3.9.72 \(\int \frac {36 x^4-36 x^5+(12 x^2-48 x^3+126 x^4) \log (\frac {e^x}{x})+(36 x^2-144 x^3) \log ^2(\frac {e^x}{x})+(2-24 x+54 x^2) \log ^3(\frac {e^x}{x})}{81 \log ^3(\frac {e^x}{x})} \, dx\)

Optimal. Leaf size=27 \[ \frac {2}{9} x \left (-\frac {1}{3}+x-\frac {x^2}{\log \left (\frac {e^x}{x}\right )}\right )^2 \]

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Rubi [F]  time = 0.60, 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 {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{81 \log ^3\left (\frac {e^x}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(36*x^4 - 36*x^5 + (12*x^2 - 48*x^3 + 126*x^4)*Log[E^x/x] + (36*x^2 - 144*x^3)*Log[E^x/x]^2 + (2 - 24*x +
54*x^2)*Log[E^x/x]^3)/(81*Log[E^x/x]^3),x]

[Out]

(2*x)/81 - (4*x^2)/27 + (2*x^3)/9 + (4*Defer[Int][x^4/Log[E^x/x]^3, x])/9 - (4*Defer[Int][x^5/Log[E^x/x]^3, x]
)/9 + (4*Defer[Int][x^2/Log[E^x/x]^2, x])/27 - (16*Defer[Int][x^3/Log[E^x/x]^2, x])/27 + (14*Defer[Int][x^4/Lo
g[E^x/x]^2, x])/9 + (4*Defer[Int][x^2/Log[E^x/x], x])/9 - (16*Defer[Int][x^3/Log[E^x/x], x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \frac {36 x^4-36 x^5+\left (12 x^2-48 x^3+126 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (36 x^2-144 x^3\right ) \log ^2\left (\frac {e^x}{x}\right )+\left (2-24 x+54 x^2\right ) \log ^3\left (\frac {e^x}{x}\right )}{\log ^3\left (\frac {e^x}{x}\right )} \, dx\\ &=\frac {1}{81} \int \left (2 \left (1-12 x+27 x^2\right )-\frac {36 (-1+x) x^4}{\log ^3\left (\frac {e^x}{x}\right )}+\frac {6 x^2 \left (2-8 x+21 x^2\right )}{\log ^2\left (\frac {e^x}{x}\right )}-\frac {36 x^2 (-1+4 x)}{\log \left (\frac {e^x}{x}\right )}\right ) \, dx\\ &=\frac {2}{81} \int \left (1-12 x+27 x^2\right ) \, dx+\frac {2}{27} \int \frac {x^2 \left (2-8 x+21 x^2\right )}{\log ^2\left (\frac {e^x}{x}\right )} \, dx-\frac {4}{9} \int \frac {(-1+x) x^4}{\log ^3\left (\frac {e^x}{x}\right )} \, dx-\frac {4}{9} \int \frac {x^2 (-1+4 x)}{\log \left (\frac {e^x}{x}\right )} \, dx\\ &=\frac {2 x}{81}-\frac {4 x^2}{27}+\frac {2 x^3}{9}+\frac {2}{27} \int \left (\frac {2 x^2}{\log ^2\left (\frac {e^x}{x}\right )}-\frac {8 x^3}{\log ^2\left (\frac {e^x}{x}\right )}+\frac {21 x^4}{\log ^2\left (\frac {e^x}{x}\right )}\right ) \, dx-\frac {4}{9} \int \left (-\frac {x^4}{\log ^3\left (\frac {e^x}{x}\right )}+\frac {x^5}{\log ^3\left (\frac {e^x}{x}\right )}\right ) \, dx-\frac {4}{9} \int \left (-\frac {x^2}{\log \left (\frac {e^x}{x}\right )}+\frac {4 x^3}{\log \left (\frac {e^x}{x}\right )}\right ) \, dx\\ &=\frac {2 x}{81}-\frac {4 x^2}{27}+\frac {2 x^3}{9}+\frac {4}{27} \int \frac {x^2}{\log ^2\left (\frac {e^x}{x}\right )} \, dx+\frac {4}{9} \int \frac {x^4}{\log ^3\left (\frac {e^x}{x}\right )} \, dx-\frac {4}{9} \int \frac {x^5}{\log ^3\left (\frac {e^x}{x}\right )} \, dx+\frac {4}{9} \int \frac {x^2}{\log \left (\frac {e^x}{x}\right )} \, dx-\frac {16}{27} \int \frac {x^3}{\log ^2\left (\frac {e^x}{x}\right )} \, dx+\frac {14}{9} \int \frac {x^4}{\log ^2\left (\frac {e^x}{x}\right )} \, dx-\frac {16}{9} \int \frac {x^3}{\log \left (\frac {e^x}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(36*x^4 - 36*x^5 + (12*x^2 - 48*x^3 + 126*x^4)*Log[E^x/x] + (36*x^2 - 144*x^3)*Log[E^x/x]^2 + (2 - 2
4*x + 54*x^2)*Log[E^x/x]^3)/(81*Log[E^x/x]^3),x]

