∫ 1_ 625(40e10x + e−ex+11x(44 − 4ex) + e−2ex+12x(12 − 2ex)) dx

Optimal. Leaf size=22 \[ \frac {1}{625} e^{10 x} \left (2+e^{-e^x+x}\right )^2 \]

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Rubi [A] time = 0.77, antiderivative size = 40, normalized size of antiderivative = 1.82, number of steps used = 59, number of rules used = 5, integrand size = 50, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.100, Rules used = {12, 2194, 2282, 2196, 2176} \begin {gather*} \frac {4 e^{10 x}}{625}+\frac {4}{625} e^{11 x-e^x}+\frac {1}{625} e^{12 x-2 e^x} \end {gather*}

Int[(40*E^(10*x) + E^(-E^x + 11*x)*(44 - 4*E^x) + E^(-2*E^x + 12*x)*(12 - 2*E^x))/625,x]

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

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m

*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x

)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] && !$UseGamma === True

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

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c

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

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \left (40 e^{10 x}+e^{-e^x+11 x} \left (44-4 e^x\right )+e^{-2 e^x+12 x} \left (12-2 e^x\right )\right ) \, dx\\ &=\frac {1}{625} \int e^{-e^x+11 x} \left (44-4 e^x\right ) \, dx+\frac {1}{625} \int e^{-2 e^x+12 x} \left (12-2 e^x\right ) \, dx+\frac {8}{125} \int e^{10 x} \, dx\\ &=\frac {4 e^{10 x}}{625}+\frac {1}{625} \operatorname {Subst}\left (\int 4 e^{-x} (11-x) x^{10} \, dx,x,e^x\right )+\frac {1}{625} \operatorname {Subst}\left (\int 2 e^{-2 x} (6-x) x^{11} \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}+\frac {2}{625} \operatorname {Subst}\left (\int e^{-2 x} (6-x) x^{11} \, dx,x,e^x\right )+\frac {4}{625} \operatorname {Subst}\left (\int e^{-x} (11-x) x^{10} \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}+\frac {2}{625} \operatorname {Subst}\left (\int \left (6 e^{-2 x} x^{11}-e^{-2 x} x^{12}\right ) \, dx,x,e^x\right )+\frac {4}{625} \operatorname {Subst}\left (\int \left (11 e^{-x} x^{10}-e^{-x} x^{11}\right ) \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {2}{625} \operatorname {Subst}\left (\int e^{-2 x} x^{12} \, dx,x,e^x\right )-\frac {4}{625} \operatorname {Subst}\left (\int e^{-x} x^{11} \, dx,x,e^x\right )+\frac {12}{625} \operatorname {Subst}\left (\int e^{-2 x} x^{11} \, dx,x,e^x\right )+\frac {44}{625} \operatorname {Subst}\left (\int e^{-x} x^{10} \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {44}{625} e^{-e^x+10 x}-\frac {6}{625} e^{-2 e^x+11 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {12}{625} \operatorname {Subst}\left (\int e^{-2 x} x^{11} \, dx,x,e^x\right )-\frac {44}{625} \operatorname {Subst}\left (\int e^{-x} x^{10} \, dx,x,e^x\right )+\frac {66}{625} \operatorname {Subst}\left (\int e^{-2 x} x^{10} \, dx,x,e^x\right )+\frac {88}{125} \operatorname {Subst}\left (\int e^{-x} x^9 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {88}{125} e^{-e^x+9 x}-\frac {33}{625} e^{-2 e^x+10 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {66}{625} \operatorname {Subst}\left (\int e^{-2 x} x^{10} \, dx,x,e^x\right )+\frac {66}{125} \operatorname {Subst}\left (\int e^{-2 x} x^9 \, dx,x,e^x\right )-\frac {88}{125} \operatorname {Subst}\left (\int e^{-x} x^9 \, dx,x,e^x\right )+\frac {792}{125} \operatorname {Subst}\left (\int e^{-x} x^8 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {792}{125} e^{-e^x+8 x}-\frac {33}{125} e^{-2 e^x+9 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {66}{125} \operatorname {Subst}\left (\int e^{-2 x} x^9 \, dx,x,e^x\right )+\frac {297}{125} \operatorname {Subst}\left (\int e^{-2 x} x^8 \, dx,x,e^x\right )-\frac {792}{125} \operatorname {Subst}\left (\int e^{-x} x^8 \, dx,x,e^x\right )+\frac {6336}{125} \operatorname {Subst}\left (\int e^{-x} x^7 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {6336}{125} e^{-e^x+7 x}-\frac {297}{250} e^{-2 e^x+8 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {297}{125} \operatorname {Subst}\left (\int e^{-2 x} x^8 \, dx,x,e^x\right )+\frac {1188}{125} \operatorname {Subst}\left (\int e^{-2 x} x^7 \, dx,x,e^x\right )-\frac {6336}{125} \operatorname {Subst}\left (\int e^{-x} x^7 \, dx,x,e^x\right )+\frac {44352}{125} \operatorname {Subst}\left (\int e^{-x} x^6 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {44352}{125} e^{-e^x+6 x}-\frac {594}{125} e^{-2 e^x+7 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {1188}{125} \operatorname {Subst}\left (\int e^{-2 x} x^7 \, dx,x,e^x\right )+\frac {4158}{125} \operatorname {Subst}\left (\int e^{-2 x} x^6 \, dx,x,e^x\right )-\frac {44352}{125} \operatorname {Subst}\left (\int e^{-x} x^6 \, dx,x,e^x\right )+\frac {266112}{125} \operatorname {Subst}\left (\int e^{-x} x^5 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {266112}{125} e^{-e^x+5 x}-\frac {2079}{125} e^{-2 e^x+6 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {4158}{125} \operatorname {Subst}\left (\int e^{-2 x} x^6 \, dx,x,e^x\right )+\frac {12474}{125} \operatorname {Subst}\left (\int e^{-2 x} x^5 \, dx,x,e^x\right )-\frac {266112}{125} \operatorname {Subst}\left (\int e^{-x} x^5 \, dx,x,e^x\right )+\frac {266112}{25} \operatorname {Subst}\left (\int e^{-x} x^4 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {266112}{25} e^{-e^x+4 x}-\frac {6237}{125} e^{-2 e^x+5 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {12474}{125} \operatorname {Subst}\left (\int e^{-2 x} x^5 \, dx,x,e^x\right )+\frac {6237}{25} \operatorname {Subst}\left (\int e^{-2 x} x^4 \, dx,x,e^x\right )-\frac {266112}{25} \operatorname {Subst}\left (\int e^{-x} x^4 \, dx,x,e^x\right )+\frac {1064448}{25} \operatorname {Subst}\left (\int e^{-x} x^3 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {1064448}{25} e^{-e^x+3 x}-\frac {6237}{50} e^{-2 e^x+4 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {6237}{25} \operatorname {Subst}\left (\int e^{-2 x} x^4 \, dx,x,e^x\right )+\frac {12474}{25} \operatorname {Subst}\left (\int e^{-2 x} x^3 \, dx,x,e^x\right )-\frac {1064448}{25} \operatorname {Subst}\left (\int e^{-x} x^3 \, dx,x,e^x\right )+\frac {3193344}{25} \operatorname {Subst}\left (\int e^{-x} x^2 \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {3193344}{25} e^{-e^x+2 x}-\frac {6237}{25} e^{-2 e^x+3 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {12474}{25} \operatorname {Subst}\left (\int e^{-2 x} x^3 \, dx,x,e^x\right )+\frac {18711}{25} \operatorname {Subst}\left (\int e^{-2 x} x^2 \, dx,x,e^x\right )-\frac {3193344}{25} \operatorname {Subst}\left (\int e^{-x} x^2 \, dx,x,e^x\right )+\frac {6386688}{25} \operatorname {Subst}\left (\int e^{-x} x \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}-\frac {6386688}{25} e^{-e^x+x}-\frac {18711}{50} e^{-2 e^x+2 x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}+\frac {18711}{25} \operatorname {Subst}\left (\int e^{-2 x} x \, dx,x,e^x\right )-\frac {18711}{25} \operatorname {Subst}\left (\int e^{-2 x} x^2 \, dx,x,e^x\right )+\frac {6386688}{25} \operatorname {Subst}\left (\int e^{-x} \, dx,x,e^x\right )-\frac {6386688}{25} \operatorname {Subst}\left (\int e^{-x} x \, dx,x,e^x\right )\\ &=-\frac {6386688 e^{-e^x}}{25}+\frac {4 e^{10 x}}{625}-\frac {18711}{50} e^{-2 e^x+x}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}+\frac {18711}{50} \operatorname {Subst}\left (\int e^{-2 x} \, dx,x,e^x\right )-\frac {18711}{25} \operatorname {Subst}\left (\int e^{-2 x} x \, dx,x,e^x\right )-\frac {6386688}{25} \operatorname {Subst}\left (\int e^{-x} \, dx,x,e^x\right )\\ &=-\frac {18711}{100} e^{-2 e^x}+\frac {4 e^{10 x}}{625}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}-\frac {18711}{50} \operatorname {Subst}\left (\int e^{-2 x} \, dx,x,e^x\right )\\ &=\frac {4 e^{10 x}}{625}+\frac {4}{625} e^{-e^x+11 x}+\frac {1}{625} e^{-2 e^x+12 x}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.12, size = 28, normalized size = 1.27 \begin {gather*} \frac {1}{625} e^{-2 e^x+10 x} \left (2 e^{e^x}+e^x\right )^2 \end {gather*}

