[x]^2))/(3*E^x*x),x]

[Out]

(3*E*Log[x] + (9 + 3*Log[x])/x^((2*(-1 + E^(-x))*x)/3))/3

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fricas [A]  time = 0.77, size = 35, normalized size = 1.40 \begin {gather*} e \log \relax (x) + \frac {\log \relax (x) + 3}{x^{\frac {2}{3} \, {\left (x e - x e^{\left (x + 1\right )}\right )} e^{\left (-x - 1\right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((2*exp(x)*x+2*x^2-2*x)*log(x)^2+(8*exp(x)*x+6*x^2-8*x)*log(x)+(6*x+3)*exp(x)-6*x)*exp(1/3*(2*e
xp(x)*x-2*x)*log(x)/exp(x))+3*exp(1)*exp(x))/exp(x)/x,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (2 \, {\left (x^{2} + x e^{x} - x\right )} \log \relax (x)^{2} + 3 \, {\left (2 \, x + 1\right )} e^{x} + 2 \, {\left (3 \, x^{2} + 4 \, x e^{x} - 4 \, x\right )} \log \relax (x) - 6 \, x\right )} x^{\frac {2}{3} \, {\left (x e^{x} - x\right )} e^{\left (-x\right )}} + 3 \, e^{\left (x + 1\right )}\right )} e^{\left (-x\right )}}{3 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((2*exp(x)*x+2*x^2-2*x)*log(x)^2+(8*exp(x)*x+6*x^2-8*x)*log(x)+(6*x+3)*exp(x)-6*x)*exp(1/3*(2*e
xp(x)*x-2*x)*log(x)/exp(x))+3*exp(1)*exp(x))/exp(x)/x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 26, normalized size = 1.04

method	result	size
risch	\({\mathrm e} \ln \relax (x )+\frac {\left (3 \ln \relax (x )+9\right ) x^{-\frac {2 x \left (-1+{\mathrm e}^{-x}\right )}{3}}}{3}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(((2*exp(x)*x+2*x^2-2*x)*ln(x)^2+(8*exp(x)*x+6*x^2-8*x)*ln(x)+(6*x+3)*exp(x)-6*x)*exp(1/3*(2*exp(x)*x-
2*x)*ln(x)/exp(x))+3*exp(1)*exp(x))/exp(x)/x,x,method=_RETURNVERBOSE)

[Out]

exp(1)*ln(x)+1/3*(3*ln(x)+9)*x^(-2/3*x*(-1+exp(-x)))

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maxima [A]  time = 0.55, size = 27, normalized size = 1.08 \begin {gather*} {\left (\log \relax (x) + 3\right )} e^{\left (-\frac {2}{3} \, x e^{\left (-x\right )} \log \relax (x) + \frac {2}{3} \, x \log \relax (x)\right )} + e \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((2*exp(x)*x+2*x^2-2*x)*log(x)^2+(8*exp(x)*x+6*x^2-8*x)*log(x)+(6*x+3)*exp(x)-6*x)*exp(1/3*(2*e
xp(x)*x-2*x)*log(x)/exp(x))+3*exp(1)*exp(x))/exp(x)/x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.28, size = 27, normalized size = 1.08 \begin {gather*} {\mathrm {e}}^{\frac {2\,x\,\ln \relax (x)}{3}-\frac {2\,x\,{\mathrm {e}}^{-x}\,\ln \relax (x)}{3}}\,\left (\ln \relax (x)+3\right )+\mathrm {e}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*((exp(-(exp(-x)*log(x)*(2*x - 2*x*exp(x)))/3)*(log(x)*(8*x*exp(x) - 8*x + 6*x^2) - 6*x + exp(x)*(
6*x + 3) + log(x)^2*(2*x*exp(x) - 2*x + 2*x^2)))/3 + exp(1)*exp(x)))/x,x)

[Out]

exp((2*x*log(x))/3 - (2*x*exp(-x)*log(x))/3)*(log(x) + 3) + exp(1)*log(x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((2*exp(x)*x+2*x**2-2*x)*ln(x)**2+(8*exp(x)*x+6*x**2-8*x)*ln(x)+(6*x+3)*exp(x)-6*x)*exp(1/3*(2*
exp(x)*x-2*x)*ln(x)/exp(x))+3*exp(1)*exp(x))/exp(x)/x,x)

[Out]

Timed out

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