[Out]

$Aborted

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fricas [B]  time = 0.75, size = 59, normalized size = 2.19 \begin {gather*} \frac {2 \, {\left (9 \, x^{5} + {\left (9 \, x^{3} - 6 \, x^{2} + x\right )} \log \left (\frac {e^{x}}{x}\right )^{2} - 6 \, {\left (3 \, x^{4} - x^{3}\right )} \log \left (\frac {e^{x}}{x}\right )\right )}}{81 \, \log \left (\frac {e^{x}}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((54*x^2-24*x+2)*log(exp(x)/x)^3+(-144*x^3+36*x^2)*log(exp(x)/x)^2+(126*x^4-48*x^3+12*x^2)*log(
exp(x)/x)-36*x^5+36*x^4)/log(exp(x)/x)^3,x, algorithm="fricas")

[Out]

2/81*(9*x^5 + (9*x^3 - 6*x^2 + x)*log(e^x/x)^2 - 6*(3*x^4 - x^3)*log(e^x/x))/log(e^x/x)^2

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giac [B]  time = 0.33, size = 56, normalized size = 2.07 \begin {gather*} \frac {2}{9} \, x^{3} - \frac {4}{27} \, x^{2} + \frac {2}{81} \, x - \frac {2 \, {\left (3 \, x^{5} - 6 \, x^{4} \log \relax (x) - 2 \, x^{4} + 2 \, x^{3} \log \relax (x)\right )}}{27 \, {\left (x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((54*x^2-24*x+2)*log(exp(x)/x)^3+(-144*x^3+36*x^2)*log(exp(x)/x)^2+(126*x^4-48*x^3+12*x^2)*log(
exp(x)/x)-36*x^5+36*x^4)/log(exp(x)/x)^3,x, algorithm="giac")

[Out]

2/9*x^3 - 4/27*x^2 + 2/81*x - 2/27*(3*x^5 - 6*x^4*log(x) - 2*x^4 + 2*x^3*log(x))/(x^2 - 2*x*log(x) + log(x)^2)

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maple [C]  time = 0.20, size = 314, normalized size = 11.63

