∫ −3+3e2−6x−3 log(9e6)___________________ x2+e4x2+2x3+x4+e2(−2x2−2x3)+(2x2−2e2x2+2x3) log(9e6)+x2 log2(9e6) dx [4044]

Optimal. Leaf size=21 \[ \frac {3}{x \left (1-e^2+x+\log \left (9 e^6\right )\right )} \]

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Rubi [A]
time = 0.09, antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 4, integrand size = 93, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6, 1694, 12, 267} \begin {gather*} \frac {3}{x \left (x-e^2+7+\log (9)\right )} \end {gather*}

Int[(-3 + 3*E^2 - 6*x - 3*Log[9*E^6])/(x^2 + E^4*x^2 + 2*x^3 + x^4 + E^2*(-2*x^2 - 2*x^3) + (2*x^2 - 2*E^2*x^2

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x

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

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ

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]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{\left (1+e^4\right ) x^2+2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \log ^2\left (9 e^6\right )} \, dx\\ &=\int \frac {-3+3 e^2-6 x-3 \log \left (9 e^6\right )}{2 x^3+x^4+e^2 \left (-2 x^2-2 x^3\right )+\left (2 x^2-2 e^2 x^2+2 x^3\right ) \log \left (9 e^6\right )+x^2 \left (1+e^4+\log ^2\left (9 e^6\right )\right )} \, dx\\ &=\text {Subst}\left (\int -\frac {96 x}{\left (49+e^4-4 x^2+12 \log (9)+\log ^2(9)-2 e^2 (7+\log (9))+\log (81)\right )^2} \, dx,x,x+\frac {1}{4} \left (2-2 e^2+2 \log \left (9 e^6\right )\right )\right )\\ &=-\left (96 \text {Subst}\left (\int \frac {x}{\left (49+e^4-4 x^2+12 \log (9)+\log ^2(9)-2 e^2 (7+\log (9))+\log (81)\right )^2} \, dx,x,x+\frac {1}{4} \left (2-2 e^2+2 \log \left (9 e^6\right )\right )\right )\right )\\ &=\frac {3}{x \left (7-e^2+x+\log (9)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 17, normalized size = 0.81 \begin {gather*} \frac {3}{x \left (7-e^2+x+\log (9)\right )} \end {gather*}

Integrate[(-3 + 3*E^2 - 6*x - 3*Log[9*E^6])/(x^2 + E^4*x^2 + 2*x^3 + x^4 + E^2*(-2*x^2 - 2*x^3) + (2*x^2 - 2*E

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method	result	size

norman	\(-\frac {3}{x \left ({\mathrm e}^{2}-2 \ln \left (3\right )-x -7\right )}\)	\(19\)
risch	\(-\frac {3}{x \left ({\mathrm e}^{2}-2 \ln \left (3\right )-x -7\right )}\)	\(19\)
gosper	\(-\frac {3}{x \left (-x -1+{\mathrm e}^{2}-\ln \left (9 \,{\mathrm e}^{6}\right )\right )}\)	\(24\)

int((-3*ln(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*ln(9*exp(3)^2)^2+(-2*x^2*exp(2)+2*x^3+2*x^2)*ln(9*exp(3)^2)+x^2*ex

p(2)^2+(-2*x^3-2*x^2)*exp(2)+x^4+2*x^3+x^2),x,method=_RETURNVERBOSE)

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Maxima [A]
time = 0.29, size = 22, normalized size = 1.05 \begin {gather*} \frac {3}{x^{2} - x {\left (e^{2} - \log \left (9 \, e^{6}\right ) - 1\right )}} \end {gather*}

integrate((-3*log(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*log(9*exp(3)^2)^2+(-2*x^2*exp(2)+2*x^3+2*x^2)*log(9*exp(3)^

2)+x^2*exp(2)^2+(-2*x^3-2*x^2)*exp(2)+x^4+2*x^3+x^2),x, algorithm="maxima")

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Fricas [A]
time = 0.41, size = 21, normalized size = 1.00 \begin {gather*} \frac {3}{x^{2} - x e^{2} + 2 \, x \log \left (3\right ) + 7 \, x} \end {gather*}

integrate((-3*log(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*log(9*exp(3)^2)^2+(-2*x^2*exp(2)+2*x^3+2*x^2)*log(9*exp(3)^

2)+x^2*exp(2)^2+(-2*x^3-2*x^2)*exp(2)+x^4+2*x^3+x^2),x, algorithm="fricas")

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Sympy [A]
time = 0.39, size = 15, normalized size = 0.71 \begin {gather*} \frac {3}{x^{2} + x \left (- e^{2} + 2 \log {\left (3 \right )} + 7\right )} \end {gather*}

integrate((-3*ln(9*exp(3)**2)+3*exp(2)-6*x-3)/(x**2*ln(9*exp(3)**2)**2+(-2*x**2*exp(2)+2*x**3+2*x**2)*ln(9*exp

(3)**2)+x**2*exp(2)**2+(-2*x**3-2*x**2)*exp(2)+x**4+2*x**3+x**2),x)

