3.78.51 \(\int \frac {e^{-4 x} (e^{4 e^4} (256-1024 x)+e^{4 e^4} (192-1024 x) \log (x)+e^{4 e^4} (48-384 x) \log ^2(x)+e^{4 e^4} (4-64 x) \log ^3(x)-4 e^{4 e^4} x \log ^4(x))}{x} \, dx\)

Optimal. Leaf size=18 \[ e^{4 e^4-4 x} (4+\log (x))^4 \]

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Rubi [A]  time = 0.52, antiderivative size = 29, normalized size of antiderivative = 1.61, number of steps used = 3, number of rules used = 3, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {6688, 12, 2288} \begin {gather*} \frac {e^{4 e^4-4 x} (\log (x)+4)^3 (4 x+x \log (x))}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4*E^4)*(256 - 1024*x) + E^(4*E^4)*(192 - 1024*x)*Log[x] + E^(4*E^4)*(48 - 384*x)*Log[x]^2 + E^(4*E^4)*
(4 - 64*x)*Log[x]^3 - 4*E^(4*E^4)*x*Log[x]^4)/(E^(4*x)*x),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} e^{4 e^4-4 x} (4+\log (x))^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*E^4)*(256 - 1024*x) + E^(4*E^4)*(192 - 1024*x)*Log[x] + E^(4*E^4)*(48 - 384*x)*Log[x]^2 + E^(4
*E^4)*(4 - 64*x)*Log[x]^3 - 4*E^(4*E^4)*x*Log[x]^4)/(E^(4*x)*x),x]

[Out]

E^(4*E^4 - 4*x)*(4 + Log[x])^4

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fricas [B]  time = 0.65, size = 69, normalized size = 3.83 \begin {gather*} e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x)^{4} + 16 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x)^{3} + 96 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x)^{2} + 256 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x) + 256 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(exp(4))^4*log(x)^4+(-64*x+4)*exp(exp(4))^4*log(x)^3+(-384*x+48)*exp(exp(4))^4*log(x)^2+(-1
024*x+192)*exp(exp(4))^4*log(x)+(-1024*x+256)*exp(exp(4))^4)/x/exp(2*x)^2,x, algorithm="fricas")

[Out]

e^(-4*x + 4*e^4)*log(x)^4 + 16*e^(-4*x + 4*e^4)*log(x)^3 + 96*e^(-4*x + 4*e^4)*log(x)^2 + 256*e^(-4*x + 4*e^4)
*log(x) + 256*e^(-4*x + 4*e^4)

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giac [B]  time = 0.24, size = 69, normalized size = 3.83 \begin {gather*} e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x)^{4} + 16 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x)^{3} + 96 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x)^{2} + 256 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x) + 256 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(exp(4))^4*log(x)^4+(-64*x+4)*exp(exp(4))^4*log(x)^3+(-384*x+48)*exp(exp(4))^4*log(x)^2+(-1
024*x+192)*exp(exp(4))^4*log(x)+(-1024*x+256)*exp(exp(4))^4)/x/exp(2*x)^2,x, algorithm="giac")

[Out]

e^(-4*x + 4*e^4)*log(x)^4 + 16*e^(-4*x + 4*e^4)*log(x)^3 + 96*e^(-4*x + 4*e^4)*log(x)^2 + 256*e^(-4*x + 4*e^4)
*log(x) + 256*e^(-4*x + 4*e^4)

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maple [B]  time = 0.04, size = 70, normalized size = 3.89

method	result	size
risch	\(\ln \relax (x )^{4} {\mathrm e}^{4 \,{\mathrm e}^{4}-4 x}+16 \ln \relax (x )^{3} {\mathrm e}^{4 \,{\mathrm e}^{4}-4 x}+96 \ln \relax (x )^{2} {\mathrm e}^{4 \,{\mathrm e}^{4}-4 x}+256 \ln \relax (x ) {\mathrm e}^{4 \,{\mathrm e}^{4}-4 x}+256 \,{\mathrm e}^{4 \,{\mathrm e}^{4}-4 x}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*exp(exp(4))^4*ln(x)^4+(-64*x+4)*exp(exp(4))^4*ln(x)^3+(-384*x+48)*exp(exp(4))^4*ln(x)^2+(-1024*x+192
)*exp(exp(4))^4*ln(x)+(-1024*x+256)*exp(exp(4))^4)/x/exp(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

ln(x)^4*exp(4*exp(4)-4*x)+16*ln(x)^3*exp(4*exp(4)-4*x)+96*ln(x)^2*exp(4*exp(4)-4*x)+256*ln(x)*exp(4*exp(4)-4*x
)+256*exp(4*exp(4)-4*x)

