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

Optimal. Leaf size=21 \[ 3+\frac {x^2}{x-2 x^2+\log \left (e^x+x\right )} \]

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Rubi [F]  time = 2.43, 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+x^3+\left (2 e^x x+2 x^2\right ) \log \left (e^x+x\right )}{x^3-4 x^4+4 x^5+e^x \left (x^2-4 x^3+4 x^4\right )+\left (2 x^2-4 x^3+e^x \left (2 x-4 x^2\right )\right ) \log \left (e^x+x\right )+\left (e^x+x\right ) \log ^2\left (e^x+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*Defer[Int][x^2/(-x + 2*x^2 - Log[E^x + x])^2, x] + 4*Defer[Int][x^3/(-x + 2*x^2 - Log[E^x + x])^2, x] - Def
er[Int][x^2/((E^x + x)*(-x + 2*x^2 - Log[E^x + x])^2), x] + Defer[Int][x^3/((E^x + x)*(-x + 2*x^2 - Log[E^x +
x])^2), x] - 2*Defer[Int][x/(-x + 2*x^2 - Log[E^x + x]), x]

Rubi steps

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

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Mathematica [A]  time = 0.29, size = 19, normalized size = 0.90 \begin {gather*} \frac {x^2}{x-2 x^2+\log \left (e^x+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/(x - 2*x^2 + Log[E^x + x])

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fricas [A]  time = 0.85, size = 23, normalized size = 1.10 \begin {gather*} -\frac {x^{2}}{2 \, x^{2} - x - \log \left (x + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/(2*x^2 - x - log(x + e^x))

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giac [A]  time = 0.33, size = 23, normalized size = 1.10 \begin {gather*} -\frac {x^{2}}{2 \, x^{2} - x - \log \left (x + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/(2*x^2 - x - log(x + e^x))

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maple [A]  time = 0.03, size = 24, normalized size = 1.14

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/(2*x^2-x-ln(exp(x)+x))

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maxima [A]  time = 0.44, size = 23, normalized size = 1.10 \begin {gather*} -\frac {x^{2}}{2 \, x^{2} - x - \log \left (x + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2/(2*x^2 - x - log(x + e^x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\ln \left (x+{\mathrm {e}}^x\right )\,\left (2\,x\,{\mathrm {e}}^x+2\,x^2\right )-x^2+x^3}{{\ln \left (x+{\mathrm {e}}^x\right )}^2\,\left (x+{\mathrm {e}}^x\right )+x^3-4\,x^4+4\,x^5+{\mathrm {e}}^x\,\left (4\,x^4-4\,x^3+x^2\right )+\ln \left (x+{\mathrm {e}}^x\right )\,\left ({\mathrm {e}}^x\,\left (2\,x-4\,x^2\right )+2\,x^2-4\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.19, size = 15, normalized size = 0.71 \begin {gather*} \frac {x^{2}}{- 2 x^{2} + x + \log {\left (x + e^{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(-2*x**2 + x + log(x + exp(x)))

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