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Giac [A]
time = 0.39, size = 21, normalized size = 1.00 \begin {gather*} \frac {3}{x^{2} - x e^{2} + x \log \left (9 \, e^{6}\right ) + x} \end {gather*}

integrate((-3*log(9*exp(3)^2)+3*exp(2)-6*x-3)/(x^2*log(9*exp(3)^2)^2+(-2*x^2*exp(2)+2*x^3+2*x^2)*log(9*exp(3)^

2)+x^2*exp(2)^2+(-2*x^3-2*x^2)*exp(2)+x^4+2*x^3+x^2),x, algorithm="giac")

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Mupad [B]
time = 18.16, size = 2500, normalized size = 119.05 \begin {gather*} \text {Too large to display} \end {gather*}

int(-(6*x + 3*log(9*exp(6)) - 3*exp(2) + 3)/(x^2*exp(4) - exp(2)*(2*x^2 + 2*x^3) + x^2*log(9*exp(6))^2 + x^2 +

2*x^3 + x^4 + log(9*exp(6))*(2*x^2 - 2*x^2*exp(2) + 2*x^3)),x)

(log(729) - 3*exp(2) + 21)/(x*(exp(4) - 14*exp(2) + 14*log(9) - 2*exp(2)*log(9) + log(9)^2 + 49)) - (log((441*

log(81) - 1764*log(9) + 294*log(729) + 504*exp(2)*log(9) - 36*exp(4)*log(9) - 126*exp(2)*log(81) + 9*exp(4)*lo

g(81) - 84*exp(2)*log(729) + 6*exp(4)*log(729) - 84*log(9)*log(729) + 42*log(81)*log(729) + 18*exp(2)*log(9)^2

- 2*exp(2)*log(729)^2 - 6*log(9)^2*log(729) + log(81)*log(729)^2 - 126*log(9)^2 + 14*log(729)^2 + 12*exp(2)*l

og(9)*log(729) - 6*exp(2)*log(81)*log(729))/(294*exp(4) - 1372*exp(2) - 28*exp(6) + exp(8) + 1372*log(9) - 588

*exp(2)*log(9) + 84*exp(4)*log(9) - 4*exp(6)*log(9) - 84*exp(2)*log(9)^2 - 4*exp(2)*log(9)^3 + 6*exp(4)*log(9)

^2 + 294*log(9)^2 + 28*log(9)^3 + log(9)^4 + 2401) + (x*(9*exp(4) - 126*exp(2) + 42*log(729) - 6*exp(2)*log(72

9) + log(729)^2 + 441))/(294*exp(4) - 1372*exp(2) - 28*exp(6) + exp(8) + 1372*log(9) - 588*exp(2)*log(9) + 84*

exp(4)*log(9) - 4*exp(6)*log(9) - 84*exp(2)*log(9)^2 - 4*exp(2)*log(9)^3 + 6*exp(4)*log(9)^2 + 294*log(9)^2 +

28*log(9)^3 + log(9)^4 + 2401) - (((36015*exp(2) - 10290*exp(4) + 1470*exp(6) - 105*exp(8) + 3*exp(10) - 86436

*log(9) + 14406*log(81) + 7203*log(729) + 49392*exp(2)*log(9) - 10584*exp(4)*log(9) + 1008*exp(6)*log(9) - 36*

exp(8)*log(9) - 8232*exp(2)*log(81) + 1764*exp(4)*log(81) - 168*exp(6)*log(81) + 6*exp(8)*log(81) - 4116*exp(2

)*log(729) + 882*exp(4)*log(729) - 84*exp(6)*log(729) + 3*exp(8)*log(729) + 1372*log(9)*log(729) + 1372*log(81

)*log(729) + 11466*exp(2)*log(9)^2 + 840*exp(2)*log(9)^3 + 15*exp(2)*log(9)^4 - 1638*exp(4)*log(9)^2 - 60*exp(

4)*log(9)^3 + 78*exp(6)*log(9)^2 - 441*exp(2)*log(81)^2 + 63*exp(4)*log(81)^2 - 3*exp(6)*log(81)^2 + 294*log(9

)*log(81)^2 - 1176*log(9)^2*log(81) - 168*log(9)^3*log(81) - 6*log(9)^4*log(81) - 98*log(9)^2*log(729) - 28*lo

g(9)^3*log(729) - log(9)^4*log(729) + 49*log(81)^2*log(729) - 26754*log(9)^2 - 2940*log(9)^3 - 105*log(9)^4 +

1029*log(81)^2 + 21*log(9)^2*log(81)^2 + 14*log(9)*log(81)^2*log(729) + 28*log(9)^2*log(81)*log(729) - 3*exp(2

)*log(9)^2*log(81)^2 + log(9)^2*log(81)^2*log(729) - 588*exp(2)*log(9)*log(729) + 84*exp(4)*log(9)*log(729) -

4*exp(6)*log(9)*log(729) - 588*exp(2)*log(81)*log(729) + 84*exp(4)*log(81)*log(729) - 4*exp(6)*log(81)*log(729

) + 392*log(9)*log(81)*log(729) - 84*exp(2)*log(9)*log(81)^2 + 336*exp(2)*log(9)^2*log(81) + 24*exp(2)*log(9)^

