3.95.86 \(\int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} (-27+27 x-11 x^2+x^3+(18-12 x+2 x^2) \log (4)+(-3+x) \log ^2(4))+e^{\frac {3-2 x}{x}} (27 x-36 x^2+15 x^3-2 x^4+(-18 x+18 x^2-4 x^3) \log (4)+(3 x-2 x^2) \log ^2(4))}{9 x-6 x^2+x^3+(-6 x+2 x^2) \log (4)+x \log ^2(4)} \, dx\)

Optimal. Leaf size=26 \[ e^{-2+\frac {3}{x}} \left (e^{\frac {2}{-3+x+\log (4)}}-x\right ) x \]

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Rubi [F]  time = 7.56, 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^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((3 - 2*x)/x + 2/(-3 + x + Log[4]))*(-27 + 27*x - 11*x^2 + x^3 + (18 - 12*x + 2*x^2)*Log[4] + (-3 + x)*
Log[4]^2) + E^((3 - 2*x)/x)*(27*x - 36*x^2 + 15*x^3 - 2*x^4 + (-18*x + 18*x^2 - 4*x^3)*Log[4] + (3*x - 2*x^2)*
Log[4]^2))/(9*x - 6*x^2 + x^3 + (-6*x + 2*x^2)*Log[4] + x*Log[4]^2),x]

[Out]

-(E^(-2 + 3/x)*x^2) + Defer[Int][E^(-2 + 3/x + 2/(-3 + x + Log[4])), x] - 3*Defer[Int][E^(-2 + 3/x + 2/(-3 + x
+ Log[4]))/x, x] - (6 - Log[16])*Defer[Int][E^(-2 + 3/x + 2/(-3 + x + Log[4]))/(-3 + x + Log[4])^2, x] - 2*De
fer[Int][E^(-2 + 3/x + 2/(-3 + x + Log[4]))/(-3 + x + Log[4]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \left (9+\log ^2(4)\right )} \, dx\\ &=\int e^{-2+\frac {3}{x}} \left (3-2 x+\frac {e^{\frac {2}{-3+x+\log (4)}} \left (x^3-3 (-3+\log (4))^2+x \left (27-12 \log (4)+\log ^2(4)\right )+x^2 (-11+\log (16))\right )}{x (-3+x+\log (4))^2}\right ) \, dx\\ &=\int \left (3 e^{-2+\frac {3}{x}}-2 e^{-2+\frac {3}{x}} x+\frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}} \left (x^3-3 (3-\log (4))^2+x (3-\log (4)) (9-\log (4))-x^2 (11-\log (16))\right )}{x (3-x-\log (4))^2}\right ) \, dx\\ &=-\left (2 \int e^{-2+\frac {3}{x}} x \, dx\right )+3 \int e^{-2+\frac {3}{x}} \, dx+\int \frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}} \left (x^3-3 (3-\log (4))^2+x (3-\log (4)) (9-\log (4))-x^2 (11-\log (16))\right )}{x (3-x-\log (4))^2} \, dx\\ &=3 e^{-2+\frac {3}{x}} x-e^{-2+\frac {3}{x}} x^2-3 \int e^{-2+\frac {3}{x}} \, dx+9 \int \frac {e^{-2+\frac {3}{x}}}{x} \, dx+\int \left (e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}-\frac {3 e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{x}-\frac {2 e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{-3+x+\log (4)}+\frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}} (-6+\log (16))}{(-3+x+\log (4))^2}\right ) \, dx\\ &=-e^{-2+\frac {3}{x}} x^2-\frac {9 \text {Ei}\left (\frac {3}{x}\right )}{e^2}-2 \int \frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{-3+x+\log (4)} \, dx-3 \int \frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{x} \, dx-9 \int \frac {e^{-2+\frac {3}{x}}}{x} \, dx+(-6+\log (16)) \int \frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{(-3+x+\log (4))^2} \, dx+\int e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}} \, dx\\ &=-e^{-2+\frac {3}{x}} x^2-2 \int \frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{-3+x+\log (4)} \, dx-3 \int \frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{x} \, dx+(-6+\log (16)) \int \frac {e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}}}{(-3+x+\log (4))^2} \, dx+\int e^{-2+\frac {3}{x}+\frac {2}{-3+x+\log (4)}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {3-2 x}{x}+\frac {2}{-3+x+\log (4)}} \left (-27+27 x-11 x^2+x^3+\left (18-12 x+2 x^2\right ) \log (4)+(-3+x) \log ^2(4)\right )+e^{\frac {3-2 x}{x}} \left (27 x-36 x^2+15 x^3-2 x^4+\left (-18 x+18 x^2-4 x^3\right ) \log (4)+\left (3 x-2 x^2\right ) \log ^2(4)\right )}{9 x-6 x^2+x^3+\left (-6 x+2 x^2\right ) \log (4)+x \log ^2(4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((3 - 2*x)/x + 2/(-3 + x + Log[4]))*(-27 + 27*x - 11*x^2 + x^3 + (18 - 12*x + 2*x^2)*Log[4] + (-3
+ x)*Log[4]^2) + E^((3 - 2*x)/x)*(27*x - 36*x^2 + 15*x^3 - 2*x^4 + (-18*x + 18*x^2 - 4*x^3)*Log[4] + (3*x - 2
*x^2)*Log[4]^2))/(9*x - 6*x^2 + x^3 + (-6*x + 2*x^2)*Log[4] + x*Log[4]^2),x]

