3.24.40 \(\int \frac {x^3+e^{e^x} (-2+e^x x)}{e^{e^x} x-5 x^3+x^4} \, dx\)

Optimal. Leaf size=13 \[ \log \left (-5+\frac {e^{e^x}}{x^2}+x\right ) \]

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Rubi [F]  time = 0.76, 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^3+e^{e^x} \left (-2+e^x x\right )}{e^{e^x} x-5 x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^3 + E^E^x*(-2 + E^x*x))/(E^E^x*x - 5*x^3 + x^4),x]

[Out]

-2*Log[x] + Defer[Int][E^(E^x + x)/(E^E^x - 5*x^2 + x^3), x] - 10*Defer[Int][x/(E^E^x - 5*x^2 + x^3), x] + 3*D
efer[Int][x^2/(E^E^x - 5*x^2 + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{e^x+x}}{e^{e^x}-5 x^2+x^3}+\frac {-2 e^{e^x}+x^3}{x \left (e^{e^x}-5 x^2+x^3\right )}\right ) \, dx\\ &=\int \frac {e^{e^x+x}}{e^{e^x}-5 x^2+x^3} \, dx+\int \frac {-2 e^{e^x}+x^3}{x \left (e^{e^x}-5 x^2+x^3\right )} \, dx\\ &=\int \frac {e^{e^x+x}}{e^{e^x}-5 x^2+x^3} \, dx+\int \left (-\frac {2}{x}+\frac {x (-10+3 x)}{e^{e^x}-5 x^2+x^3}\right ) \, dx\\ &=-2 \log (x)+\int \frac {e^{e^x+x}}{e^{e^x}-5 x^2+x^3} \, dx+\int \frac {x (-10+3 x)}{e^{e^x}-5 x^2+x^3} \, dx\\ &=-2 \log (x)+\int \frac {e^{e^x+x}}{e^{e^x}-5 x^2+x^3} \, dx+\int \left (-\frac {10 x}{e^{e^x}-5 x^2+x^3}+\frac {3 x^2}{e^{e^x}-5 x^2+x^3}\right ) \, dx\\ &=-2 \log (x)+3 \int \frac {x^2}{e^{e^x}-5 x^2+x^3} \, dx-10 \int \frac {x}{e^{e^x}-5 x^2+x^3} \, dx+\int \frac {e^{e^x+x}}{e^{e^x}-5 x^2+x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 19, normalized size = 1.46 \begin {gather*} -2 \log (x)+\log \left (e^{e^x}+(-5+x) x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3 + E^E^x*(-2 + E^x*x))/(E^E^x*x - 5*x^3 + x^4),x]

[Out]

-2*Log[x] + Log[E^E^x + (-5 + x)*x^2]

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fricas [A]  time = 0.68, size = 18, normalized size = 1.38 \begin {gather*} \log \left (x^{3} - 5 \, x^{2} + e^{\left (e^{x}\right )}\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x-2)*exp(exp(x))+x^3)/(x*exp(exp(x))+x^4-5*x^3),x, algorithm="fricas")

[Out]

log(x^3 - 5*x^2 + e^(e^x)) - 2*log(x)

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giac [B]  time = 0.15, size = 28, normalized size = 2.15 \begin {gather*} -x + \log \left (x^{3} e^{x} - 5 \, x^{2} e^{x} + e^{\left (x + e^{x}\right )}\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x-2)*exp(exp(x))+x^3)/(x*exp(exp(x))+x^4-5*x^3),x, algorithm="giac")

[Out]

-x + log(x^3*e^x - 5*x^2*e^x + e^(x + e^x)) - 2*log(x)

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maple [A]  time = 0.05, size = 19, normalized size = 1.46

method	result	size
norman	\(-2 \ln \relax (x )+\ln \left (x^{3}-5 x^{2}+{\mathrm e}^{{\mathrm e}^{x}}\right )\)	\(19\)
risch	\(-2 \ln \relax (x )+\ln \left (x^{3}-5 x^{2}+{\mathrm e}^{{\mathrm e}^{x}}\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*x-2)*exp(exp(x))+x^3)/(x*exp(exp(x))+x^4-5*x^3),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)+ln(x^3-5*x^2+exp(exp(x)))

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maxima [A]  time = 0.48, size = 18, normalized size = 1.38 \begin {gather*} \log \left (x^{3} - 5 \, x^{2} + e^{\left (e^{x}\right )}\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x-2)*exp(exp(x))+x^3)/(x*exp(exp(x))+x^4-5*x^3),x, algorithm="maxima")

[Out]

log(x^3 - 5*x^2 + e^(e^x)) - 2*log(x)

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mupad [B]  time = 0.09, size = 18, normalized size = 1.38 \begin {gather*} \ln \left ({\mathrm {e}}^{{\mathrm {e}}^x}-5\,x^2+x^3\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x))*(x*exp(x) - 2) + x^3)/(x*exp(exp(x)) - 5*x^3 + x^4),x)

[Out]

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

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sympy [A]  time = 0.16, size = 19, normalized size = 1.46 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (x^{3} - 5 x^{2} + e^{e^{x}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x-2)*exp(exp(x))+x**3)/(x*exp(exp(x))+x**4-5*x**3),x)

[Out]

-2*log(x) + log(x**3 - 5*x**2 + exp(exp(x)))

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