\(\int \frac {e^x (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2))}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} (-18 x^2+2 x^3+2 x^4)+(108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4) \log (2)+(54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx\) [5308]

Optimal result

Integrand size = 178, antiderivative size = 32 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=\frac {6 e^x}{x \left (-\sqrt {e}-x-x^2+(3+\log (2))^2\right )} \]

[Out]

6*exp(x)/((3+ln(2))^2-x^2-exp(1/2)-x)/x

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 9.66 (sec) , antiderivative size = 1976, normalized size of antiderivative = 61.75, number of steps used = 32, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6, 6820, 12, 6874, 2208, 2209, 6860} \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx =\text {Too large to display} \]

[In]

Int[(E^x*(-54 + Sqrt[E]*(6 - 6*x) + 66*x + 12*x^2 - 6*x^3 + (-36 + 36*x)*Log[2] + (-6 + 6*x)*Log[2]^2))/(81*x^
2 + E*x^2 - 18*x^3 - 17*x^4 + 2*x^5 + x^6 + Sqrt[E]*(-18*x^2 + 2*x^3 + 2*x^4) + (108*x^2 - 12*Sqrt[E]*x^2 - 12
*x^3 - 12*x^4)*Log[2] + (54*x^2 - 2*Sqrt[E]*x^2 - 2*x^3 - 2*x^4)*Log[2]^2 + 12*x^2*Log[2]^3 + x^2*Log[2]^4),x]

[Out]

(-6*E^x)/(x*(Sqrt[E] - (3 + Log[2])^2)) + (6*ExpIntegralEi[x])/(Sqrt[E] - (3 + Log[2])^2) - (6*E^((-1 + Sqrt[3
7 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2
]^2])/2])/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)^(3/2)*(Sqrt[E] - (3 + Log[2])^2)) + (6*E^((-1 - Sqrt[37 -
4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2
])/2])/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)^(3/2)*(Sqrt[E] - (3 + Log[2])^2)) + (3*E^((-1 + Sqrt[37 - 4*
Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/
2]*(1 - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)*(Sqrt[E] -
(3 + Log[2])^2)) - (6*E^x*(1 - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2]))/((37 - 4*Sqrt[E] + 24*Log[2] +
4*Log[2]^2)*(Sqrt[E] - (3 + Log[2])^2)*(1 + 2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])) + (3*E^((-1
- Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] +
4*Log[2]^2])/2]*(1 + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^
2)*(Sqrt[E] - (3 + Log[2])^2)) - (6*E^x*(1 + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2]))/((37 - 4*Sqrt[E]
+ 24*Log[2] + 4*Log[2]^2)*(Sqrt[E] - (3 + Log[2])^2)*(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])
) - (12*E^((-1 + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x - Sqrt[37 - 4*Sqrt[E
] + 24*Log[2] + 4*Log[2]^2])/2]*(171 + 2*E + 228*Log[2] + 109*Log[2]^2 + 24*Log[2]^3 + 2*Log[2]^4 - Sqrt[E]*(3
7 + 24*Log[2] + 4*Log[2]^2) - Log[64]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)^(3/2)*(Sqrt[E] - (3 + Log[2
])^2)^2) + (12*E^((-1 - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x + Sqrt[37 - 4
*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2]*(171 + 2*E + 228*Log[2] + 109*Log[2]^2 + 24*Log[2]^3 + 2*Log[2]^4 - Sqr
t[E]*(37 + 24*Log[2] + 4*Log[2]^2) - Log[64]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)^(3/2)*(Sqrt[E] - (3
+ Log[2])^2)^2) + (6*E^((-1 + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x - Sqrt[
37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2]*(171 + 2*E + 228*Log[2] + 109*Log[2]^2 + 24*Log[2]^3 + 2*Log[2]^4
- Sqrt[E]*(37 + 24*Log[2] + 4*Log[2]^2) - Log[64]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)*(Sqrt[E] - (3
+ Log[2])^2)^2) + (6*E^((-1 - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x + Sqrt[
37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2]*(171 + 2*E + 228*Log[2] + 109*Log[2]^2 + 24*Log[2]^3 + 2*Log[2]^4
- Sqrt[E]*(37 + 24*Log[2] + 4*Log[2]^2) - Log[64]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)*(Sqrt[E] - (3
+ Log[2])^2)^2) - (12*E^x*(171 + 2*E + 228*Log[2] + 109*Log[2]^2 + 24*Log[2]^3 + 2*Log[2]^4 - Sqrt[E]*(37 + 24
*Log[2] + 4*Log[2]^2) - Log[64]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)*(Sqrt[E] - (3 + Log[2])^2)^2*(1 +
2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])) - (12*E^x*(171 + 2*E + 228*Log[2] + 109*Log[2]^2 + 24*L
og[2]^3 + 2*Log[2]^4 - Sqrt[E]*(37 + 24*Log[2] + 4*Log[2]^2) - Log[64]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[
2]^2)*(Sqrt[E] - (3 + Log[2])^2)^2*(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])) - (6*ExpIntegral
Ei[x]*(81 + E + 66*Log[2] + 6*Log[2]^3 + Log[2]^4 + 7*Log[64] + Log[64]^2 + Log[2]^2*(18 + Log[64]) - 2*Sqrt[E
]*(9 + Log[2]^2 + Log[64])))/(Sqrt[E] - (3 + Log[2])^2)^3 + (3*E^((-1 + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Lo
g[2]^2])/2)*ExpIntegralEi[(1 + 2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2]*(1 - 1/Sqrt[37 - 4*Sqrt
[E] + 24*Log[2] + 4*Log[2]^2])*(81 + E + 66*Log[2] + 6*Log[2]^3 + Log[2]^4 + 7*Log[64] + Log[64]^2 + Log[2]^2*
(18 + Log[64]) - 2*Sqrt[E]*(9 + Log[2]^2 + Log[64])))/(Sqrt[E] - (3 + Log[2])^2)^3 + (3*E^((-1 - Sqrt[37 - 4*S
qrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2
]*(1 + 1/Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])*(81 + E + 66*Log[2] + 6*Log[2]^3 + Log[2]^4 + 7*Log[64
] + Log[64]^2 + Log[2]^2*(18 + Log[64]) - 2*Sqrt[E]*(9 + Log[2]^2 + Log[64])))/(Sqrt[E] - (3 + Log[2])^2)^3

