3.61.24 \(\int \frac {8-8 x+2 x^2+e^{\frac {-3 x+x^2}{-2+x}} (-156 x+104 x^2-26 x^3)+e^{\frac {-3 x+x^2}{-2+x}} (6 x-4 x^2+x^3) \log (x)}{-104 x+104 x^2-26 x^3+(4 x-4 x^2+x^3) \log (x)} \, dx\)

Optimal. Leaf size=23 \[ e^{x+\frac {x}{2-x}}+\log \left ((26-\log (x))^2\right ) \]

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Rubi [F]  time = 2.17, 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 {8-8 x+2 x^2+e^{\frac {-3 x+x^2}{-2+x}} \left (-156 x+104 x^2-26 x^3\right )+e^{\frac {-3 x+x^2}{-2+x}} \left (6 x-4 x^2+x^3\right ) \log (x)}{-104 x+104 x^2-26 x^3+\left (4 x-4 x^2+x^3\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 - 8*x + 2*x^2 + E^((-3*x + x^2)/(-2 + x))*(-156*x + 104*x^2 - 26*x^3) + E^((-3*x + x^2)/(-2 + x))*(6*x
- 4*x^2 + x^3)*Log[x])/(-104*x + 104*x^2 - 26*x^3 + (4*x - 4*x^2 + x^3)*Log[x]),x]

[Out]

E^(((3 - x)*x)/(2 - x)) - 8*Defer[Int][1/((-2 + x)^2*(-26 + Log[x])), x] + 8*Defer[Int][1/((-2 + x)^2*x*(-26 +
Log[x])), x] + 2*Defer[Int][x/((-2 + x)^2*(-26 + Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8+8 x-2 x^2-e^{\frac {-3 x+x^2}{-2+x}} \left (-156 x+104 x^2-26 x^3\right )-e^{\frac {-3 x+x^2}{-2+x}} \left (6 x-4 x^2+x^3\right ) \log (x)}{(2-x)^2 x (26-\log (x))} \, dx\\ &=\int \left (\frac {e^{\frac {(-3+x) x}{-2+x}} \left (6-4 x+x^2\right )}{(-2+x)^2}-\frac {8}{(-2+x)^2 (-26+\log (x))}+\frac {8}{(-2+x)^2 x (-26+\log (x))}+\frac {2 x}{(-2+x)^2 (-26+\log (x))}\right ) \, dx\\ &=2 \int \frac {x}{(-2+x)^2 (-26+\log (x))} \, dx-8 \int \frac {1}{(-2+x)^2 (-26+\log (x))} \, dx+8 \int \frac {1}{(-2+x)^2 x (-26+\log (x))} \, dx+\int \frac {e^{\frac {(-3+x) x}{-2+x}} \left (6-4 x+x^2\right )}{(-2+x)^2} \, dx\\ &=e^{\frac {(3-x) x}{2-x}}+2 \int \frac {x}{(-2+x)^2 (-26+\log (x))} \, dx-8 \int \frac {1}{(-2+x)^2 (-26+\log (x))} \, dx+8 \int \frac {1}{(-2+x)^2 x (-26+\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 22, normalized size = 0.96 \begin {gather*} e^{-1-\frac {2}{-2+x}+x}+2 \log (26-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 8*x + 2*x^2 + E^((-3*x + x^2)/(-2 + x))*(-156*x + 104*x^2 - 26*x^3) + E^((-3*x + x^2)/(-2 + x))
*(6*x - 4*x^2 + x^3)*Log[x])/(-104*x + 104*x^2 - 26*x^3 + (4*x - 4*x^2 + x^3)*Log[x]),x]

[Out]

E^(-1 - 2/(-2 + x) + x) + 2*Log[26 - Log[x]]

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fricas [A]  time = 0.76, size = 22, normalized size = 0.96 \begin {gather*} e^{\left (\frac {x^{2} - 3 \, x}{x - 2}\right )} + 2 \, \log \left (\log \relax (x) - 26\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-4*x^2+6*x)*exp((x^2-3*x)/(x-2))*log(x)+(-26*x^3+104*x^2-156*x)*exp((x^2-3*x)/(x-2))+2*x^2-8*x+
8)/((x^3-4*x^2+4*x)*log(x)-26*x^3+104*x^2-104*x),x, algorithm="fricas")

