3.44.32 \(\int \frac {-54 x+3 x^2+12 e^x x^2+(6 x^2+e^x (-12 x^2+12 x^3)) \log (x)}{81-54 x+9 x^2+(-36 x+12 x^2+e^x (72 x-24 x^2)) \log (x)+(4 x^2-16 e^x x^2+16 e^{2 x} x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=28 \[ 5-\frac {x^2}{3-x-\frac {1}{3} \left (2-4 e^x\right ) x \log (x)} \]

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Rubi [F]  time = 2.92, 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 {-54 x+3 x^2+12 e^x x^2+\left (6 x^2+e^x \left (-12 x^2+12 x^3\right )\right ) \log (x)}{81-54 x+9 x^2+\left (-36 x+12 x^2+e^x \left (72 x-24 x^2\right )\right ) \log (x)+\left (4 x^2-16 e^x x^2+16 e^{2 x} x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-54*x + 3*x^2 + 12*E^x*x^2 + (6*x^2 + E^x*(-12*x^2 + 12*x^3))*Log[x])/(81 - 54*x + 9*x^2 + (-36*x + 12*x^
2 + E^x*(72*x - 24*x^2))*Log[x] + (4*x^2 - 16*E^x*x^2 + 16*E^(2*x)*x^2)*Log[x]^2),x]

[Out]

-27*Defer[Int][x/(9 - 3*x - 2*x*Log[x] + 4*E^x*x*Log[x])^2, x] - 27*Defer[Int][x^2/(9 - 3*x - 2*x*Log[x] + 4*E
^x*x*Log[x])^2, x] + 9*Defer[Int][x^3/(9 - 3*x - 2*x*Log[x] + 4*E^x*x*Log[x])^2, x] - 27*Defer[Int][x/(Log[x]*
(9 - 3*x - 2*x*Log[x] + 4*E^x*x*Log[x])^2), x] + 9*Defer[Int][x^2/(Log[x]*(9 - 3*x - 2*x*Log[x] + 4*E^x*x*Log[
x])^2), x] + 6*Defer[Int][(x^3*Log[x])/(9 - 3*x - 2*x*Log[x] + 4*E^x*x*Log[x])^2, x] - 3*Defer[Int][x/(9 - 3*x
- 2*x*Log[x] + 4*E^x*x*Log[x]), x] + 3*Defer[Int][x^2/(9 - 3*x - 2*x*Log[x] + 4*E^x*x*Log[x]), x] + 3*Defer[I
nt][x/(Log[x]*(9 - 3*x - 2*x*Log[x] + 4*E^x*x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 x \left (-18+x+4 e^x x+2 \left (1+2 e^x (-1+x)\right ) x \log (x)\right )}{\left (9-3 x+2 \left (-1+2 e^x\right ) x \log (x)\right )^2} \, dx\\ &=3 \int \frac {x \left (-18+x+4 e^x x+2 \left (1+2 e^x (-1+x)\right ) x \log (x)\right )}{\left (9-3 x+2 \left (-1+2 e^x\right ) x \log (x)\right )^2} \, dx\\ &=3 \int \left (\frac {x (1-\log (x)+x \log (x))}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )}+\frac {x \left (-9+3 x-9 \log (x)-9 x \log (x)+3 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}\right ) \, dx\\ &=3 \int \frac {x (1-\log (x)+x \log (x))}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )} \, dx+3 \int \frac {x \left (-9+3 x-9 \log (x)-9 x \log (x)+3 x^2 \log (x)+2 x^2 \log ^2(x)\right )}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx\\ &=3 \int \left (-\frac {9 x}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}-\frac {9 x^2}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}+\frac {3 x^3}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}-\frac {9 x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}+\frac {3 x^2}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}+\frac {2 x^3 \log (x)}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2}\right ) \, dx+3 \int \left (-\frac {x}{9-3 x-2 x \log (x)+4 e^x x \log (x)}+\frac {x^2}{9-3 x-2 x \log (x)+4 e^x x \log (x)}+\frac {x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )}\right ) \, dx\\ &=-\left (3 \int \frac {x}{9-3 x-2 x \log (x)+4 e^x x \log (x)} \, dx\right )+3 \int \frac {x^2}{9-3 x-2 x \log (x)+4 e^x x \log (x)} \, dx+3 \int \frac {x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )} \, dx+6 \int \frac {x^3 \log (x)}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx+9 \int \frac {x^3}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx+9 \int \frac {x^2}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx-27 \int \frac {x}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx-27 \int \frac {x^2}{\left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx-27 \int \frac {x}{\log (x) \left (9-3 x-2 x \log (x)+4 e^x x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 x^2}{9-3 x-2 x \log (x)+4 e^x x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-54*x + 3*x^2 + 12*E^x*x^2 + (6*x^2 + E^x*(-12*x^2 + 12*x^3))*Log[x])/(81 - 54*x + 9*x^2 + (-36*x +
12*x^2 + E^x*(72*x - 24*x^2))*Log[x] + (4*x^2 - 16*E^x*x^2 + 16*E^(2*x)*x^2)*Log[x]^2),x]