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), 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] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 6820

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

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{(81+e) x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx \\ & = \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+x^2 \log ^4(2)+x^2 \left (81+e+12 \log ^3(2)\right )} \, dx \\ & = \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+x^2 \left (81+e+12 \log ^3(2)+\log ^4(2)\right )} \, dx \\ & = \int \frac {6 e^x \left (-9+\sqrt {e}+2 x^2-x^3-\log ^2(2)-\log (64)+x \left (11-\sqrt {e}+\log ^2(2)+\log (64)\right )\right )}{x^2 \left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2} \, dx \\ & = 6 \int \frac {e^x \left (-9+\sqrt {e}+2 x^2-x^3-\log ^2(2)-\log (64)+x \left (11-\sqrt {e}+\log ^2(2)+\log (64)\right )\right )}{x^2 \left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2} \, dx \\ & = 6 \int \left (\frac {e^x \left (-9+\sqrt {e}-\log ^2(2)-\log (64)\right )}{x^2 \left (\sqrt {e}-(3+\log (2))^2\right )^2}+\frac {e^x x \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^3 \left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )}+\frac {e^x \left (-81-e-66 \log (2)-6 \log ^3(2)-\log ^4(2)-7 \log (64)-\log ^2(64)-\log ^2(2) (18+\log (64))+2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{x \left (\sqrt {e}-(3+\log (2))^2\right )^3}+\frac {e^x \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\sqrt {e} \left (37+24 \log (2)+4 \log ^2(2)\right )-\log (64)+x \left (9-\sqrt {e}-6 \log ^3(2)-\log ^2(2) (35-\log (64))+\log (64)+\log ^2(64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^2 \left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2}\right ) \, dx \\ & = \frac {6 \int \frac {e^x \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\sqrt {e} \left (37+24 \log (2)+4 \log ^2(2)\right )-\log (64)+x \left (9-\sqrt {e}-6 \log ^3(2)-\log ^2(2) (35-\log (64))+\log (64)+\log ^2(64)\right )\right )}{\left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^2}+\frac {\left (6 \left (-9+\sqrt {e}-\log ^2(2)-\log (64)\right )\right ) \int \frac {e^x}{x^2} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {\left (6 \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )\right ) \int \frac {e^x}{x} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}+\frac {\left (6 \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )\right ) \int \frac {e^x x}{\sqrt {e}+x+x^2-(3+\log (2))^2} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^3} \\ & = \frac {6 e^x \left (9-\sqrt {e}+\log ^2(2)+\log (64)\right )}{x \left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {6 \text {Ei}(x) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}+\frac {6 \int \left (\frac {171 e^x \left (1+\frac {1}{171} \left (2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\sqrt {e} \left (37+24 \log (2)+4 \log ^2(2)\right )-\log (64)\right )\right )}{\left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2}+\frac {e^x x \left (9-\sqrt {e}-6 \log ^3(2)-\log ^2(2) (35-\log (64))+\log (64)+\log ^2(64)\right )}{\left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2}\right ) \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^2}+\frac {\left (6 \left (-9+\sqrt {e}-\log ^2(2)-\log (64)\right )\right ) \int \frac {e^x}{x} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^2}+\frac {\left (6 \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )\right ) \int \left (\frac {e^x \left (1-\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right )}{1+2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}+\frac {e^x \left (1+\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right )}{1+2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right ) \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^3} \\ & = \frac {6 e^x \left (9-\sqrt {e}+\log ^2(2)+\log (64)\right )}{x \left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {6 \text {Ei}(x) \left (9-\sqrt {e}+\log ^2(2)+\log (64)\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {6 \text {Ei}(x) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}-\frac {6 \int \frac {e^x x}{\left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2} \, dx}{\sqrt {e}-(3+\log (2))^2}+\frac {\left (6 \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\sqrt {e} \left (37+24 \log (2)+4 \log ^2(2)\right )-\log (64)\right )\right ) \int \frac {e^x}{\left (\sqrt {e}+x+x^2-(3+\log (2))^2\right )^2} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^2}+\frac {\left (6 \left (1-\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right ) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )\right ) \int \frac {e^x}{1+2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}+\frac {\left (6 \left (1+\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right ) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )\right ) \int \frac {e^x}{1+2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}} \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^3} \\ & = \frac {6 e^x \left (9-\sqrt {e}+\log ^2(2)+\log (64)\right )}{x \left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {6 \text {Ei}(x) \left (9-\sqrt {e}+\log ^2(2)+\log (64)\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {6 \text {Ei}(x) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}+\frac {3 e^{\frac {1}{2} \left (-1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \text {Ei}\left (\frac {1}{2} \left (1+2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )\right ) \left (1-\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right ) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}+\frac {3 e^{\frac {1}{2} \left (-1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \text {Ei}\left (\frac {1}{2} \left (1+2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )\right ) \left (1+\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right ) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}-\frac {6 \int \left (\frac {2 e^x \left (-1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (-1-2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )^2}-\frac {2 e^x}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (-1-2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}+\frac {2 e^x \left (-1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (1+2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )^2}-\frac {2 e^x}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (1+2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}\right ) \, dx}{\sqrt {e}-(3+\log (2))^2}+\frac {\left (6 \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\sqrt {e} \left (37+24 \log (2)+4 \log ^2(2)\right )-\log (64)\right )\right ) \int \left (\frac {4 e^x}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (-1-2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )^2}+\frac {4 e^x}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (-1-2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}+\frac {4 e^x}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (1+2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )^2}+\frac {4 e^x}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (1+2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}\right ) \, dx}{\left (\sqrt {e}-(3+\log (2))^2\right )^2} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=\int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx \]

[In]

Integrate[(E^x*(-54 + Sqrt[E]*(6 - 6*x) + 66*x + 12*x^2 - 6*x^3 + (-36 + 36*x)*Log[2] + (-6 + 6*x)*Log[2]^2))/
(81*x^2 + E*x^2 - 18*x^3 - 17*x^4 + 2*x^5 + x^6 + Sqrt[E]*(-18*x^2 + 2*x^3 + 2*x^4) + (108*x^2 - 12*Sqrt[E]*x^
2 - 12*x^3 - 12*x^4)*Log[2] + (54*x^2 - 2*Sqrt[E]*x^2 - 2*x^3 - 2*x^4)*Log[2]^2 + 12*x^2*Log[2]^3 + x^2*Log[2]
^4),x]

[Out]

Integrate[(E^x*(-54 + Sqrt[E]*(6 - 6*x) + 66*x + 12*x^2 - 6*x^3 + (-36 + 36*x)*Log[2] + (-6 + 6*x)*Log[2]^2))/
(81*x^2 + E*x^2 - 18*x^3 - 17*x^4 + 2*x^5 + x^6 + Sqrt[E]*(-18*x^2 + 2*x^3 + 2*x^4) + (108*x^2 - 12*Sqrt[E]*x^
2 - 12*x^3 - 12*x^4)*Log[2] + (54*x^2 - 2*Sqrt[E]*x^2 - 2*x^3 - 2*x^4)*Log[2]^2 + 12*x^2*Log[2]^3 + x^2*Log[2]
^4), x]

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88

method	result	size
gosper	\(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\)	\(28\)
norman	\(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\)	\(28\)
risch	\(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\)	\(28\)
parallelrisch	\(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\)	\(28\)
default	\(\text {Expression too large to display}\)	\(40397\)

