∫ e−x4 (8ee2e−x4 _________ log (5) +2e−x4 _________log(5) x3−ex4 log(5))__________________________

Optimal. Leaf size=28 \[ \frac {1}{2} \left (-1-e^{e^{\frac {2 e^{-x^4}}{\log (5)}}}-x\right ) \]

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Rubi [A] time = 0.53, antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 5, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {12, 6742, 6715, 2282, 2194} \begin {gather*} -\frac {1}{2} e^{e^{\frac {2 e^{-x^4}}{\log (5)}}}-\frac {x}{2} \end {gather*}

Int[(8*E^(E^(2/(E^x^4*Log[5])) + 2/(E^x^4*Log[5]))*x^3 - E^x^4*Log[5])/(2*E^x^4*Log[5]),x]

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

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu

nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F

reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /

; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int e^{-x^4} \left (8 e^{e^{\frac {2 e^{-x^4}}{\log (5)}}+\frac {2 e^{-x^4}}{\log (5)}} x^3-e^{x^4} \log (5)\right ) \, dx}{2 \log (5)}\\ &=\frac {\int \left (8 e^{e^{\frac {2 e^{-x^4}}{\log (5)}}-x^4+\frac {2 e^{-x^4}}{\log (5)}} x^3-\log (5)\right ) \, dx}{2 \log (5)}\\ &=-\frac {x}{2}+\frac {4 \int e^{e^{\frac {2 e^{-x^4}}{\log (5)}}-x^4+\frac {2 e^{-x^4}}{\log (5)}} x^3 \, dx}{\log (5)}\\ &=-\frac {x}{2}+\frac {\operatorname {Subst}\left (\int e^{e^{\frac {2 e^{-x}}{\log (5)}}-x+\frac {2 e^{-x}}{\log (5)}} \, dx,x,x^4\right )}{\log (5)}\\ &=-\frac {x}{2}-\frac {\operatorname {Subst}\left (\int e^{e^{\frac {2 x}{\log (5)}}+\frac {2 x}{\log (5)}} \, dx,x,e^{-x^4}\right )}{\log (5)}\\ &=-\frac {x}{2}-\frac {1}{2} \operatorname {Subst}\left (\int e^x \, dx,x,e^{\frac {2 e^{-x^4}}{\log (5)}}\right )\\ &=-\frac {1}{2} e^{e^{\frac {2 e^{-x^4}}{\log (5)}}}-\frac {x}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.14, size = 32, normalized size = 1.14 \begin {gather*} \frac {-e^{e^{\frac {2 e^{-x^4}}{\log (5)}}} \log (5)-x \log (5)}{\log (25)} \end {gather*}

Integrate[(8*E^(E^(2/(E^x^4*Log[5])) + 2/(E^x^4*Log[5]))*x^3 - E^x^4*Log[5])/(2*E^x^4*Log[5]),x]

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fricas [B] time = 0.66, size = 65, normalized size = 2.32 \begin {gather*} -\frac {1}{2} \, {\left (x e^{\left (\frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )} + e^{\left (\frac {{\left (e^{\left (x^{4} + \frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )} \log \relax (5) + 2\right )} e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )}\right )} e^{\left (-\frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )} \end {gather*}

integrate(1/2*(8*x^3*exp(2/log(5)/exp(x^4))*exp(exp(2/log(5)/exp(x^4)))-log(5)*exp(x^4))/log(5)/exp(x^4),x, al

-1/2*(x*e^(2*e^(-x^4)/log(5)) + e^((e^(x^4 + 2*e^(-x^4)/log(5))*log(5) + 2)*e^(-x^4)/log(5)))*e^(-2*e^(-x^4)/l

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

integrate(1/2*(8*x^3*exp(2/log(5)/exp(x^4))*exp(exp(2/log(5)/exp(x^4)))-log(5)*exp(x^4))/log(5)/exp(x^4),x, al

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method	result	size

risch	\(-\frac {x}{2}-\frac {{\mathrm e}^{{\mathrm e}^{\frac {2 \,{\mathrm e}^{-x^{4}}}{\ln \relax (5)}}}}{2}\)	\(21\)
default	\(\frac {-{\mathrm e}^{{\mathrm e}^{\frac {2 \,{\mathrm e}^{-x^{4}}}{\ln \relax (5)}}} \ln \relax (5)-x \ln \relax (5)}{2 \ln \relax (5)}\)	\(32\)

int(1/2*(8*x^3*exp(2/ln(5)/exp(x^4))*exp(exp(2/ln(5)/exp(x^4)))-ln(5)*exp(x^4))/ln(5)/exp(x^4),x,method=_RETUR

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maxima [A] time = 0.46, size = 28, normalized size = 1.00 \begin {gather*} -\frac {x \log \relax (5) + e^{\left (e^{\left (\frac {2 \, e^{\left (-x^{4}\right )}}{\log \relax (5)}\right )}\right )} \log \relax (5)}{2 \, \log \relax (5)} \end {gather*}

integrate(1/2*(8*x^3*exp(2/log(5)/exp(x^4))*exp(exp(2/log(5)/exp(x^4)))-log(5)*exp(x^4))/log(5)/exp(x^4),x, al

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mupad [B] time = 5.98, size = 20, normalized size = 0.71 \begin {gather*} -\frac {x}{2}-\frac {{\mathrm {e}}^{{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-x^4}}{\ln \relax (5)}}}}{2} \end {gather*}

int(-(exp(-x^4)*((exp(x^4)*log(5))/2 - 4*x^3*exp((2*exp(-x^4))/log(5))*exp(exp((2*exp(-x^4))/log(5)))))/log(5)

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sympy [A] time = 0.49, size = 19, normalized size = 0.68 \begin {gather*} - \frac {x}{2} - \frac {e^{e^{\frac {2 e^{- x^{4}}}{\log {\relax (5 )}}}}}{2} \end {gather*}

integrate(1/2*(8*x**3*exp(2/ln(5)/exp(x**4))*exp(exp(2/ln(5)/exp(x**4)))-ln(5)*exp(x**4))/ln(5)/exp(x**4),x)

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3.82.1 \(\int \frac {e^{-x^4} (8 e^{e^{\frac {2 e^{-x^4}}{\log (5)}}+\frac {2 e^{-x^4}}{\log (5)}} x^3-e^{x^4} \log (5))}{2 \log (5)} \, dx\)