3.61.22 \(\int \frac {-216+216 x-54 x^2+e^x (36-144 x+99 x^2-18 x^3)+(-108+216 x-81 x^2) \log (9)+(108-216 x+81 x^2) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+(9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)) \log (x)+(27 e^x-81 \log (9)) \log ^2(x)+27 \log ^3(x)} \, dx\)

Optimal. Leaf size=31 \[ \frac {\left (2 x-x^2\right )^2}{x \left (\frac {e^x}{3}-\log (9)+\log (x)\right )^2} \]

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Rubi [F]  time = 3.53, 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 {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-216 + 216*x - 54*x^2 + E^x*(36 - 144*x + 99*x^2 - 18*x^3) + (-108 + 216*x - 81*x^2)*Log[9] + (108 - 216*
x + 81*x^2)*Log[x])/(E^(3*x) - 9*E^(2*x)*Log[9] + 27*E^x*Log[9]^2 - 27*Log[9]^3 + (9*E^(2*x) - 54*E^x*Log[9] +
81*Log[9]^2)*Log[x] + (27*E^x - 81*Log[9])*Log[x]^2 + 27*Log[x]^3),x]

[Out]

-216*Defer[Int][(E^x - 3*Log[9] + 3*Log[x])^(-3), x] + 216*Defer[Int][x/(E^x - 3*Log[9] + 3*Log[x])^3, x] - 10
8*Log[81]*Defer[Int][x/(E^x - 3*Log[9] + 3*Log[x])^3, x] - 54*Defer[Int][x^2/(E^x - 3*Log[9] + 3*Log[x])^3, x]
+ 108*Log[81]*Defer[Int][x^2/(E^x - 3*Log[9] + 3*Log[x])^3, x] - 27*Log[81]*Defer[Int][x^3/(E^x - 3*Log[9] +
3*Log[x])^3, x] + 216*Defer[Int][(x*Log[x])/(E^x - 3*Log[9] + 3*Log[x])^3, x] - 216*Defer[Int][(x^2*Log[x])/(E
^x - 3*Log[9] + 3*Log[x])^3, x] + 54*Defer[Int][(x^3*Log[x])/(E^x - 3*Log[9] + 3*Log[x])^3, x] + 36*Defer[Int]
[(E^x - 3*Log[9] + 3*Log[x])^(-2), x] - 144*Defer[Int][x/(E^x - 3*Log[9] + 3*Log[x])^2, x] + 99*Defer[Int][x^2
/(E^x - 3*Log[9] + 3*Log[x])^2, x] - 18*Defer[Int][x^3/(E^x - 3*Log[9] + 3*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 (2-x) \left (e^x \left (2-7 x+2 x^2\right )-12 (1+\log (3))+6 x (1+\log (27))+(6-9 x) \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=9 \int \frac {(2-x) \left (e^x \left (2-7 x+2 x^2\right )-12 (1+\log (3))+6 x (1+\log (27))+(6-9 x) \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=9 \int \left (-\frac {-4+16 x-11 x^2+2 x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}+\frac {3 (2-x) \left (-4+2 x (1-\log (81))+x^2 \log (81)+4 x \log (x)-2 x^2 \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx\\ &=-\left (9 \int \frac {-4+16 x-11 x^2+2 x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \frac {(2-x) \left (-4+2 x (1-\log (81))+x^2 \log (81)+4 x \log (x)-2 x^2 \log (x)\right )}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=-\left (9 \int \left (-\frac {4}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}+\frac {16 x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}-\frac {11 x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}+\frac {2 x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2}\right ) \, dx\right )+27 \int \frac {(2-x)^2 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \left (\frac {4 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {4 x (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {x^2 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \frac {x^2 (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+108 \int \frac {-2-x \log (81)+2 x \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-108 \int \frac {x (-2-x \log (81)+2 x \log (x))}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+27 \int \left (-\frac {2 x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {x^3 \log (81)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {2 x^3 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx+108 \int \left (-\frac {2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {x \log (81)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {2 x \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx-108 \int \left (-\frac {2 x}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}-\frac {x^2 \log (81)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}+\frac {2 x^2 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3}\right ) \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\\ &=-\left (18 \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx\right )+36 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-54 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+54 \int \frac {x^3 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+99 \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-144 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \, dx-216 \int \frac {1}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+216 \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+216 \int \frac {x \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-216 \int \frac {x^2 \log (x)}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-(27 \log (81)) \int \frac {x^3}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx-(108 \log (81)) \int \frac {x}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx+(108 \log (81)) \int \frac {x^2}{\left (e^x-3 \log (9)+3 \log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.79, size = 22, normalized size = 0.71 \begin {gather*} \frac {9 (-2+x)^2 x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-216 + 216*x - 54*x^2 + E^x*(36 - 144*x + 99*x^2 - 18*x^3) + (-108 + 216*x - 81*x^2)*Log[9] + (108
- 216*x + 81*x^2)*Log[x])/(E^(3*x) - 9*E^(2*x)*Log[9] + 27*E^x*Log[9]^2 - 27*Log[9]^3 + (9*E^(2*x) - 54*E^x*Lo
g[9] + 81*Log[9]^2)*Log[x] + (27*E^x - 81*Log[9])*Log[x]^2 + 27*Log[x]^3),x]

