3.61.64 \(\int \frac {-x^2+\frac {1}{3} e^{5-e^x+x} (-3-3 e^x x)}{e^{5-e^x+x} x+x^2} \, dx\)

Optimal. Leaf size=22 \[ 2-x+\log \left (\frac {e^{5-e^x+x}+x}{x}\right ) \]

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Rubi [F]  time = 1.13, 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 {-x^2+\frac {1}{3} e^{5-e^x+x} \left (-3-3 e^x x\right )}{e^{5-e^x+x} x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 + (E^(5 - E^x + x)*(-3 - 3*E^x*x))/3)/(E^(5 - E^x + x)*x + x^2),x]

[Out]

-E^x - Log[x] + Defer[Int][E^E^x*x, x]/E^5 + Defer[Int][E^E^x/(E^(5 + x) + E^E^x*x), x] - Defer[Int][(E^E^x*x)
/(E^(5 + x) + E^E^x*x), x] - Defer[Int][(E^(-5 + 2*E^x)*x^2)/(E^(5 + x) + E^E^x*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^x-\frac {e^5-e^{e^x} x^2}{e^5 x}-\frac {e^{-5+e^x} \left (-e^5+e^5 x+e^{e^x} x^2\right )}{e^{5+x}+e^{e^x} x}\right ) \, dx\\ &=-\frac {\int \frac {e^5-e^{e^x} x^2}{x} \, dx}{e^5}-\int e^x \, dx-\int \frac {e^{-5+e^x} \left (-e^5+e^5 x+e^{e^x} x^2\right )}{e^{5+x}+e^{e^x} x} \, dx\\ &=-e^x-\frac {\int \left (\frac {e^5}{x}-e^{e^x} x\right ) \, dx}{e^5}-\int \left (-\frac {e^{e^x}}{e^{5+x}+e^{e^x} x}+\frac {e^{e^x} x}{e^{5+x}+e^{e^x} x}+\frac {e^{-5+2 e^x} x^2}{e^{5+x}+e^{e^x} x}\right ) \, dx\\ &=-e^x-\log (x)+\frac {\int e^{e^x} x \, dx}{e^5}+\int \frac {e^{e^x}}{e^{5+x}+e^{e^x} x} \, dx-\int \frac {e^{e^x} x}{e^{5+x}+e^{e^x} x} \, dx-\int \frac {e^{-5+2 e^x} x^2}{e^{5+x}+e^{e^x} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 27, normalized size = 1.23 \begin {gather*} -e^x-x-\log (x)+\log \left (e^{5+x}+e^{e^x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + (E^(5 - E^x + x)*(-3 - 3*E^x*x))/3)/(E^(5 - E^x + x)*x + x^2),x]

[Out]

-E^x - x - Log[x] + Log[E^(5 + x) + E^E^x*x]

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fricas [A]  time = 1.26, size = 25, normalized size = 1.14 \begin {gather*} -x + \log \left (x + 3 \, e^{\left (x - e^{x} - \log \relax (3) + 5\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-3)*exp(-exp(x)+x+5-log(3))-x^2)/(3*x*exp(-exp(x)+x+5-log(3))+x^2),x, algorithm="fricas
")

[Out]

-x + log(x + 3*e^(x - e^x - log(3) + 5)) - log(x)

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giac [A]  time = 0.21, size = 19, normalized size = 0.86 \begin {gather*} -x + \log \left (x + e^{\left (x - e^{x} + 5\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-3)*exp(-exp(x)+x+5-log(3))-x^2)/(3*x*exp(-exp(x)+x+5-log(3))+x^2),x, algorithm="giac")

[Out]

-x + log(x + e^(x - e^x + 5)) - log(x)

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maple [A]  time = 0.08, size = 26, normalized size = 1.18

method	result	size
norman	\(-x -\ln \relax (x )+\ln \left (3 \,{\mathrm e}^{-{\mathrm e}^{x}+x +5-\ln \relax (3)}+x \right )\)	\(26\)
risch	\(-\ln \relax (x )-x -5+\ln \relax (3)+\ln \left (\frac {x}{3}+\frac {{\mathrm e}^{x -{\mathrm e}^{x}+5}}{3}\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*exp(x)*x-3)*exp(-exp(x)+x+5-ln(3))-x^2)/(3*x*exp(-exp(x)+x+5-ln(3))+x^2),x,method=_RETURNVERBOSE)

[Out]

-x-ln(x)+ln(3*exp(-exp(x)+x+5-ln(3))+x)

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maxima [A]  time = 0.39, size = 23, normalized size = 1.05 \begin {gather*} -x - e^{x} + \log \left (\frac {x e^{\left (e^{x}\right )} + e^{\left (x + 5\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-3)*exp(-exp(x)+x+5-log(3))-x^2)/(3*x*exp(-exp(x)+x+5-log(3))+x^2),x, algorithm="maxima
")

[Out]

-x - e^x + log((x*e^(e^x) + e^(x + 5))/x)

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mupad [B]  time = 4.40, size = 21, normalized size = 0.95 \begin {gather*} \ln \left (x+{\mathrm {e}}^5\,{\mathrm {e}}^{-{\mathrm {e}}^x}\,{\mathrm {e}}^x\right )-x-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x - log(3) - exp(x) + 5)*(3*x*exp(x) + 3) + x^2)/(3*x*exp(x - log(3) - exp(x) + 5) + x^2),x)

[Out]

log(x + exp(5)*exp(-exp(x))*exp(x)) - x - log(x)

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sympy [A]  time = 0.16, size = 15, normalized size = 0.68 \begin {gather*} - x - \log {\relax (x )} + \log {\left (x + e^{x - e^{x} + 5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*exp(x)*x-3)*exp(-exp(x)+x+5-ln(3))-x**2)/(3*x*exp(-exp(x)+x+5-ln(3))+x**2),x)

[Out]

-x - log(x) + log(x + exp(x - exp(x) + 5))

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