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

Optimal. Leaf size=20 \[ \log \left (3 e+\frac {2 \left (1+e^{2 e^x}\right )}{x}+x\right ) \]

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Rubi [F]  time = 0.79, 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^2+e^{2 e^x} \left (-2+4 e^x x\right )}{2 x+2 e^{2 e^x} x+3 e x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-Log[x] + 3*E*Defer[Int][(2 + 2*E^(2*E^x) + 3*E*x + x^2)^(-1), x] + 4*Defer[Int][E^(2*E^x + x)/(2 + 2*E^(2*E^x
) + 3*E*x + x^2), x] + 2*Defer[Int][x/(2 + 2*E^(2*E^x) + 3*E*x + x^2), x]

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 24, normalized size = 1.20 \begin {gather*} -\log (x)+\log \left (2+2 e^{2 e^x}+3 e x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 23, normalized size = 1.15 \begin {gather*} \log \left (x^{2} + 3 \, x e + 2 \, e^{\left (2 \, e^{x}\right )} + 2\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x)*x-2)*exp(exp(x))^2+x^2-2)/(2*x*exp(exp(x))^2+3*x^2*exp(1)+x^3+2*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 23, normalized size = 1.15 \begin {gather*} \log \left (x^{2} + 3 \, x e + 2 \, e^{\left (2 \, e^{x}\right )} + 2\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x)*x-2)*exp(exp(x))^2+x^2-2)/(2*x*exp(exp(x))^2+3*x^2*exp(1)+x^3+2*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.19, size = 24, normalized size = 1.20

method	result	size
norman	\(-\ln \relax (x )+\ln \left (2 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+3 x \,{\mathrm e}+x^{2}+2\right )\)	\(24\)
risch	\(-\ln \relax (x )+\ln \left (\frac {3 x \,{\mathrm e}}{2}+\frac {x^{2}}{2}+{\mathrm e}^{2 \,{\mathrm e}^{x}}+1\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*exp(x)*x-2)*exp(exp(x))^2+x^2-2)/(2*x*exp(exp(x))^2+3*x^2*exp(1)+x^3+2*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.39, size = 23, normalized size = 1.15 \begin {gather*} \log \left (\frac {1}{2} \, x^{2} + \frac {3}{2} \, x e + e^{\left (2 \, e^{x}\right )} + 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x)*x-2)*exp(exp(x))^2+x^2-2)/(2*x*exp(exp(x))^2+3*x^2*exp(1)+x^3+2*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.27, size = 23, normalized size = 1.15 \begin {gather*} \ln \left (2\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}+3\,x\,\mathrm {e}+x^2+2\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*exp(x))*(4*x*exp(x) - 2) + x^2 - 2)/(2*x + 2*x*exp(2*exp(x)) + 3*x^2*exp(1) + x^3),x)

[Out]

log(2*exp(2*exp(x)) + 3*x*exp(1) + x^2 + 2) - log(x)

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sympy [A]  time = 0.18, size = 26, normalized size = 1.30 \begin {gather*} - \log {\relax (x )} + \log {\left (\frac {x^{2}}{2} + \frac {3 e x}{2} + e^{2 e^{x}} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x)*x-2)*exp(exp(x))**2+x**2-2)/(2*x*exp(exp(x))**2+3*x**2*exp(1)+x**3+2*x),x)

[Out]

-log(x) + log(x**2/2 + 3*E*x/2 + exp(2*exp(x)) + 1)

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