3.11.55 \(\int e^{16-6 e^x+20 x} (-20+6 e^x) \, dx\) [1055]

Optimal. Leaf size=28 \[ 1-e^{-3 \left (2 e^x-4 x\right )-x^2+(4+x)^2} \]

[Out]

1-exp((4+x)^2-6*exp(x)+12*x-x^2)

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Rubi [A]
time = 0.62, antiderivative size = 14, normalized size of antiderivative = 0.50, number of steps used = 44, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2320, 2227, 2207, 2225} \begin {gather*} -e^{20 x-6 e^x+16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(16 - 6*E^x + 20*x)*(-20 + 6*E^x),x]

[Out]

-E^(16 - 6*E^x + 20*x)

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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

Rule 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 2320

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
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int e^{16-6 x} x^{19} (-20+6 x) \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (-20 e^{16-6 x} x^{19}+6 e^{16-6 x} x^{20}\right ) \, dx,x,e^x\right )\\ &=6 \text {Subst}\left (\int e^{16-6 x} x^{20} \, dx,x,e^x\right )-20 \text {Subst}\left (\int e^{16-6 x} x^{19} \, dx,x,e^x\right )\\ &=\frac {10}{3} e^{16-6 e^x+19 x}-e^{16-6 e^x+20 x}+20 \text {Subst}\left (\int e^{16-6 x} x^{19} \, dx,x,e^x\right )-\frac {190}{3} \text {Subst}\left (\int e^{16-6 x} x^{18} \, dx,x,e^x\right )\\ &=\frac {95}{9} e^{16-6 e^x+18 x}-e^{16-6 e^x+20 x}+\frac {190}{3} \text {Subst}\left (\int e^{16-6 x} x^{18} \, dx,x,e^x\right )-190 \text {Subst}\left (\int e^{16-6 x} x^{17} \, dx,x,e^x\right )\\ &=\frac {95}{3} e^{16-6 e^x+17 x}-e^{16-6 e^x+20 x}+190 \text {Subst}\left (\int e^{16-6 x} x^{17} \, dx,x,e^x\right )-\frac {1615}{3} \text {Subst}\left (\int e^{16-6 x} x^{16} \, dx,x,e^x\right )\\ &=\frac {1615}{18} e^{16-6 e^x+16 x}-e^{16-6 e^x+20 x}+\frac {1615}{3} \text {Subst}\left (\int e^{16-6 x} x^{16} \, dx,x,e^x\right )-\frac {12920}{9} \text {Subst}\left (\int e^{16-6 x} x^{15} \, dx,x,e^x\right )\\ &=\frac {6460}{27} e^{16-6 e^x+15 x}-e^{16-6 e^x+20 x}+\frac {12920}{9} \text {Subst}\left (\int e^{16-6 x} x^{15} \, dx,x,e^x\right )-\frac {32300}{9} \text {Subst}\left (\int e^{16-6 x} x^{14} \, dx,x,e^x\right )\\ &=\frac {16150}{27} e^{16-6 e^x+14 x}-e^{16-6 e^x+20 x}+\frac {32300}{9} \text {Subst}\left (\int e^{16-6 x} x^{14} \, dx,x,e^x\right )-\frac {226100}{27} \text {Subst}\left (\int e^{16-6 x} x^{13} \, dx,x,e^x\right )\\ &=\frac {113050}{81} e^{16-6 e^x+13 x}-e^{16-6 e^x+20 x}+\frac {226100}{27} \text {Subst}\left (\int e^{16-6 x} x^{13} \, dx,x,e^x\right )-\frac {1469650}{81} \text {Subst}\left (\int e^{16-6 x} x^{12} \, dx,x,e^x\right )\\ &=\frac {734825}{243} e^{16-6 e^x+12 x}-e^{16-6 e^x+20 x}+\frac {1469650}{81} \text {Subst}\left (\int e^{16-6 x} x^{12} \, dx,x,e^x\right )-\frac {2939300}{81} \text {Subst}\left (\int e^{16-6 x} x^{11} \, dx,x,e^x\right )\\ &=\frac {1469650}{243} e^{16-6 e^x+11 