[Out]

Integrate[(E^((3 - 2*x)/x + 2/(-3 + x + Log[4]))*(-27 + 27*x - 11*x^2 + x^3 + (18 - 12*x + 2*x^2)*Log[4] + (-3
+ x)*Log[4]^2) + E^((3 - 2*x)/x)*(27*x - 36*x^2 + 15*x^3 - 2*x^4 + (-18*x + 18*x^2 - 4*x^3)*Log[4] + (3*x - 2
*x^2)*Log[4]^2))/(9*x - 6*x^2 + x^3 + (-6*x + 2*x^2)*Log[4] + x*Log[4]^2), x]

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fricas [B]  time = 0.56, size = 55, normalized size = 2.12 \begin {gather*} -x^{2} e^{\left (-\frac {2 \, x - 3}{x}\right )} + x e^{\left (-\frac {2 \, x^{2} + 2 \, {\left (2 \, x - 3\right )} \log \relax (2) - 11 \, x + 9}{x^{2} + 2 \, x \log \relax (2) - 3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-3)*log(2)^2+2*(2*x^2-12*x+18)*log(2)+x^3-11*x^2+27*x-27)*exp((3-2*x)/x)*exp(2/(2*log(2)+x-3))
+(4*(-2*x^2+3*x)*log(2)^2+2*(-4*x^3+18*x^2-18*x)*log(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*log(2)^
2+2*(2*x^2-6*x)*log(2)+x^3-6*x^2+9*x),x, algorithm="fricas")

[Out]

-x^2*e^(-(2*x - 3)/x) + x*e^(-(2*x^2 + 2*(2*x - 3)*log(2) - 11*x + 9)/(x^2 + 2*x*log(2) - 3*x))

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giac [B]  time = 0.63, size = 55, normalized size = 2.12 \begin {gather*} -x^{2} e^{\left (-\frac {2 \, x - 3}{x}\right )} + x e^{\left (-\frac {2 \, x^{2} + 4 \, x \log \relax (2) - 11 \, x - 6 \, \log \relax (2) + 9}{x^{2} + 2 \, x \log \relax (2) - 3 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-3)*log(2)^2+2*(2*x^2-12*x+18)*log(2)+x^3-11*x^2+27*x-27)*exp((3-2*x)/x)*exp(2/(2*log(2)+x-3))
+(4*(-2*x^2+3*x)*log(2)^2+2*(-4*x^3+18*x^2-18*x)*log(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*log(2)^
2+2*(2*x^2-6*x)*log(2)+x^3-6*x^2+9*x),x, algorithm="giac")

[Out]

-x^2*e^(-(2*x - 3)/x) + x*e^(-(2*x^2 + 4*x*log(2) - 11*x - 6*log(2) + 9)/(x^2 + 2*x*log(2) - 3*x))

