3.42.8 \(\int \frac {e^4 (-207360-810 x)+e^9 (17694720+138240 x+270 x^2)}{81+e^4 (-14155776-110592 x-216 x^2)+e^{15} (-201326592-2359296 x-9216 x^2-12 x^3)+e^{20} (4294967296+67108864 x+393216 x^2+1024 x^3+x^4)+e^8 (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4)+e^3 (e^2 (-27648-108 x)+e^6 (2415919104+28311552 x+110592 x^2+144 x^3))+e^6 (e^4 (3538944+27648 x+54 x^2)+e^8 (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4))} \, dx\)

Optimal. Leaf size=30 \[ \frac {5}{\frac {4}{3}-\left (\frac {e^3}{3}-\frac {1}{e^2 (256+x)}\right )^2} \]

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Rubi [B]  time = 0.18, antiderivative size = 120, normalized size of antiderivative = 4.00, number of steps used = 4, number of rules used = 4, integrand size = 184, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {1680, 12, 1814, 8} \begin {gather*} -\frac {135 \left (3 \left (144-e^{12}\right )-2 e^5 \left (12-e^6\right ) \left (\left (12-e^6\right ) x-256 e^6+3 e+3072\right )\right )}{\left (12-e^6\right )^2 \left (108-e^4 \left (12-e^6\right )^2 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(-207360 - 810*x) + E^9*(17694720 + 138240*x + 270*x^2))/(81 + E^4*(-14155776 - 110592*x - 216*x^2) +
E^15*(-201326592 - 2359296*x - 9216*x^2 - 12*x^3) + E^20*(4294967296 + 67108864*x + 393216*x^2 + 1024*x^3 + x
^4) + E^8*(618475290624 + 9663676416*x + 56623104*x^2 + 147456*x^3 + 144*x^4) + E^3*(E^2*(-27648 - 108*x) + E^
6*(2415919104 + 28311552*x + 110592*x^2 + 144*x^3)) + E^6*(E^4*(3538944 + 27648*x + 54*x^2) + E^8*(-1030792151
04 - 1610612736*x - 9437184*x^2 - 24576*x^3 - 24*x^4))),x]

[Out]

(-135*(3*(144 - E^12) - 2*E^5*(12 - E^6)*(3072 + 3*E - 256*E^6 + (12 - E^6)*x)))/((12 - E^6)^2*(108 - E^4*(12
- E^6)^2*((147456*E^8 + 144*E^9 - 24576*E^14 - 12*E^15 + 1024*E^20)/(4*(144*E^8 - 24*E^14 + E^20)) + x)^2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {270 e^4 \left (108 e-3 \left (144-e^{12}\right ) x+e^5 \left (12-e^6\right )^2 x^2\right )}{\left (108-e^4 \left (12-e^6\right )^2 x^2\right )^2} \, dx,x,\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}+x\right )\\ &=\left (270 e^4\right ) \operatorname {Subst}\left (\int \frac {108 e-3 \left (144-e^{12}\right ) x+e^5 \left (12-e^6\right )^2 x^2}{\left (108-e^4 \left (12-e^6\right )^2 x^2\right )^2} \, dx,x,\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}+x\right )\\ &=-\frac {135 \left (3 \left (144-e^{12}\right )-2 e^5 \left (12-e^6\right ) \left (3072+3 e-256 e^6+\left (12-e^6\right ) x\right )\right )}{\left (12-e^6\right )^2 \left (108-e^4 \left (12-e^6\right )^2 \left (\frac {3072+3 e-256 e^6}{12-e^6}+x\right )^2\right )}-\frac {1}{4} \left (5 e^4\right ) \operatorname {Subst}\left (\int 0 \, dx,x,\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}+x\right )\\ &=-\frac {135 \left (3 \left (144-e^{12}\right )-2 e^5 \left (12-e^6\right ) \left (3072+3 e-256 e^6+\left (12-e^6\right ) x\right )\right )}{\left (12-e^6\right )^2 \left (108-e^4 \left (12-e^6\right )^2 \left (\frac {3072+3 e-256 e^6}{12-e^6}+x\right )^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 50, normalized size = 1.67 \begin {gather*} \frac {135 \left (3-2 e^5 (256+x)\right )}{\left (-12+e^6\right ) \left (9-6 e^5 (256+x)-12 e^4 (256+x)^2+e^{10} (256+x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(-207360 - 810*x) + E^9*(17694720 + 138240*x + 270*x^2))/(81 + E^4*(-14155776 - 110592*x - 216*
x^2) + E^15*(-201326592 - 2359296*x - 9216*x^2 - 12*x^3) + E^20*(4294967296 + 67108864*x + 393216*x^2 + 1024*x
^3 + x^4) + E^8*(618475290624 + 9663676416*x + 56623104*x^2 + 147456*x^3 + 144*x^4) + E^3*(E^2*(-27648 - 108*x
) + E^6*(2415919104 + 28311552*x + 110592*x^2 + 144*x^3)) + E^6*(E^4*(3538944 + 27648*x + 54*x^2) + E^8*(-1030
79215104 - 1610612736*x - 9437184*x^2 - 24576*x^3 - 24*x^4))),x]