[Out]

e^((x^2 - 3*x)/(x - 2)) + 2*log(log(x) - 26)

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giac [A]  time = 0.25, size = 22, normalized size = 0.96 \begin {gather*} e^{\left (\frac {x^{2} - 3 \, x}{x - 2}\right )} + 2 \, \log \left (\log \relax (x) - 26\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-4*x^2+6*x)*exp((x^2-3*x)/(x-2))*log(x)+(-26*x^3+104*x^2-156*x)*exp((x^2-3*x)/(x-2))+2*x^2-8*x+
8)/((x^3-4*x^2+4*x)*log(x)-26*x^3+104*x^2-104*x),x, algorithm="giac")

[Out]

e^((x^2 - 3*x)/(x - 2)) + 2*log(log(x) - 26)

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maple [A]  time = 0.14, size = 20, normalized size = 0.87

method	result	size
risch	\({\mathrm e}^{\frac {x \left (x -3\right )}{x -2}}+2 \ln \left (-26+\ln \relax (x )\right )\)	\(20\)
default	\(2 \ln \left (-26+\ln \relax (x )\right )+\frac {x \,{\mathrm e}^{\frac {x^{2}-3 x}{x -2}}-2 \,{\mathrm e}^{\frac {x^{2}-3 x}{x -2}}}{x -2}\)	\(48\)
norman	\(2 \ln \left (-26+\ln \relax (x )\right )+\frac {x \,{\mathrm e}^{\frac {x^{2}-3 x}{x -2}}-2 \,{\mathrm e}^{\frac {x^{2}-3 x}{x -2}}}{x -2}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-4*x^2+6*x)*exp((x^2-3*x)/(x-2))*ln(x)+(-26*x^3+104*x^2-156*x)*exp((x^2-3*x)/(x-2))+2*x^2-8*x+8)/((x^
3-4*x^2+4*x)*ln(x)-26*x^3+104*x^2-104*x),x,method=_RETURNVERBOSE)

[Out]

exp(x*(x-3)/(x-2))+2*ln(-26+ln(x))

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maxima [A]  time = 0.40, size = 19, normalized size = 0.83 \begin {gather*} e^{\left (x - \frac {2}{x - 2} - 1\right )} + 2 \, \log \left (\log \relax (x) - 26\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-4*x^2+6*x)*exp((x^2-3*x)/(x-2))*log(x)+(-26*x^3+104*x^2-156*x)*exp((x^2-3*x)/(x-2))+2*x^2-8*x+
8)/((x^3-4*x^2+4*x)*log(x)-26*x^3+104*x^2-104*x),x, algorithm="maxima")

[Out]

e^(x - 2/(x - 2) - 1) + 2*log(log(x) - 26)

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mupad [B]  time = 4.64, size = 28, normalized size = 1.22 \begin {gather*} 2\,\ln \left (\ln \relax (x)-26\right )+{\mathrm {e}}^{-\frac {3\,x}{x-2}}\,{\mathrm {e}}^{\frac {x^2}{x-2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^2 - exp(-(3*x - x^2)/(x - 2))*(156*x - 104*x^2 + 26*x^3) - 8*x + exp(-(3*x - x^2)/(x - 2))*log(x)*(6
*x - 4*x^2 + x^3) + 8)/(104*x - log(x)*(4*x - 4*x^2 + x^3) - 104*x^2 + 26*x^3),x)

[Out]

2*log(log(x) - 26) + exp(-(3*x)/(x - 2))*exp(x^2/(x - 2))

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sympy [A]  time = 0.46, size = 19, normalized size = 0.83 \begin {gather*} e^{\frac {x^{2} - 3 x}{x - 2}} + 2 \log {\left (\log {\relax (x )} - 26 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-4*x**2+6*x)*exp((x**2-3*x)/(x-2))*ln(x)+(-26*x**3+104*x**2-156*x)*exp((x**2-3*x)/(x-2))+2*x**
2-8*x+8)/((x**3-4*x**2+4*x)*ln(x)-26*x**3+104*x**2-104*x),x)

[Out]

exp((x**2 - 3*x)/(x - 2)) + 2*log(log(x) - 26)

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