3.15.13 \(\int \frac {(-72+120 x-56 x^2+8 x^3) \log (\frac {e^x}{x})+(36-48 x+12 x^2) \log ^2(\frac {e^x}{x})}{e^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {4 (3-x)^2 x \log ^2\left (\frac {e^x}{x}\right )}{e^2} \]

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Rubi [F]  time = 0.20, 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 {\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )}{e^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-72 + 120*x - 56*x^2 + 8*x^3)*Log[E^x/x] + (36 - 48*x + 12*x^2)*Log[E^x/x]^2)/E^2,x]

[Out]

(-72*x)/E^2 + (66*x^2)/E^2 - (236*x^3)/(9*E^2) + (31*x^4)/(6*E^2) - (2*x^5)/(5*E^2) - (72*x*Log[E^x/x])/E^2 +
(60*x^2*Log[E^x/x])/E^2 - (56*x^3*Log[E^x/x])/(3*E^2) + (2*x^4*Log[E^x/x])/E^2 + (36*Defer[Int][Log[E^x/x]^2,
x])/E^2 - (48*Defer[Int][x*Log[E^x/x]^2, x])/E^2 + (12*Defer[Int][x^2*Log[E^x/x]^2, x])/E^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (\left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right )+\left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right )\right ) \, dx}{e^2}\\ &=\frac {\int \left (-72+120 x-56 x^2+8 x^3\right ) \log \left (\frac {e^x}{x}\right ) \, dx}{e^2}+\frac {\int \left (36-48 x+12 x^2\right ) \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}\\ &=-\frac {72 x \log \left (\frac {e^x}{x}\right )}{e^2}+\frac {60 x^2 \log \left (\frac {e^x}{x}\right )}{e^2}-\frac {56 x^3 \log \left (\frac {e^x}{x}\right )}{3 e^2}+\frac {2 x^4 \log \left (\frac {e^x}{x}\right )}{e^2}-\frac {\int \frac {2}{3} (1-x) \left (108-90 x+28 x^2-3 x^3\right ) \, dx}{e^2}+\frac {\int \left (36 \log ^2\left (\frac {e^x}{x}\right )-48 x \log ^2\left (\frac {e^x}{x}\right )+12 x^2 \log ^2\left (\frac {e^x}{x}\right )\right ) \, dx}{e^2}\\ &=-\frac {72 x \log \left (\frac {e^x}{x}\right )}{e^2}+\frac {60 x^2 \log \left (\frac {e^x}{x}\right )}{e^2}-\frac {56 x^3 \log \left (\frac {e^x}{x}\right )}{3 e^2}+\frac {2 x^4 \log \left (\frac {e^x}{x}\right )}{e^2}-\frac {2 \int (1-x) \left (108-90 x+28 x^2-3 x^3\right ) \, dx}{3 e^2}+\frac {12 \int x^2 \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}+\frac {36 \int \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}-\frac {48 \int x \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}\\ &=-\frac {72 x \log \left (\frac {e^x}{x}\right )}{e^2}+\frac {60 x^2 \log \left (\frac {e^x}{x}\right )}{e^2}-\frac {56 x^3 \log \left (\frac {e^x}{x}\right )}{3 e^2}+\frac {2 x^4 \log \left (\frac {e^x}{x}\right )}{e^2}-\frac {2 \int \left (108-198 x+118 x^2-31 x^3+3 x^4\right ) \, dx}{3 e^2}+\frac {12 \int x^2 \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}+\frac {36 \int \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}-\frac {48 \int x \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}\\ &=-\frac {72 x}{e^2}+\frac {66 x^2}{e^2}-\frac {236 x^3}{9 e^2}+\frac {31 x^4}{6 e^2}-\frac {2 x^5}{5 e^2}-\frac {72 x \log \left (\frac {e^x}{x}\right )}{e^2}+\frac {60 x^2 \log \left (\frac {e^x}{x}\right )}{e^2}-\frac {56 x^3 \log \left (\frac {e^x}{x}\right )}{3 e^2}+\frac {2 x^4 \log \left (\frac {e^x}{x}\right )}{e^2}+\frac {12 \int x^2 \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}+\frac {36 \int \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}-\frac {48 \int x \log ^2\left (\frac {e^x}{x}\right ) \, dx}{e^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 21, normalized size = 0.91 \begin {gather*} \frac {4 (-3+x)^2 x \log ^2\left (\frac {e^x}{x}\right )}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-72 + 120*x - 56*x^2 + 8*x^3)*Log[E^x/x] + (36 - 48*x + 12*x^2)*Log[E^x/x]^2)/E^2,x]