[Out]

(135*(3 - 2*E^5*(256 + x)))/((-12 + E^6)*(9 - 6*E^5*(256 + x) - 12*E^4*(256 + x)^2 + E^10*(256 + x)^2))

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fricas [B]  time = 0.60, size = 68, normalized size = 2.27 \begin {gather*} -\frac {135 \, {\left (2 \, {\left (x + 256\right )} e^{5} - 3\right )}}{{\left (x^{2} + 512 \, x + 65536\right )} e^{16} - 6 \, {\left (x + 256\right )} e^{11} - 24 \, {\left (x^{2} + 512 \, x + 65536\right )} e^{10} + 72 \, {\left (x + 256\right )} e^{5} + 144 \, {\left (x^{2} + 512 \, x + 65536\right )} e^{4} + 9 \, e^{6} - 108} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp(1)^4)/((x^4+1024*x^3+393216*x^2+671
08864*x+4294967296)*exp(1)^8*exp(3)^4+(-12*x^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576
*x^3-9437184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944)*exp(1)^4)*exp(3)^2+((144*x^3+110
592*x^2+28311552*x+2415919104)*exp(1)^6+(-108*x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+96636
76416*x+618475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x, algorithm="fricas")

[Out]

-135*(2*(x + 256)*e^5 - 3)/((x^2 + 512*x + 65536)*e^16 - 6*(x + 256)*e^11 - 24*(x^2 + 512*x + 65536)*e^10 + 72
*(x + 256)*e^5 + 144*(x^2 + 512*x + 65536)*e^4 + 9*e^6 - 108)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp(1)^4)/((x^4+1024*x^3+393216*x^2+671
08864*x+4294967296)*exp(1)^8*exp(3)^4+(-12*x^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576
*x^3-9437184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944)*exp(1)^4)*exp(3)^2+((144*x^3+110
592*x^2+28311552*x+2415919104)*exp(1)^6+(-108*x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+96636
76416*x+618475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.51, size = 72, normalized size = 2.40

