3.71.34 \(\int \frac {9+4 e^3+e^{2 e}+e^e (6+4 e^3-2 x)-6 x+x^2+(6+4 e^3+2 e^e-2 x) \log (x^2)+\log ^2(x^2)}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+(6+2 e^e-2 x) \log (x^2)+\log ^2(x^2)} \, dx\)

Optimal. Leaf size=22 \[ x+\frac {4 e^3 x}{3+e^e-x+\log \left (x^2\right )} \]

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Rubi [F]  time = 0.73, 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 {9+4 e^3+e^{2 e}+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{9+e^{2 e}+e^e (6-2 x)-6 x+x^2+\left (6+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 + 4*E^3 + E^(2*E) + E^E*(6 + 4*E^3 - 2*x) - 6*x + x^2 + (6 + 4*E^3 + 2*E^E - 2*x)*Log[x^2] + Log[x^2]^2
)/(9 + E^(2*E) + E^E*(6 - 2*x) - 6*x + x^2 + (6 + 2*E^E - 2*x)*Log[x^2] + Log[x^2]^2),x]

[Out]

x - 8*E^3*Defer[Int][(3*(1 + E^E/3) - x + Log[x^2])^(-2), x] + 4*E^3*Defer[Int][x/(3*(1 + E^E/3) - x + Log[x^2
])^2, x] + 4*E^3*Defer[Int][(3*(1 + E^E/3) - x + Log[x^2])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 \left (1+\frac {1}{9} \left (4 e^3+e^{2 e}\right )\right )+e^e \left (6+4 e^3-2 x\right )-6 x+x^2+\left (6+4 e^3+2 e^e-2 x\right ) \log \left (x^2\right )+\log ^2\left (x^2\right )}{\left (3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )\right )^2} \, dx\\ &=\int \left (1+\frac {4 e^3 (-2+x)}{\left (3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )\right )^2}+\frac {4 e^3}{3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )}\right ) \, dx\\ &=x+\left (4 e^3\right ) \int \frac {-2+x}{\left (3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )\right )^2} \, dx+\left (4 e^3\right ) \int \frac {1}{3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )} \, dx\\ &=x+\left (4 e^3\right ) \int \frac {1}{3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )} \, dx+\left (4 e^3\right ) \int \left (-\frac {2}{\left (3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )\right )^2}+\frac {x}{\left (3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )\right )^2}\right ) \, dx\\ &=x+\left (4 e^3\right ) \int \frac {x}{\left (3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )\right )^2} \, dx+\left (4 e^3\right ) \int \frac {1}{3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )} \, dx-\left (8 e^3\right ) \int \frac {1}{\left (3 \left (1+\frac {e^e}{3}\right )-x+\log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 22, normalized size = 1.00 \begin {gather*} x+\frac {4 e^3 x}{3+e^e-x+\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + 4*E^3 + E^(2*E) + E^E*(6 + 4*E^3 - 2*x) - 6*x + x^2 + (6 + 4*E^3 + 2*E^E - 2*x)*Log[x^2] + Log[
x^2]^2)/(9 + E^(2*E) + E^E*(6 - 2*x) - 6*x + x^2 + (6 + 2*E^E - 2*x)*Log[x^2] + Log[x^2]^2),x]

[Out]

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

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fricas [A]  time = 0.62, size = 42, normalized size = 1.91 \begin {gather*} \frac {x^{2} - 4 \, x e^{3} - x e^{e} - x \log \left (x^{2}\right ) - 3 \, x}{x - e^{e} - \log \left (x^{2}\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*log(x^2)+exp(exp(1))^2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp
(3)+x^2-6*x+9)/(log(x^2)^2+(2*exp(exp(1))+6-2*x)*log(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x, algo
rithm="fricas")

[Out]

(x^2 - 4*x*e^3 - x*e^e - x*log(x^2) - 3*x)/(x - e^e - log(x^2) - 3)

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giac [A]  time = 0.21, size = 42, normalized size = 1.91 \begin {gather*} \frac {x^{2} - 4 \, x e^{3} - x e^{e} - x \log \left (x^{2}\right ) - 3 \, x}{x - e^{e} - \log \left (x^{2}\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*log(x^2)+exp(exp(1))^2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp
(3)+x^2-6*x+9)/(log(x^2)^2+(2*exp(exp(1))+6-2*x)*log(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x, algo
rithm="giac")

[Out]