[Out]

(4*(-3 + x)^2*x*Log[E^x/x]^2)/E^2

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fricas [A]  time = 0.68, size = 25, normalized size = 1.09 \begin {gather*} 4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{\left (-2\right )} \log \left (\frac {e^{x}}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-48*x+36)*log(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*log(exp(x)/x))/exp(1)^2,x, algorithm="fric
as")

[Out]

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

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giac [B]  time = 0.38, size = 63, normalized size = 2.74 \begin {gather*} 4 \, {\left (x^{5} - 2 \, x^{4} \log \relax (x) + x^{3} \log \relax (x)^{2} - 6 \, x^{4} + 12 \, x^{3} \log \relax (x) - 6 \, x^{2} \log \relax (x)^{2} + 9 \, x^{3} - 18 \, x^{2} \log \relax (x) + 9 \, x \log \relax (x)^{2}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-48*x+36)*log(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*log(exp(x)/x))/exp(1)^2,x, algorithm="giac
")

[Out]

4*(x^5 - 2*x^4*log(x) + x^3*log(x)^2 - 6*x^4 + 12*x^3*log(x) - 6*x^2*log(x)^2 + 9*x^3 - 18*x^2*log(x) + 9*x*lo
g(x)^2)*e^(-2)

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maple [B]  time = 0.16, size = 310, normalized size = 13.48

method	result	size
default	\({\mathrm e}^{-2} \left (-12 x^{2} \ln \relax (x )-24 x^{2} \ln \relax (x )^{2}-\frac {16 x^{4}}{3}-24 x^{3}+72 x^{2}+2 x^{5}+\frac {88 x^{3} \ln \relax (x )}{3}-6 x^{4} \ln \relax (x )+36 x \ln \relax (x )^{2}-72 x \ln \relax (x )+60 x^{2} \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+4 x^{3} \ln \relax (x )^{2}-72 x \ln \left (\frac {{\mathrm e}^{x}}{x}\right )+2 \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{4}+6 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right ) x^{4}+4 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right )^{2} x^{3}-\frac {88 x^{3} \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right )}{3}+12 x^{2} \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right )+72 x \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right )-24 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right )^{2} x^{2}+36 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right )^{2} x -\frac {56 \ln \left (\frac {{\mathrm e}^{x}}{x}\right ) x^{3}}{3}-8 \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right ) x^{3} \ln \relax (x )+48 \ln \relax (x ) \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right ) x^{2}-72 \ln \relax (x ) \left (\ln \left (\frac {{\mathrm e}^{x}}{x}\right )+\ln \relax (x )-x \right ) x \right )\)	\(310\)
risch	\(-9 \,{\mathrm e}^{-2} \pi ^{2} x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{6}-{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{6}+6 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{6}+6 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}-12 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-12 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}+24 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \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 )^{4}-9 \,{\mathrm e}^{-2} \pi ^{2} x \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+18 \,{\mathrm e}^{-2} \pi ^{2} x \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}+18 \,{\mathrm e}^{-2} \pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-36 \,{\mathrm e}^{-2} \pi ^{2} 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 )^{4}-{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+2 \,{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}+2 \,{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-4 \,{\mathrm e}^{-2} \pi ^{2} x^{3} \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 )^{4}-36 i {\mathrm e}^{-2} \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \ln \relax (x )+24 i {\mathrm e}^{-2} \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \ln \relax (x )-36 i {\mathrm e}^{-2} \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \ln \relax (x )-4 i {\mathrm e}^{-2} \pi \,x^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \ln \relax (x )-4 i {\mathrm e}^{-2} \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \ln \relax (x )+24 i {\mathrm e}^{-2} \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2} \ln \relax (x )-{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{4}+2 \,{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{5}+6 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{4}-12 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{5}+6 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{4}-12 \,{\mathrm e}^{-2} \pi ^{2} x^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{5}-9 \,{\mathrm e}^{-2} \pi ^{2} x \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{4}+18 \,{\mathrm e}^{-2} \pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{5}-9 \,{\mathrm e}^{-2} \pi ^{2} x \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{4}+18 \,{\mathrm e}^{-2} \pi ^{2} x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{5}+36 i {\mathrm e}^{-2} \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3} \ln \relax (x )-24 i {\mathrm e}^{-2} \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3} \ln \relax (x )+4 i {\mathrm e}^{-2} \pi \,x^{3} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3} \ln \relax (x )-{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{4}+2 \,{\mathrm e}^{-2} \pi ^{2} x^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{5}+4 i {\mathrm e}^{-2} \pi \,x^{3} \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 ) \ln \relax (x )-24 i {\mathrm e}^{-2} \pi \,x^{2} \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 ) \ln \relax (x )+36 i {\mathrm e}^{-2} \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 ) \ln \relax (x )+{\mathrm e}^{-2} \left (-4 i \pi \,x^{3} \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 )+4 i \pi \,x^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+4 i \pi \,x^{3} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}-4 i \pi \,x^{3} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}+24 i \pi \,x^{2} \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 )-24 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}-24 i \pi \,x^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+24 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-36 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 )+36 i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}+36 i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{2}-36 i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )^{3}-8 x^{3} \ln \relax (x )+48 x^{2} \ln \relax (x )-72 x \ln \relax (x )\right ) \ln \left ({\mathrm e}^{x}\right )+4 \,{\mathrm e}^{-2} x^{3} \ln \relax (x )^{2}+{\mathrm e}^{-2} \left (4 x^{3}-24 x^{2}+36 x \right ) \ln \left ({\mathrm e}^{x}\right )^{2}-24 \,{\mathrm e}^{-2} x^{2} \ln \relax (x )^{2}+36 \,{\mathrm e}^{-2} x \ln \relax (x )^{2}\)	\(1508\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((12*x^2-48*x+36)*ln(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*ln(exp(x)/x))/exp(1)^2,x,method=_RETURNVERBOSE)