method	result	size
risch	\(\frac {-\frac {270 \,{\mathrm e}^{5} x}{{\mathrm e}^{6}-12}-\frac {135 \left (512 \,{\mathrm e}^{5}-3\right )}{{\mathrm e}^{6}-12}}{x^{2} {\mathrm e}^{10}+512 x \,{\mathrm e}^{10}+65536 \,{\mathrm e}^{10}-6 x \,{\mathrm e}^{5}-12 x^{2} {\mathrm e}^{4}-1536 \,{\mathrm e}^{5}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}+9}\)	\(72\)
gosper	\(-\frac {135 \left (2 x \,{\mathrm e}^{2} {\mathrm e}^{3}+512 \,{\mathrm e}^{2} {\mathrm e}^{3}-3\right )}{\left ({\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9\right ) \left ({\mathrm e}^{6}-12\right )}\)	\(106\)
norman	\(\frac {-\frac {270 \,{\mathrm e}^{2} {\mathrm e}^{3} x}{{\mathrm e}^{6}-12}-\frac {135 \left (512 \,{\mathrm e}^{2} {\mathrm e}^{3}-3\right )}{{\mathrm e}^{6}-12}}{{\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9}\)	\(116\)
default	\(-\frac {135 \,{\mathrm e}^{4} \left (\munderset {\textit {\_R} =\RootOf \left (-\left (-{\mathrm e}^{20}-144 \,{\mathrm e}^{8}+24 \,{\mathrm e}^{14}\right ) \textit {\_Z}^{4}-\left (-1024 \,{\mathrm e}^{20}-147456 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{15}-144 \,{\mathrm e}^{9}+24576 \,{\mathrm e}^{14}\right ) \textit {\_Z}^{3}-\left (216 \,{\mathrm e}^{4}-393216 \,{\mathrm e}^{20}-56623104 \,{\mathrm e}^{8}+9216 \,{\mathrm e}^{15}-110592 \,{\mathrm e}^{9}+9437184 \,{\mathrm e}^{14}-54 \,{\mathrm e}^{10}\right ) \textit {\_Z}^{2}-\left (108 \,{\mathrm e}^{5}+110592 \,{\mathrm e}^{4}-67108864 \,{\mathrm e}^{20}-9663676416 \,{\mathrm e}^{8}+2359296 \,{\mathrm e}^{15}-28311552 \,{\mathrm e}^{9}[phone] \,{\mathrm e}^{14}-27648 \,{\mathrm e}^{10}\right ) \textit {\_Z} +81+4294967296 \,{\mathrm e}^{20}[phone] \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{9}+3538944 \,{\mathrm e}^{10}-14155776 \,{\mathrm e}^{4}-27648 \,{\mathrm e}^{5}-103079215104 \,{\mathrm e}^{14}-201326592 \,{\mathrm e}^{15}\right )}{\sum }\frac {\left (\textit {\_R}^{2} {\mathrm e}^{5}+\left (512 \,{\mathrm e}^{5}-3\right ) \textit {\_R} -768+65536 \,{\mathrm e}^{5}\right ) \ln \left (x -\textit {\_R} \right )}{-6912 \,{\mathrm e}^{10}-196608 \textit {\_R} \,{\mathrm e}^{20}-2415919104 \,{\mathrm e}^{8}+589824 \,{\mathrm e}^{15}-28311552 \textit {\_R} \,{\mathrm e}^{8}+27648 \,{\mathrm e}^{4}+27 \,{\mathrm e}^{5}-7077888 \,{\mathrm e}^{9}+108 \textit {\_R} \,{\mathrm e}^{4}+4608 \textit {\_R} \,{\mathrm e}^{15}[phone] \,{\mathrm e}^{14}-55296 \textit {\_R} \,{\mathrm e}^{9}-16777216 \,{\mathrm e}^{20}-108 \textit {\_R}^{2} {\mathrm e}^{9}-\textit {\_R}^{3} {\mathrm e}^{20}-110592 \textit {\_R}^{2} {\mathrm e}^{8}-144 \,{\mathrm e}^{8} \textit {\_R}^{3}+24 \textit {\_R}^{3} {\mathrm e}^{14}-768 \textit {\_R}^{2} {\mathrm e}^{20}+18432 \textit {\_R}^{2} {\mathrm e}^{14}+4718592 \,{\mathrm e}^{14} \textit {\_R} -27 \textit {\_R} \,{\mathrm e}^{10}+9 \textit {\_R}^{2} {\mathrm e}^{15}}\right )}{2}\)	\(312\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp(1)^4)/((x^4+1024*x^3+393216*x^2+67108864*
x+4294967296)*exp(1)^8*exp(3)^4+(-12*x^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576*x^3-9
437184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944)*exp(1)^4)*exp(3)^2+((144*x^3+110592*x^
2+28311552*x+2415919104)*exp(1)^6+(-108*x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+9663676416*
x+618475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x,method=_RETURNVERBOSE)

[Out]

(-270*exp(5)/(exp(6)-12)*x-135*(512*exp(5)-3)/(exp(6)-12))/(x^2*exp(10)+512*x*exp(10)+65536*exp(10)-6*x*exp(5)
-12*x^2*exp(4)-1536*exp(5)-6144*x*exp(4)-786432*exp(4)+9)

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maxima [B]  time = 0.36, size = 80, normalized size = 2.67 \begin {gather*} -\frac {135 \, {\left (2 \, x e^{5} + 512 \, e^{5} - 3\right )}}{x^{2} {\left (e^{16} - 24 \, e^{10} + 144 \, e^{4}\right )} + 2 \, x {\left (256 \, e^{16} - 3 \, e^{11} - 6144 \, e^{10} + 36 \, e^{5} + 36864 \, e^{4}\right )} + 65536 \, e^{16} - 1536 \, e^{11} - 1572864 \, e^{10} + 9 \, e^{6} + 18432 \, e^{5} + 9437184 \, e^{4} - 108} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp(1)^4)/((x^4+1024*x^3+393216*x^2+671
08864*x+4294967296)*exp(1)^8*exp(3)^4+(-12*x^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576
*x^3-9437184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944)*exp(1)^4)*exp(3)^2+((144*x^3+110
592*x^2+28311552*x+2415919104)*exp(1)^6+(-108*x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+96636
76416*x+618475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x, algorithm="maxima")