[In]

int(((6*x-6)*ln(2)^2+(36*x-36)*ln(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2+66*x-54)*exp(x)/(x^2*ln(2)^4+12*x^2*ln(2)^3
+(-2*x^2*exp(1/2)-2*x^4-2*x^3+54*x^2)*ln(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*ln(2)+x^2*exp(1/2)^2+(2
*x^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x,method=_RETURNVERBOSE)

[Out]

-6*exp(x)/x/(-ln(2)^2+x^2+exp(1/2)-6*ln(2)+x-9)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {6 \, e^{x}}{x^{3} - x \log \left (2\right )^{2} + x^{2} + x e^{\frac {1}{2}} - 6 \, x \log \left (2\right ) - 9 \, x} \]

[In]

integrate(((6*x-6)*log(2)^2+(36*x-36)*log(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2+66*x-54)*exp(x)/(x^2*log(2)^4+12*x^
2*log(2)^3+(-2*x^2*exp(1/2)-2*x^4-2*x^3+54*x^2)*log(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*log(2)+x^2*e
xp(1/2)^2+(2*x^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x, algorithm="fricas")

[Out]

-6*e^x/(x^3 - x*log(2)^2 + x^2 + x*e^(1/2) - 6*x*log(2) - 9*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=\text {Timed out} \]

[In]

integrate(((6*x-6)*ln(2)**2+(36*x-36)*ln(2)+(6-6*x)*exp(1/2)-6*x**3+12*x**2+66*x-54)*exp(x)/(x**2*ln(2)**4+12*
x**2*ln(2)**3+(-2*x**2*exp(1/2)-2*x**4-2*x**3+54*x**2)*ln(2)**2+(-12*x**2*exp(1/2)-12*x**4-12*x**3+108*x**2)*l
n(2)+x**2*exp(1/2)**2+(2*x**4+2*x**3-18*x**2)*exp(1/2)+x**6+2*x**5-17*x**4-18*x**3+81*x**2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {6 \, e^{x}}{x^{3} - {\left (\log \left (2\right )^{2} - e^{\frac {1}{2}} + 6 \, \log \left (2\right ) + 9\right )} x + x^{2}} \]

[In]

integrate(((6*x-6)*log(2)^2+(36*x-36)*log(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2+66*x-54)*exp(x)/(x^2*log(2)^4+12*x^
2*log(2)^3+(-2*x^2*exp(1/2)-2*x^4-2*x^3+54*x^2)*log(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*log(2)+x^2*e
xp(1/2)^2+(2*x^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x, algorithm="maxima")

[Out]

-6*e^x/(x^3 - (log(2)^2 - e^(1/2) + 6*log(2) + 9)*x + x^2)

Giac [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {12 \, e^{x}}{x^{3} - x \log \left (2\right )^{2} + x^{2} + x e^{\frac {1}{2}} - 6 \, x \log \left (2\right ) - 9 \, x} \]

[In]

integrate(((6*x-6)*log(2)^2+(36*x-36)*log(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2+66*x-54)*exp(x)/(x^2*log(2)^4+12*x^
2*log(2)^3+(-2*x^2*exp(1/2)-2*x^4-2*x^3+54*x^2)*log(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*log(2)+x^2*e
xp(1/2)^2+(2*x^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x, algorithm="giac")

[Out]

-12*e^x/(x^3 - x*log(2)^2 + x^2 + x*e^(1/2) - 6*x*log(2) - 9*x)

Mupad [B] (verification not implemented)

Time = 15.58 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {6\,{\mathrm {e}}^x}{x^3+x^2+\left (\sqrt {\mathrm {e}}-\ln \left (64\right )-{\ln \left (2\right )}^2-9\right )\,x} \]

[In]

int((exp(x)*(66*x + log(2)*(36*x - 36) + log(2)^2*(6*x - 6) + 12*x^2 - 6*x^3 - exp(1/2)*(6*x - 6) - 54))/(12*x
^2*log(2)^3 + x^2*log(2)^4 - log(2)*(12*x^2*exp(1/2) - 108*x^2 + 12*x^3 + 12*x^4) + x^2*exp(1) + exp(1/2)*(2*x
^3 - 18*x^2 + 2*x^4) - log(2)^2*(2*x^2*exp(1/2) - 54*x^2 + 2*x^3 + 2*x^4) + 81*x^2 - 18*x^3 - 17*x^4 + 2*x^5 +
x^6),x)

[Out]

-(6*exp(x))/(x^2 - x*(log(64) - exp(1/2) + log(2)^2 + 9) + x^3)