3*log(81) + 6*exp(4)*log(9)*log(81)^2 - 24*exp(4)*log(9)^2*log(81) + 28*exp(2)*log(9)^2*log(729) + 4*exp(2)*lo

g(9)^3*log(729) - 2*exp(4)*log(9)^2*log(729) - 14*exp(2)*log(81)^2*log(729) + exp(4)*log(81)^2*log(729) - 112*

exp(2)*log(9)*log(81)*log(729) + 8*exp(4)*log(9)*log(81)*log(729) - 2*exp(2)*log(9)*log(81)^2*log(729) - 4*exp

(2)*log(9)^2*log(81)*log(729) - 50421)/(294*exp(4) - 1372*exp(2) - 28*exp(6) + exp(8) + 1372*log(9) - 588*exp(

2)*log(9) + 84*exp(4)*log(9) - 4*exp(6)*log(9) - 84*exp(2)*log(9)^2 - 4*exp(2)*log(9)^3 + 6*exp(4)*log(9)^2 +

294*log(9)^2 + 28*log(9)^3 + log(9)^4 + 2401) + (((705894*exp(4) - 1647086*exp(2) - 168070*exp(6) + 24010*exp(

8) - 2058*exp(10) + 98*exp(12) - 2*exp(14) + 1411788*log(9) + 117649*log(81) - 1210104*exp(2)*log(9) + 432180*

exp(4)*log(9) - 82320*exp(6)*log(9) + 8820*exp(8)*log(9) - 504*exp(10)*log(9) + 12*exp(12)*log(9) - 100842*exp

(2)*log(81) + 36015*exp(4)*log(81) - 6860*exp(6)*log(81) + 735*exp(8)*log(81) - 42*exp(10)*log(81) + exp(12)*l

og(81) + 100842*log(9)*log(81) - 360150*exp(2)*log(9)^2 - 54880*exp(2)*log(9)^3 - 4410*exp(2)*log(9)^4 + 10290

0*exp(4)*log(9)^2 - 168*exp(2)*log(9)^5 + 11760*exp(4)*log(9)^3 - 2*exp(2)*log(9)^6 + 630*exp(4)*log(9)^4 - 14

700*exp(6)*log(9)^2 + 12*exp(4)*log(9)^5 - 1120*exp(6)*log(9)^3 - 30*exp(6)*log(9)^4 + 1050*exp(8)*log(9)^2 +

40*exp(8)*log(9)^3 - 30*exp(10)*log(9)^2 + 36015*log(9)^2*log(81) + 6860*log(9)^3*log(81) + 735*log(9)^4*log(8

1) + 42*log(9)^5*log(81) + log(9)^6*log(81) + 504210*log(9)^2 + 96040*log(9)^3 + 10290*log(9)^4 + 588*log(9)^5

+ 14*log(9)^6 - 72030*exp(2)*log(9)*log(81) + 20580*exp(4)*log(9)*log(81) - 2940*exp(6)*log(9)*log(81) + 210*

exp(8)*log(9)*log(81) - 6*exp(10)*log(9)*log(81) - 20580*exp(2)*log(9)^2*log(81) - 2940*exp(2)*log(9)^3*log(81

) - 210*exp(2)*log(9)^4*log(81) + 4410*exp(4)*log(9)^2*log(81) - 6*exp(2)*log(9)^5*log(81) + 420*exp(4)*log(9)

^3*log(81) + 15*exp(4)*log(9)^4*log(81) - 420*exp(6)*log(9)^2*log(81) - 20*exp(6)*log(9)^3*log(81) + 15*exp(8)

*log(9)^2*log(81) + 1647086)/(294*exp(4) - 1372*exp(2) - 28*exp(6) + exp(8) + 1372*log(9) - 588*exp(2)*log(9)

+ 84*exp(4)*log(9) - 4*exp(6)*log(9) - 84*exp(2)*log(9)^2 - 4*exp(2)*log(9)^3 + 6*exp(4)*log(9)^2 + 294*log(9)

^2 + 28*log(9)^3 + log(9)^4 + 2401) + (x*(72030*exp(4) - 201684*exp(2) - 13720*exp(6) + 1470*exp(8) - 84*exp(1

0) + 2*exp(12) - 67228*log(9) + 134456*log(81) + 48020*exp(2)*log(9) - 13720*exp(4)*log(9) + 1960*exp(6)*log(9

) - 140*exp(8)*log(9) + 4*exp(10)*log(9) - 96040*exp(2)*log(81) + 27440*exp(4)*log(81) - 3920*exp(6)*log(81) +

280*exp(8)*log(81) - 8*exp(10)*log(81) + 76832*log(9)*log(81) + 57624*exp(2)*log(9)^2 + 12936*exp(2)*log(9)^3

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3.41.44 \(\int \frac {-3+3 e^2-6 x-3 \log (9 e^6)}{x^2+e^4 x^2+2 x^3+x^4+e^2 (-2 x^2-2 x^3)+(2 x^2-2 e^2 x^2+2 x^3) \log (9 e^6)+x^2 \log ^2(9 e^6)} \, dx\) [4044]