Integrate[(40*E^(10*x) + E^(-E^x + 11*x)*(44 - 4*E^x) + E^(-2*E^x + 12*x)*(12 - 2*E^x))/625,x]

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fricas [A] time = 0.66, size = 33, normalized size = 1.50 \begin {gather*} \frac {1}{625} \, {\left (4 \, e^{\left (20 \, x\right )} + e^{\left (22 \, x - 2 \, e^{x}\right )} + 4 \, e^{\left (21 \, x - e^{x}\right )}\right )} e^{\left (-10 \, x\right )} \end {gather*}

integrate(1/625*(-2*exp(x)+12)*exp(5*x)^2*exp(x-exp(x))^2+1/625*(-4*exp(x)+44)*exp(5*x)^2*exp(x-exp(x))+8/125*

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giac [A] time = 0.25, size = 29, normalized size = 1.32 \begin {gather*} \frac {4}{625} \, e^{\left (10 \, x\right )} + \frac {1}{625} \, e^{\left (12 \, x - 2 \, e^{x}\right )} + \frac {4}{625} \, e^{\left (11 \, x - e^{x}\right )} \end {gather*}

integrate(1/625*(-2*exp(x)+12)*exp(5*x)^2*exp(x-exp(x))^2+1/625*(-4*exp(x)+44)*exp(5*x)^2*exp(x-exp(x))+8/125*

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method	result	size

risch	$\frac {{\mathrm e}^{12 x -2 \,{\mathrm e}^{x}}}{625}+\frac {4 \,{\mathrm e}^{11 x -{\mathrm e}^{x}}}{625}+\frac {4 \,{\mathrm e}^{10 x}}{625}$	$30$
default	$\frac {4 \,{\mathrm e}^{10 x}}{625}+\frac {4 \,{\mathrm e}^{11 x} {\mathrm e}^{-{\mathrm e}^{x}}}{625}+\frac {{\mathrm e}^{12 x} {\mathrm e}^{-2 \,{\mathrm e}^{x}}}{625}$	$32$

int(1/625*(-2*exp(x)+12)*exp(5*x)^2*exp(x-exp(x))^2+1/625*(-4*exp(x)+44)*exp(5*x)^2*exp(x-exp(x))+8/125*exp(5*

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maxima [A] time = 0.38, size = 29, normalized size = 1.32 \begin {gather*} \frac {4}{625} \, e^{\left (10 \, x\right )} + \frac {1}{625} \, e^{\left (12 \, x - 2 \, e^{x}\right )} + \frac {4}{625} \, e^{\left (11 \, x - e^{x}\right )} \end {gather*}

integrate(1/625*(-2*exp(x)+12)*exp(5*x)^2*exp(x-exp(x))^2+1/625*(-4*exp(x)+44)*exp(5*x)^2*exp(x-exp(x))+8/125*

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mupad [B] time = 5.25, size = 29, normalized size = 1.32 \begin {gather*} \frac {4\,{\mathrm {e}}^{10\,x}}{625}+\frac {4\,{\mathrm {e}}^{11\,x-{\mathrm {e}}^x}}{625}+\frac {{\mathrm {e}}^{12\,x-2\,{\mathrm {e}}^x}}{625} \end {gather*}

int((8*exp(10*x))/125 - (exp(10*x)*exp(2*x - 2*exp(x))*(2*exp(x) - 12))/625 - (exp(10*x)*exp(x - exp(x))*(4*ex

(4*exp(10*x))/625 + (4*exp(11*x - exp(x)))/625 + exp(12*x - 2*exp(x))/625

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sympy [B] time = 0.29, size = 39, normalized size = 1.77 \begin {gather*} \frac {4 e^{10 x} e^{x - e^{x}}}{625} + \frac {e^{10 x} e^{2 x - 2 e^{x}}}{625} + \frac {4 e^{10 x}}{625} \end {gather*}

integrate(1/625*(-2*exp(x)+12)*exp(5*x)**2*exp(x-exp(x))**2+1/625*(-4*exp(x)+44)*exp(5*x)**2*exp(x-exp(x))+8/1

4*exp(10*x)*exp(x - exp(x))/625 + exp(10*x)*exp(2*x - 2*exp(x))/625 + 4*exp(10*x)/625

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3.89.49 \(\int \frac {1}{625} (40 e^{10 x}+e^{-e^x+11 x} (44-4 e^x)+e^{-2 e^x+12 x} (12-2 e^x)) \, dx\)