[Out]

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

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fricas [A]  time = 0.59, size = 52, normalized size = 1.68 \begin {gather*} -\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{12 \, e^{x} \log \relax (3) - 36 \, \log \relax (3)^{2} - 6 \, {\left (e^{x} - 6 \, \log \relax (3)\right )} \log \relax (x) - 9 \, \log \relax (x)^{2} - e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^2-216*x+108)*log(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(-81*x^2+216*x-108)*log(3)-54*x^2+216*
x-216)/(27*log(x)^3+(27*exp(x)-162*log(3))*log(x)^2+(9*exp(x)^2-108*log(3)*exp(x)+324*log(3)^2)*log(x)+exp(x)^
3-18*log(3)*exp(x)^2+108*log(3)^2*exp(x)-216*log(3)^3),x, algorithm="fricas")

[Out]

-9*(x^3 - 4*x^2 + 4*x)/(12*e^x*log(3) - 36*log(3)^2 - 6*(e^x - 6*log(3))*log(x) - 9*log(x)^2 - e^(2*x))

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giac [A]  time = 0.30, size = 53, normalized size = 1.71 \begin {gather*} -\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{12 \, e^{x} \log \relax (3) - 36 \, \log \relax (3)^{2} - 6 \, e^{x} \log \relax (x) + 36 \, \log \relax (3) \log \relax (x) - 9 \, \log \relax (x)^{2} - e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^2-216*x+108)*log(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(-81*x^2+216*x-108)*log(3)-54*x^2+216*
x-216)/(27*log(x)^3+(27*exp(x)-162*log(3))*log(x)^2+(9*exp(x)^2-108*log(3)*exp(x)+324*log(3)^2)*log(x)+exp(x)^
3-18*log(3)*exp(x)^2+108*log(3)^2*exp(x)-216*log(3)^3),x, algorithm="giac")

[Out]

-9*(x^3 - 4*x^2 + 4*x)/(12*e^x*log(3) - 36*log(3)^2 - 6*e^x*log(x) + 36*log(3)*log(x) - 9*log(x)^2 - e^(2*x))

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maple [A]  time = 0.05, size = 27, normalized size = 0.87

method	result	size
risch	\(\frac {9 x \left (x^{2}-4 x +4\right )}{\left (6 \ln \relax (3)-{\mathrm e}^{x}-3 \ln \relax (x )\right )^{2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((81*x^2-216*x+108)*ln(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(-81*x^2+216*x-108)*ln(3)-54*x^2+216*x-216)/(
27*ln(x)^3+(27*exp(x)-162*ln(3))*ln(x)^2+(9*exp(x)^2-108*ln(3)*exp(x)+324*ln(3)^2)*ln(x)+exp(x)^3-18*ln(3)*exp
(x)^2+108*ln(3)^2*exp(x)-216*ln(3)^3),x,method=_RETURNVERBOSE)

[Out]

