3.96.17 \(\int \frac {e^{-e^{14}+e^x} (-1+e^x x)}{3 e^{-e^{14}+e^x} x+3 x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{3} \log \left (\frac {-e^{-e^{14}+e^x}-x}{x}\right ) \]

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Rubi [F]  time = 0.62, 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^{-e^{14}+e^x} \left (-1+e^x x\right )}{3 e^{-e^{14}+e^x} x+3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-E^14 + E^x)*(-1 + E^x*x))/(3*E^(-E^14 + E^x)*x + 3*x^2),x]

[Out]

Defer[Int][E^(E^x + x)/(E^E^x + E^E^14*x), x]/3 - Defer[Int][E^E^x/(x*(E^E^x + E^E^14*x)), x]/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{e^x+x}}{3 \left (e^{e^x}+e^{e^{14}} x\right )}-\frac {e^{e^x}}{3 x \left (e^{e^x}+e^{e^{14}} x\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{e^x+x}}{e^{e^x}+e^{e^{14}} x} \, dx-\frac {1}{3} \int \frac {e^{e^x}}{x \left (e^{e^x}+e^{e^{14}} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{3} \left (-\log (x)+\log \left (e^{e^x}+e^{e^{14}} x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^14 + E^x)*(-1 + E^x*x))/(3*E^(-E^14 + E^x)*x + 3*x^2),x]

[Out]

(-Log[x] + Log[E^E^x + E^E^14*x])/3

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fricas [A]  time = 0.50, size = 18, normalized size = 0.69 \begin {gather*} \frac {1}{3} \, \log \left (x + e^{\left (-e^{14} + e^{x}\right )}\right ) - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)*exp(exp(x)-exp(14))/(3*x*exp(exp(x)-exp(14))+3*x^2),x, algorithm="fricas")

[Out]

1/3*log(x + e^(-e^14 + e^x)) - 1/3*log(x)

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giac [A]  time = 0.32, size = 25, normalized size = 0.96 \begin {gather*} -\frac {1}{3} \, x + \frac {1}{3} \, \log \left (x e^{x} + e^{\left (x - e^{14} + e^{x}\right )}\right ) - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)*exp(exp(x)-exp(14))/(3*x*exp(exp(x)-exp(14))+3*x^2),x, algorithm="giac")

[Out]

-1/3*x + 1/3*log(x*e^x + e^(x - e^14 + e^x)) - 1/3*log(x)

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maple [A]  time = 0.08, size = 19, normalized size = 0.73

method	result	size
norman	\(-\frac {\ln \relax (x )}{3}+\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}-{\mathrm e}^{14}}+x \right )}{3}\)	\(19\)
risch	\(-\frac {\ln \relax (x )}{3}+\frac {{\mathrm e}^{14}}{3}+\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}-{\mathrm e}^{14}}+x \right )}{3}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*x-1)*exp(exp(x)-exp(14))/(3*x*exp(exp(x)-exp(14))+3*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/3*ln(x)+1/3*ln(exp(exp(x)-exp(14))+x)

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maxima [A]  time = 0.40, size = 17, normalized size = 0.65 \begin {gather*} \frac {1}{3} \, \log \left (x e^{\left (e^{14}\right )} + e^{\left (e^{x}\right )}\right ) - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)*exp(exp(x)-exp(14))/(3*x*exp(exp(x)-exp(14))+3*x^2),x, algorithm="maxima")

[Out]

1/3*log(x*e^(e^14) + e^(e^x)) - 1/3*log(x)

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mupad [B]  time = 0.25, size = 19, normalized size = 0.73 \begin {gather*} \frac {\ln \left (x+{\mathrm {e}}^{-{\mathrm {e}}^{14}}\,{\mathrm {e}}^{{\mathrm {e}}^x}\right )}{3}-\frac {\ln \relax (x)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x) - exp(14))*(x*exp(x) - 1))/(3*x*exp(exp(x) - exp(14)) + 3*x^2),x)

[Out]

log(x + exp(-exp(14))*exp(exp(x)))/3 - log(x)/3

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sympy [A]  time = 0.13, size = 17, normalized size = 0.65 \begin {gather*} - \frac {\log {\relax (x )}}{3} + \frac {\log {\left (x + e^{e^{x} - e^{14}} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*x-1)*exp(exp(x)-exp(14))/(3*x*exp(exp(x)-exp(14))+3*x**2),x)

[Out]

-log(x)/3 + log(x + exp(exp(x) - exp(14)))/3

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