x}-e^{16-6 e^x+20 x}+\frac {2939300}{81} \text {Subst}\left (\int e^{16-6 x} x^{11} \, dx,x,e^x\right )-\frac {16166150}{243} \text {Subst}\left (\int e^{16-6 x} x^{10} \, dx,x,e^x\right )\\ &=\frac {8083075}{729} e^{16-6 e^x+10 x}-e^{16-6 e^x+20 x}+\frac {16166150}{243} \text {Subst}\left (\int e^{16-6 x} x^{10} \, dx,x,e^x\right )-\frac {80830750}{729} \text {Subst}\left (\int e^{16-6 x} x^9 \, dx,x,e^x\right )\\ &=\frac {40415375 e^{16-6 e^x+9 x}}{2187}-e^{16-6 e^x+20 x}+\frac {80830750}{729} \text {Subst}\left (\int e^{16-6 x} x^9 \, dx,x,e^x\right )-\frac {40415375}{243} \text {Subst}\left (\int e^{16-6 x} x^8 \, dx,x,e^x\right )\\ &=\frac {40415375 e^{16-6 e^x+8 x}}{1458}-e^{16-6 e^x+20 x}+\frac {40415375}{243} \text {Subst}\left (\int e^{16-6 x} x^8 \, dx,x,e^x\right )-\frac {161661500}{729} \text {Subst}\left (\int e^{16-6 x} x^7 \, dx,x,e^x\right )\\ &=\frac {80830750 e^{16-6 e^x+7 x}}{2187}-e^{16-6 e^x+20 x}+\frac {161661500}{729} \text {Subst}\left (\int e^{16-6 x} x^7 \, dx,x,e^x\right )-\frac {565815250 \text {Subst}\left (\int e^{16-6 x} x^6 \, dx,x,e^x\right )}{2187}\\ &=\frac {282907625 e^{16-6 e^x+6 x}}{6561}-e^{16-6 e^x+20 x}-\frac {565815250 \text {Subst}\left (\int e^{16-6 x} x^5 \, dx,x,e^x\right )}{2187}+\frac {565815250 \text {Subst}\left (\int e^{16-6 x} x^6 \, dx,x,e^x\right )}{2187}\\ &=\frac {282907625 e^{16-6 e^x+5 x}}{6561}-e^{16-6 e^x+20 x}-\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} x^4 \, dx,x,e^x\right )}{6561}+\frac {565815250 \text {Subst}\left (\int e^{16-6 x} x^5 \, dx,x,e^x\right )}{2187}\\ &=\frac {1414538125 e^{16-6 e^x+4 x}}{39366}-e^{16-6 e^x+20 x}-\frac {2829076250 \text {Subst}\left (\int e^{16-6 x} x^3 \, dx,x,e^x\right )}{19683}+\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} x^4 \, dx,x,e^x\right )}{6561}\\ &=\frac {1414538125 e^{16-6 e^x+3 x}}{59049}-e^{16-6 e^x+20 x}-\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} x^2 \, dx,x,e^x\right )}{19683}+\frac {2829076250 \text {Subst}\left (\int e^{16-6 x} x^3 \, dx,x,e^x\right )}{19683}\\ &=\frac {1414538125 e^{16-6 e^x+2 x}}{118098}-e^{16-6 e^x+20 x}-\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} x \, dx,x,e^x\right )}{59049}+\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} x^2 \, dx,x,e^x\right )}{19683}\\ &=\frac {1414538125 e^{16-6 e^x+x}}{354294}-e^{16-6 e^x+20 x}-\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} \, dx,x,e^x\right )}{354294}+\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} x \, dx,x,e^x\right )}{59049}\\ &=\frac {1414538125 e^{16-6 e^x}}{2125764}-e^{16-6 e^x+20 x}+\frac {1414538125 \text {Subst}\left (\int e^{16-6 x} \, dx,x,e^x\right )}{354294}\\ &=-e^{16-6 e^x+20 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 14, normalized size = 0.50 \begin {gather*} -e^{16-6 e^x+20 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(16 - 6*E^x + 20*x)*(-20 + 6*E^x),x]

