3.101.23 \(\int \frac {-2 x^3+e^{\frac {e^x}{x}} (3 x-3 x^2+e^x (3-4 x+x^2))}{-9 x^3+3 x^4+e^{\frac {e^x}{x}} (-9 x^2+3 x^3)} \, dx\)

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

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Rubi [F]  time = 2.67, 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 {-2 x^3+e^{\frac {e^x}{x}} \left (3 x-3 x^2+e^x \left (3-4 x+x^2\right )\right )}{-9 x^3+3 x^4+e^{\frac {e^x}{x}} \left (-9 x^2+3 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x^3 + E^(E^x/x)*(3*x - 3*x^2 + E^x*(3 - 4*x + x^2)))/(-9*x^3 + 3*x^4 + E^(E^x/x)*(-9*x^2 + 3*x^3)),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.30, size = 29, normalized size = 1.16 \begin {gather*} \frac {1}{3} \left (-2 \log (3-x)-\log (x)+\log \left (e^{\frac {e^x}{x}}+x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^3 + E^(E^x/x)*(3*x - 3*x^2 + E^x*(3 - 4*x + x^2)))/(-9*x^3 + 3*x^4 + E^(E^x/x)*(-9*x^2 + 3*x^3
)),x]

[Out]

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

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fricas [A]  time = 0.90, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, \log \left (x + e^{\left (\frac {e^{x}}{x}\right )}\right ) - \frac {2}{3} \, \log \left (x - 3\right ) - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-4*x+3)*exp(x)-3*x^2+3*x)*exp(exp(x)/x)-2*x^3)/((3*x^3-9*x^2)*exp(exp(x)/x)+3*x^4-9*x^3),x, al
gorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 33, normalized size = 1.32 \begin {gather*} -\frac {1}{3} \, x + \frac {1}{3} \, \log \left (x e^{x} + e^{\left (\frac {x^{2} + e^{x}}{x}\right )}\right ) - \frac {2}{3} \, \log \left (x - 3\right ) - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-4*x+3)*exp(x)-3*x^2+3*x)*exp(exp(x)/x)-2*x^3)/((3*x^3-9*x^2)*exp(exp(x)/x)+3*x^4-9*x^3),x, al
gorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 24, normalized size = 0.96

method	result	size
norman	\(-\frac {\ln \relax (x )}{3}-\frac {2 \ln \left (x -3\right )}{3}+\frac {\ln \left (x +{\mathrm e}^{\frac {{\mathrm e}^{x}}{x}}\right )}{3}\)	\(24\)
risch	\(-\frac {\ln \relax (x )}{3}-\frac {2 \ln \left (x -3\right )}{3}+\frac {\ln \left (x +{\mathrm e}^{\frac {{\mathrm e}^{x}}{x}}\right )}{3}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-4*x+3)*exp(x)-3*x^2+3*x)*exp(exp(x)/x)-2*x^3)/((3*x^3-9*x^2)*exp(exp(x)/x)+3*x^4-9*x^3),x,method=_R
ETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, \log \left (x + e^{\left (\frac {e^{x}}{x}\right )}\right ) - \frac {2}{3} \, \log \left (x - 3\right ) - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-4*x+3)*exp(x)-3*x^2+3*x)*exp(exp(x)/x)-2*x^3)/((3*x^3-9*x^2)*exp(exp(x)/x)+3*x^4-9*x^3),x, al
gorithm="maxima")

[Out]

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

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mupad [B]  time = 6.67, size = 23, normalized size = 0.92 \begin {gather*} \frac {\ln \left (x+{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\right )}{3}-\frac {2\,\ln \left (x-3\right )}{3}-\frac {\ln \relax (x)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x)/x)*(3*x + exp(x)*(x^2 - 4*x + 3) - 3*x^2) - 2*x^3)/(exp(exp(x)/x)*(9*x^2 - 3*x^3) + 9*x^3 - 3
*x^4),x)

[Out]

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

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sympy [A]  time = 0.28, size = 24, normalized size = 0.96 \begin {gather*} - \frac {\log {\relax (x )}}{3} - \frac {2 \log {\left (x - 3 \right )}}{3} + \frac {\log {\left (x + e^{\frac {e^{x}}{x}} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-4*x+3)*exp(x)-3*x**2+3*x)*exp(exp(x)/x)-2*x**3)/((3*x**3-9*x**2)*exp(exp(x)/x)+3*x**4-9*x**3
),x)

[Out]

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

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