[Out]

-135*(2*x*e^5 + 512*e^5 - 3)/(x^2*(e^16 - 24*e^10 + 144*e^4) + 2*x*(256*e^16 - 3*e^11 - 6144*e^10 + 36*e^5 + 3
6864*e^4) + 65536*e^16 - 1536*e^11 - 1572864*e^10 + 9*e^6 + 18432*e^5 + 9437184*e^4 - 108)

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mupad [B]  time = 3.70, size = 63, normalized size = 2.10 \begin {gather*} \frac {135\,\left (512\,{\mathrm {e}}^5+2\,x\,{\mathrm {e}}^5-3\right )}{\left ({\mathrm {e}}^6-12\right )\,\left (\left (12\,{\mathrm {e}}^4-{\mathrm {e}}^{10}\right )\,x^2+\left (6144\,{\mathrm {e}}^4+6\,{\mathrm {e}}^5-512\,{\mathrm {e}}^{10}\right )\,x+786432\,{\mathrm {e}}^4+1536\,{\mathrm {e}}^5-65536\,{\mathrm {e}}^{10}-9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(9)*(138240*x + 270*x^2 + 17694720) - exp(4)*(810*x + 207360))/(exp(3)*(exp(6)*(28311552*x + 110592*x^
2 + 144*x^3 + 2415919104) - exp(2)*(108*x + 27648)) - exp(4)*(110592*x + 216*x^2 + 14155776) + exp(20)*(671088
64*x + 393216*x^2 + 1024*x^3 + x^4 + 4294967296) - exp(6)*(exp(8)*(1610612736*x + 9437184*x^2 + 24576*x^3 + 24
*x^4 + 103079215104) - exp(4)*(27648*x + 54*x^2 + 3538944)) - exp(15)*(2359296*x + 9216*x^2 + 12*x^3 + 2013265
92) + exp(8)*(9663676416*x + 56623104*x^2 + 147456*x^3 + 144*x^4 + 618475290624) + 81),x)

[Out]

(135*(512*exp(5) + 2*x*exp(5) - 3))/((exp(6) - 12)*(786432*exp(4) + 1536*exp(5) - 65536*exp(10) + x^2*(12*exp(
4) - exp(10)) + x*(6144*exp(4) + 6*exp(5) - 512*exp(10)) - 9))

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sympy [B]  time = 1.93, size = 88, normalized size = 2.93 \begin {gather*} \frac {- 270 x e^{5} - 69120 e^{5} + 405}{x^{2} \left (- 24 e^{10} + 144 e^{4} + e^{16}\right ) + x \left (- 12288 e^{10} - 6 e^{11} + 72 e^{5} + 73728 e^{4} + 512 e^{16}\right ) - 1572864 e^{10} - 1536 e^{11} - 108 + 9 e^{6} + 18432 e^{5} + 9437184 e^{4} + 65536 e^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((270*x**2+138240*x+17694720)*exp(1)**6*exp(3)+(-810*x-207360)*exp(1)**4)/((x**4+1024*x**3+393216*x*
*2+67108864*x+4294967296)*exp(1)**8*exp(3)**4+(-12*x**3-9216*x**2-2359296*x-201326592)*exp(1)**6*exp(3)**3+((-
24*x**4-24576*x**3-9437184*x**2-1610612736*x-103079215104)*exp(1)**8+(54*x**2+27648*x+3538944)*exp(1)**4)*exp(
3)**2+((144*x**3+110592*x**2+28311552*x+2415919104)*exp(1)**6+(-108*x-27648)*exp(1)**2)*exp(3)+(144*x**4+14745
6*x**3+56623104*x**2+9663676416*x+618475290624)*exp(1)**8+(-216*x**2-110592*x-14155776)*exp(1)**4+81),x)

[Out]

(-270*x*exp(5) - 69120*exp(5) + 405)/(x**2*(-24*exp(10) + 144*exp(4) + exp(16)) + x*(-12288*exp(10) - 6*exp(11
) + 72*exp(5) + 73728*exp(4) + 512*exp(16)) - 1572864*exp(10) - 1536*exp(11) - 108 + 9*exp(6) + 18432*exp(5) +
9437184*exp(4) + 65536*exp(16))

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