9*x*(x^2-4*x+4)/(6*ln(3)-exp(x)-3*ln(x))^2

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maxima [A]  time = 0.51, size = 54, normalized size = 1.74 \begin {gather*} -\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{6 \, {\left (2 \, \log \relax (3) - \log \relax (x)\right )} e^{x} - 36 \, \log \relax (3)^{2} + 36 \, \log \relax (3) \log \relax (x) - 9 \, \log \relax (x)^{2} - e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^2-216*x+108)*log(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(-81*x^2+216*x-108)*log(3)-54*x^2+216*
x-216)/(27*log(x)^3+(27*exp(x)-162*log(3))*log(x)^2+(9*exp(x)^2-108*log(3)*exp(x)+324*log(3)^2)*log(x)+exp(x)^
3-18*log(3)*exp(x)^2+108*log(3)^2*exp(x)-216*log(3)^3),x, algorithm="maxima")

[Out]

-9*(x^3 - 4*x^2 + 4*x)/(6*(2*log(3) - log(x))*e^x - 36*log(3)^2 + 36*log(3)*log(x) - 9*log(x)^2 - e^(2*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,\ln \relax (3)\,\left (81\,x^2-216\,x+108\right )-216\,x-\ln \relax (x)\,\left (81\,x^2-216\,x+108\right )+54\,x^2+{\mathrm {e}}^x\,\left (18\,x^3-99\,x^2+144\,x-36\right )+216}{27\,{\ln \relax (x)}^3+\left (27\,{\mathrm {e}}^x-162\,\ln \relax (3)\right )\,{\ln \relax (x)}^2+\left (9\,{\mathrm {e}}^{2\,x}-108\,{\mathrm {e}}^x\,\ln \relax (3)+324\,{\ln \relax (3)}^2\right )\,\ln \relax (x)+{\mathrm {e}}^{3\,x}-18\,{\mathrm {e}}^{2\,x}\,\ln \relax (3)+108\,{\mathrm {e}}^x\,{\ln \relax (3)}^2-216\,{\ln \relax (3)}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*log(3)*(81*x^2 - 216*x + 108) - 216*x - log(x)*(81*x^2 - 216*x + 108) + 54*x^2 + exp(x)*(144*x - 99*x^
2 + 18*x^3 - 36) + 216)/(exp(3*x) - 18*exp(2*x)*log(3) + 108*exp(x)*log(3)^2 - log(x)^2*(162*log(3) - 27*exp(x
)) + log(x)*(9*exp(2*x) - 108*exp(x)*log(3) + 324*log(3)^2) + 27*log(x)^3 - 216*log(3)^3),x)

[Out]

int(-(2*log(3)*(81*x^2 - 216*x + 108) - 216*x - log(x)*(81*x^2 - 216*x + 108) + 54*x^2 + exp(x)*(144*x - 99*x^
2 + 18*x^3 - 36) + 216)/(exp(3*x) - 18*exp(2*x)*log(3) + 108*exp(x)*log(3)^2 - log(x)^2*(162*log(3) - 27*exp(x
)) + log(x)*(9*exp(2*x) - 108*exp(x)*log(3) + 324*log(3)^2) + 27*log(x)^3 - 216*log(3)^3), x)

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sympy [B]  time = 0.34, size = 53, normalized size = 1.71 \begin {gather*} \frac {9 x^{3} - 36 x^{2} + 36 x}{\left (6 \log {\relax (x )} - 12 \log {\relax (3 )}\right ) e^{x} + e^{2 x} + 9 \log {\relax (x )}^{2} - 36 \log {\relax (3 )} \log {\relax (x )} + 36 \log {\relax (3 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x**2-216*x+108)*ln(x)+(-18*x**3+99*x**2-144*x+36)*exp(x)+2*(-81*x**2+216*x-108)*ln(3)-54*x**2+2
16*x-216)/(27*ln(x)**3+(27*exp(x)-162*ln(3))*ln(x)**2+(9*exp(x)**2-108*ln(3)*exp(x)+324*ln(3)**2)*ln(x)+exp(x)
**3-18*ln(3)*exp(x)**2+108*ln(3)**2*exp(x)-216*ln(3)**3),x)

[Out]

(9*x**3 - 36*x**2 + 36*x)/((6*log(x) - 12*log(3))*exp(x) + exp(2*x) + 9*log(x)**2 - 36*log(3)*log(x) + 36*log(
3)**2)

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