3.86.4 \(\int \frac {80 x-320 x^2+360 x^3-160 x^4+25 x^5+(4-4 x) \log (e^{-x} x)}{x} \, dx\)

Optimal. Leaf size=24 \[ e^5+5 (-2+x)^4 x+2 \log ^2\left (e^{-x} x\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 34, 43, 6686} \begin {gather*} 5 x (2-x)^4+2 \log ^2\left (e^{-x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(80*x - 320*x^2 + 360*x^3 - 160*x^4 + 25*x^5 + (4 - 4*x)*Log[x/E^x])/x,x]

[Out]

5*(2 - x)^4*x + 2*Log[x/E^x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 34

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[(d*x*(a + b*x)^(m + 1))/(b*(m + 2)), x] /
; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5 (-2+x)^3 (-2+5 x)-\frac {4 (-1+x) \log \left (e^{-x} x\right )}{x}\right ) \, dx\\ &=-\left (4 \int \frac {(-1+x) \log \left (e^{-x} x\right )}{x} \, dx\right )+5 \int (-2+x)^3 (-2+5 x) \, dx\\ &=5 (2-x)^4 x+2 \log ^2\left (e^{-x} x\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 71, normalized size = 2.96 \begin {gather*} 76 x-158 x^2+120 x^3-40 x^4+5 x^5-4 x \log (x)+2 \log ^2(x)-4 x \left (-1+x-\log (x)+\log \left (e^{-x} x\right )\right )+4 \log (x) \left (x-\log (x)+\log \left (e^{-x} x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(80*x - 320*x^2 + 360*x^3 - 160*x^4 + 25*x^5 + (4 - 4*x)*Log[x/E^x])/x,x]

[Out]

76*x - 158*x^2 + 120*x^3 - 40*x^4 + 5*x^5 - 4*x*Log[x] + 2*Log[x]^2 - 4*x*(-1 + x - Log[x] + Log[x/E^x]) + 4*L
og[x]*(x - Log[x] + Log[x/E^x])

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fricas [A]  time = 0.56, size = 35, normalized size = 1.46 \begin {gather*} 5 \, x^{5} - 40 \, x^{4} + 120 \, x^{3} - 160 \, x^{2} + 2 \, \log \left (x e^{\left (-x\right )}\right )^{2} + 80 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*log(x/exp(x))+25*x^5-160*x^4+360*x^3-320*x^2+80*x)/x,x, algorithm="fricas")

[Out]

5*x^5 - 40*x^4 + 120*x^3 - 160*x^2 + 2*log(x*e^(-x))^2 + 80*x

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giac [A]  time = 0.14, size = 35, normalized size = 1.46 \begin {gather*} 5 \, x^{5} - 40 \, x^{4} + 120 \, x^{3} - 158 \, x^{2} - 4 \, x \log \relax (x) + 2 \, \log \relax (x)^{2} + 80 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*log(x/exp(x))+25*x^5-160*x^4+360*x^3-320*x^2+80*x)/x,x, algorithm="giac")

[Out]

5*x^5 - 40*x^4 + 120*x^3 - 158*x^2 - 4*x*log(x) + 2*log(x)^2 + 80*x

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maple [B]  time = 0.11, size = 57, normalized size = 2.38

method	result	size
default	\(5 x^{5}-40 x^{4}+120 x^{3}-162 x^{2}+80 x +4 \ln \left (x \,{\mathrm e}^{-x}\right ) \ln \relax (x )-4 x \ln \left (x \,{\mathrm e}^{-x}\right )+4 x \ln \relax (x )-2 \ln \relax (x )^{2}\)	\(57\)
risch	\(\left (4 x -4 \ln \relax (x )\right ) \ln \left ({\mathrm e}^{x}\right )+2 \ln \relax (x )^{2}-2 i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}+2 i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+2 i x \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}-2 i x \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )+5 x^{5}-40 x^{4}+120 x^{3}-162 x^{2}+80 x +2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )-2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}+2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right )\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x+4)*ln(x/exp(x))+25*x^5-160*x^4+360*x^3-320*x^2+80*x)/x,x,method=_RETURNVERBOSE)

[Out]

5*x^5-40*x^4+120*x^3-162*x^2+80*x+4*ln(x/exp(x))*ln(x)-4*x*ln(x/exp(x))+4*x*ln(x)-2*ln(x)^2

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maxima [B]  time = 0.37, size = 61, normalized size = 2.54 \begin {gather*} 5 \, x^{5} - 40 \, x^{4} + 120 \, x^{3} - 162 \, x^{2} - 4 \, x \log \left (x e^{\left (-x\right )}\right ) + 4 \, {\left (x - \log \relax (x)\right )} \log \relax (x) + 4 \, \log \left (x e^{\left (-x\right )}\right ) \log \relax (x) + 2 \, \log \relax (x)^{2} + 80 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*log(x/exp(x))+25*x^5-160*x^4+360*x^3-320*x^2+80*x)/x,x, algorithm="maxima")

[Out]

5*x^5 - 40*x^4 + 120*x^3 - 162*x^2 - 4*x*log(x*e^(-x)) + 4*(x - log(x))*log(x) + 4*log(x*e^(-x))*log(x) + 2*lo
g(x)^2 + 80*x

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mupad [B]  time = 5.53, size = 35, normalized size = 1.46 \begin {gather*} 5\,x^5-40\,x^4+120\,x^3-158\,x^2-4\,x\,\ln \relax (x)+80\,x+2\,{\ln \relax (x)}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((80*x - log(x*exp(-x))*(4*x - 4) - 320*x^2 + 360*x^3 - 160*x^4 + 25*x^5)/x,x)

[Out]

80*x + 2*log(x)^2 - 4*x*log(x) - 158*x^2 + 120*x^3 - 40*x^4 + 5*x^5

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sympy [A]  time = 0.16, size = 32, normalized size = 1.33 \begin {gather*} 5 x^{5} - 40 x^{4} + 120 x^{3} - 160 x^{2} + 80 x + 2 \log {\left (x e^{- x} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x+4)*ln(x/exp(x))+25*x**5-160*x**4+360*x**3-320*x**2+80*x)/x,x)

[Out]

5*x**5 - 40*x**4 + 120*x**3 - 160*x**2 + 80*x + 2*log(x*exp(-x))**2

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