[Out]

-E^(16 - 6*E^x + 20*x)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(450\) vs. \(2(23)=46\).
time = 0.05, size = 451, normalized size = 16.11

method	result	size
norman	\(-{\mathrm e}^{-6 \,{\mathrm e}^{x}+20 x +16}\)	\(13\)
risch	\(-{\mathrm e}^{-6 \,{\mathrm e}^{x}+20 x +16}\)	\(13\)
default	\(-20 \,{\mathrm e}^{16} \left (-\frac {{\mathrm e}^{-6 \,{\mathrm e}^{x}} {\mathrm e}^{19 x}}{6}-\frac {19 \,{\mathrm e}^{18 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{36}-\frac {19 \,{\mathrm e}^{17 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{12}-\frac {323 \,{\mathrm e}^{16 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{72}-\frac {323 \,{\mathrm e}^{15 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{27}-\frac {1615 \,{\mathrm e}^{14 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{54}-\frac {11305 \,{\mathrm e}^{13 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{162}-\frac {146965 \,{\mathrm e}^{12 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{972}-\frac {146965 \,{\mathrm e}^{11 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{486}-\frac {1616615 \,{\mathrm e}^{10 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{2916}-\frac {8083075 \,{\mathrm e}^{9 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{8748}-\frac {8083075 \,{\mathrm e}^{8 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{5832}-\frac {8083075 \,{\mathrm e}^{7 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{4374}-\frac {56581525 \,{\mathrm e}^{6 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{26244}-\frac {56581525 \,{\mathrm e}^{5 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{26244}-\frac {282907625 \,{\mathrm e}^{4 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{157464}-\frac {282907625 \,{\mathrm e}^{3 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{236196}-\frac {282907625 \,{\mathrm e}^{2 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{472392}-\frac {282907625 \,{\mathrm e}^{x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{1417176}-\frac {282907625 \,{\mathrm e}^{-6 \,{\mathrm e}^{x}}}{8503056}\right )+6 \,{\mathrm e}^{16} \left (-\frac {{\mathrm e}^{-6 \,{\mathrm e}^{x}} {\mathrm e}^{20 x}}{6}-\frac {5 \,{\mathrm e}^{-6 \,{\mathrm e}^{x}} {\mathrm e}^{19 x}}{9}-\frac {95 \,{\mathrm e}^{18 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{54}-\frac {95 \,{\mathrm e}^{17 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{18}-\frac {1615 \,{\mathrm e}^{16 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{108}-\frac {3230 \,{\mathrm e}^{15 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{81}-\frac {8075 \,{\mathrm e}^{14 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{81}-\frac {56525 \,{\mathrm e}^{13 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{243}-\frac {734825 \,{\mathrm e}^{12 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{1458}-\frac {734825 \,{\mathrm e}^{11 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{729}-\frac {8083075 \,{\mathrm e}^{10 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{4374}-\frac {40415375 \,{\mathrm e}^{9 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{13122}-\frac {40415375 \,{\mathrm e}^{8 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{8748}-\frac {40415375 \,{\mathrm e}^{7 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{6561}-\frac {282907625 \,{\mathrm e}^{6 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{39366}-\frac {282907625 \,{\mathrm e}^{5 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{39366}-\frac {1414538125 \,{\mathrm e}^{4 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{236196}-\frac {1414538125 \,{\mathrm e}^{3 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{354294}-\frac {1414538125 \,{\mathrm e}^{2 x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{708588}-\frac {1414538125 \,{\mathrm e}^{x} {\mathrm e}^{-6 \,{\mathrm e}^{x}}}{2125764}-\frac {1414538125 \,{\mathrm e}^{-6 \,{\mathrm e}^{x}}}{12754584}\right )\)	\(451\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*exp(x)-20)*exp(-6*exp(x)+20*x+16),x,method=_RETURNVERBOSE)

[Out]