(x^2 - 4*x*e^3 - x*e^e - x*log(x^2) - 3*x)/(x - e^e - log(x^2) - 3)

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maple [A]  time = 0.09, size = 22, normalized size = 1.00

method	result	size
risch	\(\frac {4 \,{\mathrm e}^{3} x}{3-x +\ln \left (x^{2}\right )+{\mathrm e}^{{\mathrm e}}}+x\)	\(22\)
norman	\(\frac {x \ln \left (x^{2}\right )+\left (3+{\mathrm e}^{{\mathrm e}}+4 \,{\mathrm e}^{3}\right ) \ln \left (x^{2}\right )-x^{2}+\left ({\mathrm e}^{{\mathrm e}}+3\right ) \left (3+{\mathrm e}^{{\mathrm e}}+4 \,{\mathrm e}^{3}\right )}{3-x +\ln \left (x^{2}\right )+{\mathrm e}^{{\mathrm e}}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*ln(x^2)+exp(exp(1))^2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp(3)+x^2-
6*x+9)/(ln(x^2)^2+(2*exp(exp(1))+6-2*x)*ln(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x,method=_RETURNV
ERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 36, normalized size = 1.64 \begin {gather*} \frac {x^{2} - x {\left (4 \, e^{3} + e^{e} + 3\right )} - 2 \, x \log \relax (x)}{x - e^{e} - 2 \, \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x^2)^2+(2*exp(exp(1))+4*exp(3)+6-2*x)*log(x^2)+exp(exp(1))^2+(4*exp(3)+6-2*x)*exp(exp(1))+4*exp
(3)+x^2-6*x+9)/(log(x^2)^2+(2*exp(exp(1))+6-2*x)*log(x^2)+exp(exp(1))^2+(6-2*x)*exp(exp(1))+x^2-6*x+9),x, algo
rithm="maxima")

[Out]

(x^2 - x*(4*e^3 + e^e + 3) - 2*x*log(x))/(x - e^e - 2*log(x) - 3)

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mupad [B]  time = 4.60, size = 75, normalized size = 3.41 \begin {gather*} x+\frac {x\,\left ({\mathrm {e}}^{2\,\mathrm {e}}-3\,x+4\,{\mathrm {e}}^{\mathrm {e}+3}+12\,{\mathrm {e}}^3+6\,{\mathrm {e}}^{\mathrm {e}}-x\,{\mathrm {e}}^{\mathrm {e}}+9\right )-x\,\left ({\mathrm {e}}^{\mathrm {e}}+3\right )\,\left ({\mathrm {e}}^{\mathrm {e}}-x+3\right )}{\left ({\mathrm {e}}^{\mathrm {e}}+3\right )\,\left (\ln \left (x^2\right )-x+{\mathrm {e}}^{\mathrm {e}}+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*exp(1)) - 6*x + 4*exp(3) + exp(exp(1))*(4*exp(3) - 2*x + 6) + log(x^2)^2 + log(x^2)*(4*exp(3) - 2*x
+ 2*exp(exp(1)) + 6) + x^2 + 9)/(exp(2*exp(1)) - 6*x + log(x^2)*(2*exp(exp(1)) - 2*x + 6) - exp(exp(1))*(2*x
- 6) + log(x^2)^2 + x^2 + 9),x)

[Out]

x + (x*(exp(2*exp(1)) - 3*x + 4*exp(exp(1) + 3) + 12*exp(3) + 6*exp(exp(1)) - x*exp(exp(1)) + 9) - x*(exp(exp(
1)) + 3)*(exp(exp(1)) - x + 3))/((exp(exp(1)) + 3)*(log(x^2) - x + exp(exp(1)) + 3))

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sympy [A]  time = 0.14, size = 20, normalized size = 0.91 \begin {gather*} x + \frac {4 x e^{3}}{- x + \log {\left (x^{2} \right )} + 3 + e^{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x**2)**2+(2*exp(exp(1))+4*exp(3)+6-2*x)*ln(x**2)+exp(exp(1))**2+(4*exp(3)+6-2*x)*exp(exp(1))+4*e
xp(3)+x**2-6*x+9)/(ln(x**2)**2+(2*exp(exp(1))+6-2*x)*ln(x**2)+exp(exp(1))**2+(6-2*x)*exp(exp(1))+x**2-6*x+9),x
)

[Out]

x + 4*x*exp(3)/(-x + log(x**2) + 3 + exp(E))

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