method	result	size
risch	\(\frac {2 x^{3}}{9}-\frac {4 x^{2}}{27}+\frac {2 x}{81}+\frac {8 x^{3} \left (3 i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )-3 i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}-3 i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+3 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}+3 x^{2}+6 x \ln \relax (x )-6 \ln \left ({\mathrm e}^{x}\right ) x -2 \ln \relax (x )+2 \ln \left ({\mathrm e}^{x}\right )\right )}{27 \left (2 \ln \relax (x )-2 \ln \left ({\mathrm e}^{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}\right )^{2}}\)	\(314\)
default	\(\frac {\left (18 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{2}-12 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+12 x -12 \ln \relax (x )+2\right ) x^{3}+\left (12 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{2}-4 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+4 x -4 \ln \relax (x )\right ) \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{2}+\left (12 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{2}-4 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+4 x -4 \ln \relax (x )\right ) \ln \relax (x )^{2}+\left (24 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{3}-6 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{2}\right ) x +\left (12-36 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+36 x -36 \ln \relax (x )\right ) x^{3} \ln \relax (x )+\left (24 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )-24 x +24 \ln \relax (x )-4\right ) x^{2} \ln \relax (x )+\left (-24 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{2}+4 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )-4 x +4 \ln \relax (x )\right ) x \ln \relax (x )+18 x^{3} \ln \relax (x )^{2}+2 x \ln \relax (x )^{2}-12 x^{2} \ln \relax (x )^{2}-2 \left (12 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right )^{2}-4 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+4 x -4 \ln \relax (x )\right ) \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )-x +\ln \relax (x )\right ) \ln \relax (x )}{81 \ln \left (\frac {{\mathrm e}^{x}}{x}\right )^{2}}\)	\(340\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/81*((54*x^2-24*x+2)*ln(exp(x)/x)^3+(-144*x^3+36*x^2)*ln(exp(x)/x)^2+(126*x^4-48*x^3+12*x^2)*ln(exp(x)/x)
-36*x^5+36*x^4)/ln(exp(x)/x)^3,x,method=_RETURNVERBOSE)

[Out]

2/9*x^3-4/27*x^2+2/81*x+8/27*x^3*(3*I*Pi*x*csgn(I/x)*csgn(I*exp(x))*csgn(I/x*exp(x))-3*I*Pi*x*csgn(I/x)*csgn(I
/x*exp(x))^2-3*I*Pi*x*csgn(I*exp(x))*csgn(I/x*exp(x))^2+3*I*Pi*x*csgn(I/x*exp(x))^3-I*Pi*csgn(I/x)*csgn(I*exp(
x))*csgn(I/x*exp(x))+I*Pi*csgn(I/x)*csgn(I/x*exp(x))^2+I*Pi*csgn(I*exp(x))*csgn(I/x*exp(x))^2-I*Pi*csgn(I/x*ex
p(x))^3+3*x^2+6*x*ln(x)-6*ln(exp(x))*x-2*ln(x)+2*ln(exp(x)))/(2*ln(x)-2*ln(exp(x))+I*Pi*csgn(I/x)*csgn(I*exp(x
))*csgn(I/x*exp(x))-I*Pi*csgn(I/x)*csgn(I/x*exp(x))^2-I*Pi*csgn(I*exp(x))*csgn(I/x*exp(x))^2+I*Pi*csgn(I/x*exp
(x))^3)^2

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maxima [B]  time = 0.70, size = 57, normalized size = 2.11 \begin {gather*} \frac {2}{9} \, x^{3} - \frac {4}{27} \, x^{2} + \frac {2}{81} \, x - \frac {2 \, {\left (3 \, x^{5} - 2 \, x^{4} - 2 \, {\left (3 \, x^{4} - x^{3}\right )} \log \relax (x)\right )}}{27 \, {\left (x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((54*x^2-24*x+2)*log(exp(x)/x)^3+(-144*x^3+36*x^2)*log(exp(x)/x)^2+(126*x^4-48*x^3+12*x^2)*log(
exp(x)/x)-36*x^5+36*x^4)/log(exp(x)/x)^3,x, algorithm="maxima")

[Out]

2/9*x^3 - 4/27*x^2 + 2/81*x - 2/27*(3*x^5 - 2*x^4 - 2*(3*x^4 - x^3)*log(x))/(x^2 - 2*x*log(x) + log(x)^2)