-20*exp(16)*(-1/6/exp(exp(x))^6*exp(x)^19-19/36*exp(x)^18/exp(exp(x))^6-19/12*exp(x)^17/exp(exp(x))^6-323/72*e
xp(x)^16/exp(exp(x))^6-323/27*exp(x)^15/exp(exp(x))^6-1615/54*exp(x)^14/exp(exp(x))^6-11305/162*exp(x)^13/exp(
exp(x))^6-146965/972*exp(x)^12/exp(exp(x))^6-146965/486*exp(x)^11/exp(exp(x))^6-1616615/2916*exp(x)^10/exp(exp
(x))^6-8083075/8748*exp(x)^9/exp(exp(x))^6-8083075/5832*exp(x)^8/exp(exp(x))^6-8083075/4374*exp(x)^7/exp(exp(x
))^6-56581525/26244*exp(x)^6/exp(exp(x))^6-56581525/26244*exp(x)^5/exp(exp(x))^6-282907625/157464*exp(x)^4/exp
(exp(x))^6-282907625/236196*exp(x)^3/exp(exp(x))^6-282907625/472392*exp(x)^2/exp(exp(x))^6-282907625/1417176*e
xp(x)/exp(exp(x))^6-282907625/8503056/exp(exp(x))^6)+6*exp(16)*(-1/6/exp(exp(x))^6*exp(x)^20-5/9/exp(exp(x))^6
*exp(x)^19-95/54*exp(x)^18/exp(exp(x))^6-95/18*exp(x)^17/exp(exp(x))^6-1615/108*exp(x)^16/exp(exp(x))^6-3230/8
1*exp(x)^15/exp(exp(x))^6-8075/81*exp(x)^14/exp(exp(x))^6-56525/243*exp(x)^13/exp(exp(x))^6-734825/1458*exp(x)
^12/exp(exp(x))^6-734825/729*exp(x)^11/exp(exp(x))^6-8083075/4374*exp(x)^10/exp(exp(x))^6-40415375/13122*exp(x
)^9/exp(exp(x))^6-40415375/8748*exp(x)^8/exp(exp(x))^6-40415375/6561*exp(x)^7/exp(exp(x))^6-282907625/39366*ex
p(x)^6/exp(exp(x))^6-282907625/39366*exp(x)^5/exp(exp(x))^6-1414538125/236196*exp(x)^4/exp(exp(x))^6-141453812
5/354294*exp(x)^3/exp(exp(x))^6-1414538125/708588*exp(x)^2/exp(exp(x))^6-1414538125/2125764*exp(x)/exp(exp(x))
^6-1414538125/12754584/exp(exp(x))^6)

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Maxima [A]
time = 0.29, size = 12, normalized size = 0.43 \begin {gather*} -e^{\left (20 \, x - 6 \, e^{x} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*exp(x)-20)*exp(-6*exp(x)+20*x+16),x, algorithm="maxima")

[Out]

-e^(20*x - 6*e^x + 16)

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Fricas [A]
time = 0.38, size = 12, normalized size = 0.43 \begin {gather*} -e^{\left (20 \, x - 6 \, e^{x} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*exp(x)-20)*exp(-6*exp(x)+20*x+16),x, algorithm="fricas")

[Out]

-e^(20*x - 6*e^x + 16)

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Sympy [A]
time = 0.05, size = 12, normalized size = 0.43 \begin {gather*} - e^{20 x - 6 e^{x} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*exp(x)-20)*exp(-6*exp(x)+20*x+16),x)

[Out]

-exp(20*x - 6*exp(x) + 16)

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Giac [A]
time = 0.38, size = 12, normalized size = 0.43 \begin {gather*} -e^{\left (20 \, x - 6 \, e^{x} + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*exp(x)-20)*exp(-6*exp(x)+20*x+16),x, algorithm="giac")

[Out]

-e^(20*x - 6*e^x + 16)

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Mupad [B]
time = 0.05, size = 13, normalized size = 0.46 \begin {gather*} -{\mathrm {e}}^{20\,x}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{-6\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(20*x - 6*exp(x) + 16)*(6*exp(x) - 20),x)

[Out]

-exp(20*x)*exp(16)*exp(-6*exp(x))

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