[Out]

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

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fricas [A]  time = 0.52, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \, x^{2}}{2 \, {\left (2 \, x e^{x} - x\right )} \log \relax (x) - 3 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x^3-12*x^2)*exp(x)+6*x^2)*log(x)+12*exp(x)*x^2+3*x^2-54*x)/((16*exp(x)^2*x^2-16*exp(x)*x^2+4*x
^2)*log(x)^2+((-24*x^2+72*x)*exp(x)+12*x^2-36*x)*log(x)+9*x^2-54*x+81),x, algorithm="fricas")

[Out]

-3*x^2/(2*(2*x*e^x - x)*log(x) - 3*x + 9)

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giac [A]  time = 0.27, size = 24, normalized size = 0.86 \begin {gather*} -\frac {3 \, x^{2}}{4 \, x e^{x} \log \relax (x) - 2 \, x \log \relax (x) - 3 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x^3-12*x^2)*exp(x)+6*x^2)*log(x)+12*exp(x)*x^2+3*x^2-54*x)/((16*exp(x)^2*x^2-16*exp(x)*x^2+4*x
^2)*log(x)^2+((-24*x^2+72*x)*exp(x)+12*x^2-36*x)*log(x)+9*x^2-54*x+81),x, algorithm="giac")

[Out]

-3*x^2/(4*x*e^x*log(x) - 2*x*log(x) - 3*x + 9)

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maple [A]  time = 0.03, size = 25, normalized size = 0.89

method	result	size
risch	\(-\frac {3 x^{2}}{4 x \,{\mathrm e}^{x} \ln \relax (x )-2 x \ln \relax (x )-3 x +9}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((12*x^3-12*x^2)*exp(x)+6*x^2)*ln(x)+12*exp(x)*x^2+3*x^2-54*x)/((16*exp(x)^2*x^2-16*exp(x)*x^2+4*x^2)*ln(
x)^2+((-24*x^2+72*x)*exp(x)+12*x^2-36*x)*ln(x)+9*x^2-54*x+81),x,method=_RETURNVERBOSE)

[Out]

-3*x^2/(4*x*exp(x)*ln(x)-2*x*ln(x)-3*x+9)

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maxima [A]  time = 0.40, size = 24, normalized size = 0.86 \begin {gather*} -\frac {3 \, x^{2}}{4 \, x e^{x} \log \relax (x) - 2 \, x \log \relax (x) - 3 \, x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x^3-12*x^2)*exp(x)+6*x^2)*log(x)+12*exp(x)*x^2+3*x^2-54*x)/((16*exp(x)^2*x^2-16*exp(x)*x^2+4*x
^2)*log(x)^2+((-24*x^2+72*x)*exp(x)+12*x^2-36*x)*log(x)+9*x^2-54*x+81),x, algorithm="maxima")

[Out]

-3*x^2/(4*x*e^x*log(x) - 2*x*log(x) - 3*x + 9)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {54\,x-12\,x^2\,{\mathrm {e}}^x-3\,x^2+\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (12\,x^2-12\,x^3\right )-6\,x^2\right )}{\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (72\,x-24\,x^2\right )-36\,x+12\,x^2\right )-54\,x+{\ln \relax (x)}^2\,\left (16\,x^2\,{\mathrm {e}}^{2\,x}-16\,x^2\,{\mathrm {e}}^x+4\,x^2\right )+9\,x^2+81} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(54*x - 12*x^2*exp(x) - 3*x^2 + log(x)*(exp(x)*(12*x^2 - 12*x^3) - 6*x^2))/(log(x)*(exp(x)*(72*x - 24*x^2
) - 36*x + 12*x^2) - 54*x + log(x)^2*(16*x^2*exp(2*x) - 16*x^2*exp(x) + 4*x^2) + 9*x^2 + 81),x)

[Out]

int(-(54*x - 12*x^2*exp(x) - 3*x^2 + log(x)*(exp(x)*(12*x^2 - 12*x^3) - 6*x^2))/(log(x)*(exp(x)*(72*x - 24*x^2
) - 36*x + 12*x^2) - 54*x + log(x)^2*(16*x^2*exp(2*x) - 16*x^2*exp(x) + 4*x^2) + 9*x^2 + 81), x)

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sympy [A]  time = 0.38, size = 27, normalized size = 0.96 \begin {gather*} - \frac {3 x^{2}}{4 x e^{x} \log {\relax (x )} - 2 x \log {\relax (x )} - 3 x + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((12*x**3-12*x**2)*exp(x)+6*x**2)*ln(x)+12*exp(x)*x**2+3*x**2-54*x)/((16*exp(x)**2*x**2-16*exp(x)*x
**2+4*x**2)*ln(x)**2+((-24*x**2+72*x)*exp(x)+12*x**2-36*x)*ln(x)+9*x**2-54*x+81),x)

[Out]

-3*x**2/(4*x*exp(x)*log(x) - 2*x*log(x) - 3*x + 9)

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