3.19.57 \(\int \frac {6+2 e^x+2 x+e^x (-2+2 x) \log (x)+e^{3+e^e} (-3-e^x-x+e^x (1-x) \log (x))}{9+(-6 e^x-6 x) \log (x)+(e^{2 x}+2 e^x x+x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {\left (-2+e^{3+e^e}\right ) x}{-3+\left (e^x+x\right ) \log (x)} \]

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Rubi [F]  time = 1.66, 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 {6+2 e^x+2 x+e^x (-2+2 x) \log (x)+e^{3+e^e} \left (-3-e^x-x+e^x (1-x) \log (x)\right )}{9+\left (-6 e^x-6 x\right ) \log (x)+\left (e^{2 x}+2 e^x x+x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 + 2*E^x + 2*x + E^x*(-2 + 2*x)*Log[x] + E^(3 + E^E)*(-3 - E^x - x + E^x*(1 - x)*Log[x]))/(9 + (-6*E^x -
6*x)*Log[x] + (E^(2*x) + 2*E^x*x + x^2)*Log[x]^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 25, normalized size = 1.09 \begin {gather*} \frac {\left (-2+e^{3+e^e}\right ) x}{-3+e^x \log (x)+x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 2*E^x + 2*x + E^x*(-2 + 2*x)*Log[x] + E^(3 + E^E)*(-3 - E^x - x + E^x*(1 - x)*Log[x]))/(9 + (-6
*E^x - 6*x)*Log[x] + (E^(2*x) + 2*E^x*x + x^2)*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.78, size = 24, normalized size = 1.04 \begin {gather*} \frac {x e^{\left (e^{e} + 3\right )} - 2 \, x}{{\left (x + e^{x}\right )} \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.39, size = 26, normalized size = 1.13 \begin {gather*} \frac {x e^{\left (e^{e} + 3\right )} - 2 \, x}{x \log \relax (x) + e^{x} \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 24, normalized size = 1.04

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.76, size = 23, normalized size = 1.00 \begin {gather*} \frac {x {\left (e^{\left (e^{e} + 3\right )} - 2\right )}}{x \log \relax (x) + e^{x} \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,x+2\,{\mathrm {e}}^x-{\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}+3}\,\left (x+{\mathrm {e}}^x+{\mathrm {e}}^x\,\ln \relax (x)\,\left (x-1\right )+3\right )+{\mathrm {e}}^x\,\ln \relax (x)\,\left (2\,x-2\right )+6}{\left ({\mathrm {e}}^{2\,x}+2\,x\,{\mathrm {e}}^x+x^2\right )\,{\ln \relax (x)}^2+\left (-6\,x-6\,{\mathrm {e}}^x\right )\,\ln \relax (x)+9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.35, size = 27, normalized size = 1.17 \begin {gather*} \frac {- 2 x + x e^{3} e^{e^{e}}}{x \log {\relax (x )} + e^{x} \log {\relax (x )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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