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maxima [B]  time = 0.45, size = 63, normalized size = 3.50 \begin {gather*} {\left (e^{\left (4 \, e^{4}\right )} \log \relax (x)^{4} + 16 \, e^{\left (4 \, e^{4}\right )} \log \relax (x)^{3} + 96 \, e^{\left (4 \, e^{4}\right )} \log \relax (x)^{2}\right )} e^{\left (-4 \, x\right )} + 256 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \log \relax (x) + 256 \, e^{\left (-4 \, x + 4 \, e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(exp(4))^4*log(x)^4+(-64*x+4)*exp(exp(4))^4*log(x)^3+(-384*x+48)*exp(exp(4))^4*log(x)^2+(-1
024*x+192)*exp(exp(4))^4*log(x)+(-1024*x+256)*exp(exp(4))^4)/x/exp(2*x)^2,x, algorithm="maxima")

[Out]

(e^(4*e^4)*log(x)^4 + 16*e^(4*e^4)*log(x)^3 + 96*e^(4*e^4)*log(x)^2)*e^(-4*x) + 256*e^(-4*x + 4*e^4)*log(x) +
256*e^(-4*x + 4*e^4)

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mupad [B]  time = 6.06, size = 69, normalized size = 3.83 \begin {gather*} {\mathrm {e}}^{4\,{\mathrm {e}}^4-4\,x}\,{\ln \relax (x)}^4+16\,{\mathrm {e}}^{4\,{\mathrm {e}}^4-4\,x}\,{\ln \relax (x)}^3+96\,{\mathrm {e}}^{4\,{\mathrm {e}}^4-4\,x}\,{\ln \relax (x)}^2+256\,{\mathrm {e}}^{4\,{\mathrm {e}}^4-4\,x}\,\ln \relax (x)+256\,{\mathrm {e}}^{4\,{\mathrm {e}}^4-4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4*x)*(exp(4*exp(4))*(1024*x - 256) + exp(4*exp(4))*log(x)^3*(64*x - 4) + exp(4*exp(4))*log(x)^2*(38
4*x - 48) + 4*x*exp(4*exp(4))*log(x)^4 + exp(4*exp(4))*log(x)*(1024*x - 192)))/x,x)

[Out]

256*exp(4*exp(4) - 4*x) + 96*exp(4*exp(4) - 4*x)*log(x)^2 + 16*exp(4*exp(4) - 4*x)*log(x)^3 + exp(4*exp(4) - 4
*x)*log(x)^4 + 256*exp(4*exp(4) - 4*x)*log(x)

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sympy [B]  time = 0.63, size = 63, normalized size = 3.50 \begin {gather*} \left (e^{4 e^{4}} \log {\relax (x )}^{4} + 16 e^{4 e^{4}} \log {\relax (x )}^{3} + 96 e^{4 e^{4}} \log {\relax (x )}^{2} + 256 e^{4 e^{4}} \log {\relax (x )} + 256 e^{4 e^{4}}\right ) e^{- 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*exp(exp(4))**4*ln(x)**4+(-64*x+4)*exp(exp(4))**4*ln(x)**3+(-384*x+48)*exp(exp(4))**4*ln(x)**2+
(-1024*x+192)*exp(exp(4))**4*ln(x)+(-1024*x+256)*exp(exp(4))**4)/x/exp(2*x)**2,x)

[Out]

(exp(4*exp(4))*log(x)**4 + 16*exp(4*exp(4))*log(x)**3 + 96*exp(4*exp(4))*log(x)**2 + 256*exp(4*exp(4))*log(x)
+ 256*exp(4*exp(4)))*exp(-4*x)

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