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maple [B]  time = 0.50, size = 54, normalized size = 2.08

method	result	size
risch	\(-x^{2} {\mathrm e}^{-\frac {2 x -3}{x}}+x \,{\mathrm e}^{-\frac {4 x \ln \relax (2)+2 x^{2}-6 \ln \relax (2)-11 x +9}{x \left (2 \ln \relax (2)+x -3\right )}}\)	\(54\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {3-2 x}{x}} {\mathrm e}^{\frac {2}{2 \ln \relax (2)+x -3}}+\left (3-2 \ln \relax (2)\right ) x^{2} {\mathrm e}^{\frac {3-2 x}{x}}+\left (2 \ln \relax (2)-3\right ) x \,{\mathrm e}^{\frac {3-2 x}{x}} {\mathrm e}^{\frac {2}{2 \ln \relax (2)+x -3}}-x^{3} {\mathrm e}^{\frac {3-2 x}{x}}}{2 \ln \relax (2)+x -3}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*(x-3)*ln(2)^2+2*(2*x^2-12*x+18)*ln(2)+x^3-11*x^2+27*x-27)*exp((3-2*x)/x)*exp(2/(2*ln(2)+x-3))+(4*(-2*x
^2+3*x)*ln(2)^2+2*(-4*x^3+18*x^2-18*x)*ln(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*ln(2)^2+2*(2*x^2-6
*x)*ln(2)+x^3-6*x^2+9*x),x,method=_RETURNVERBOSE)

[Out]

-x^2*exp(-(2*x-3)/x)+x*exp(-(4*x*ln(2)+2*x^2-6*ln(2)-11*x+9)/x/(2*ln(2)+x-3))

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maxima [A]  time = 0.50, size = 36, normalized size = 1.38 \begin {gather*} -{\left (x^{2} e^{\frac {3}{x}} - x e^{\left (\frac {2}{x + 2 \, \log \relax (2) - 3} + \frac {3}{x}\right )}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-3)*log(2)^2+2*(2*x^2-12*x+18)*log(2)+x^3-11*x^2+27*x-27)*exp((3-2*x)/x)*exp(2/(2*log(2)+x-3))
+(4*(-2*x^2+3*x)*log(2)^2+2*(-4*x^3+18*x^2-18*x)*log(2)-2*x^4+15*x^3-36*x^2+27*x)*exp((3-2*x)/x))/(4*x*log(2)^
2+2*(2*x^2-6*x)*log(2)+x^3-6*x^2+9*x),x, algorithm="maxima")

[Out]

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

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(2*x - 3)/x)*(27*x - 2*log(2)*(18*x - 18*x^2 + 4*x^3) + 4*log(2)^2*(3*x - 2*x^2) - 36*x^2 + 15*x^3 -
2*x^4) + exp(2/(x + 2*log(2) - 3))*exp(-(2*x - 3)/x)*(27*x + 4*log(2)^2*(x - 3) + 2*log(2)*(2*x^2 - 12*x + 18
) - 11*x^2 + x^3 - 27))/(9*x - 2*log(2)*(6*x - 2*x^2) + 4*x*log(2)^2 - 6*x^2 + x^3),x)

[Out]

\text{Hanged}

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sympy [A]  time = 2.01, size = 32, normalized size = 1.23 \begin {gather*} - x^{2} e^{\frac {3 - 2 x}{x}} + x e^{\frac {3 - 2 x}{x}} e^{\frac {2}{x - 3 + 2 \log {\relax (2 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-3)*ln(2)**2+2*(2*x**2-12*x+18)*ln(2)+x**3-11*x**2+27*x-27)*exp((3-2*x)/x)*exp(2/(2*ln(2)+x-3)
)+(4*(-2*x**2+3*x)*ln(2)**2+2*(-4*x**3+18*x**2-18*x)*ln(2)-2*x**4+15*x**3-36*x**2+27*x)*exp((3-2*x)/x))/(4*x*l
n(2)**2+2*(2*x**2-6*x)*ln(2)+x**3-6*x**2+9*x),x)

[Out]

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

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