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mupad [B]  time = 0.87, size = 303, normalized size = 11.22 \begin {gather*} \frac {245\,x}{81}+\frac {2\,\ln \relax (x)}{3}+\frac {\frac {x\,\left (9\,x^4+8\,x^3-2\,x^2\right )}{27\,\left (x-1\right )}+\frac {x^3\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (-12\,x^3+7\,x^2+12\,x-2\right )}{9\,{\left (x-1\right )}^3}+\frac {2\,x^3\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,\left (4\,x^2-6\,x+1\right )}{3\,{\left (x-1\right )}^3}}{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}+\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (\frac {24\,x^5-92\,x^4+128\,x^3-\frac {230\,x^2}{3}+18\,x-\frac {2}{3}}{x^3-3\,x^2+3\,x-1}-\frac {\frac {80\,x^5}{3}-96\,x^4+128\,x^3-\frac {224\,x^2}{3}+16\,x}{x^3-3\,x^2+3\,x-1}\right )+\frac {\frac {38\,x^2}{9}-9\,x+\frac {38}{9}}{x^3-3\,x^2+3\,x-1}+\frac {\frac {2\,x^5}{9}-\frac {x^3\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (21\,x^2-8\,x+2\right )}{27\,\left (x-1\right )}+\frac {2\,x^3\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,\left (4\,x-1\right )}{9\,\left (x-1\right )}}{{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2}+\frac {65\,x^2}{27}+\frac {2\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(exp(x)/x)^2*(36*x^2 - 144*x^3))/81 + (log(exp(x)/x)^3*(54*x^2 - 24*x + 2))/81 + (4*x^4)/9 - (4*x^5)/
9 + (log(exp(x)/x)*(12*x^2 - 48*x^3 + 126*x^4))/81)/log(exp(x)/x)^3,x)

[Out]

(245*x)/81 + (2*log(x))/3 + ((x*(8*x^3 - 2*x^2 + 9*x^4))/(27*(x - 1)) + (x^3*log(exp(x)/x)*(12*x + 7*x^2 - 12*
x^3 - 2))/(9*(x - 1)^3) + (2*x^3*log(exp(x)/x)^2*(4*x^2 - 6*x + 1))/(3*(x - 1)^3))/log(exp(x)/x) + log(exp(x)/
x)*((18*x - (230*x^2)/3 + 128*x^3 - 92*x^4 + 24*x^5 - 2/3)/(3*x - 3*x^2 + x^3 - 1) - (16*x - (224*x^2)/3 + 128
*x^3 - 96*x^4 + (80*x^5)/3)/(3*x - 3*x^2 + x^3 - 1)) + ((38*x^2)/9 - 9*x + 38/9)/(3*x - 3*x^2 + x^3 - 1) + ((2
*x^5)/9 - (x^3*log(exp(x)/x)*(21*x^2 - 8*x + 2))/(27*(x - 1)) + (2*x^3*log(exp(x)/x)^2*(4*x - 1))/(9*(x - 1)))
/log(exp(x)/x)^2 + (65*x^2)/27 + (2*x^3)/3

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sympy [B]  time = 0.15, size = 49, normalized size = 1.81 \begin {gather*} \frac {2 x^{3}}{9} - \frac {4 x^{2}}{27} + \frac {2 x}{81} + \frac {6 x^{5} + \left (- 12 x^{4} + 4 x^{3}\right ) \log {\left (\frac {e^{x}}{x} \right )}}{27 \log {\left (\frac {e^{x}}{x} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/81*((54*x**2-24*x+2)*ln(exp(x)/x)**3+(-144*x**3+36*x**2)*ln(exp(x)/x)**2+(126*x**4-48*x**3+12*x**2
)*ln(exp(x)/x)-36*x**5+36*x**4)/ln(exp(x)/x)**3,x)

[Out]

2*x**3/9 - 4*x**2/27 + 2*x/81 + (6*x**5 + (-12*x**4 + 4*x**3)*log(exp(x)/x))/(27*log(exp(x)/x)**2)

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