[Out]

1/exp(1)^2*(-12*x^2*ln(x)-24*x^2*ln(x)^2-16/3*x^4-24*x^3+72*x^2+2*x^5+88/3*x^3*ln(x)-6*x^4*ln(x)+36*x*ln(x)^2-
72*x*ln(x)+60*x^2*ln(exp(x)/x)+4*x^3*ln(x)^2-72*x*ln(exp(x)/x)+2*ln(exp(x)/x)*x^4+6*(ln(exp(x)/x)+ln(x)-x)*x^4
+4*(ln(exp(x)/x)+ln(x)-x)^2*x^3-88/3*x^3*(ln(exp(x)/x)+ln(x)-x)+12*x^2*(ln(exp(x)/x)+ln(x)-x)+72*x*(ln(exp(x)/
x)+ln(x)-x)-24*(ln(exp(x)/x)+ln(x)-x)^2*x^2+36*(ln(exp(x)/x)+ln(x)-x)^2*x-56/3*ln(exp(x)/x)*x^3-8*(ln(exp(x)/x
)+ln(x)-x)*x^3*ln(x)+48*ln(x)*(ln(exp(x)/x)+ln(x)-x)*x^2-72*ln(x)*(ln(exp(x)/x)+ln(x)-x)*x)

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maxima [A]  time = 0.37, size = 25, normalized size = 1.09 \begin {gather*} 4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{\left (-2\right )} \log \left (\frac {e^{x}}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x^2-48*x+36)*log(exp(x)/x)^2+(8*x^3-56*x^2+120*x-72)*log(exp(x)/x))/exp(1)^2,x, algorithm="maxi
ma")

[Out]

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

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mupad [B]  time = 1.20, size = 19, normalized size = 0.83 \begin {gather*} 4\,x\,{\mathrm {e}}^{-2}\,{\ln \left (\frac {{\mathrm {e}}^x}{x}\right )}^2\,{\left (x-3\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2)*(log(exp(x)/x)*(120*x - 56*x^2 + 8*x^3 - 72) + log(exp(x)/x)^2*(12*x^2 - 48*x + 36)),x)

[Out]

4*x*exp(-2)*log(exp(x)/x)^2*(x - 3)^2

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sympy [A]  time = 0.21, size = 24, normalized size = 1.04 \begin {gather*} \frac {\left (4 x^{3} - 24 x^{2} + 36 x\right ) \log {\left (\frac {e^{x}}{x} \right )}^{2}}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((12*x**2-48*x+36)*ln(exp(x)/x)**2+(8*x**3-56*x**2+120*x-72)*ln(exp(x)/x))/exp(1)**2,x)

[Out]

(4*x**3 - 24*x**2 + 36*x)*exp(-2)*log(exp(x)/x)**2

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