∫ 4x2+24x4+8x5+e8(24x2+8x3)+e4(−8+48x3+16x4)+ex(−8+4x2+e4(4+4x))_______________________________________________________ 4x2+4x3+x4+e8(4+4x+x2)+e4(8x+8x2+2x3) dx [8497]

Optimal. Leaf size=29 \[ \frac {4 x \left (x^2+\frac {-1+\frac {e^x}{x}}{e^4+x}\right )}{2+x} \]

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. $487$ vs. $2(29)=58$.
time = 1.07, antiderivative size = 487, normalized size of antiderivative = 16.79, number of steps used = 22, number of rules used = 8, integrand size = 114, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.070, Rules used = {6820, 12, 6874, 90, 153, 2208, 2209, 1634} \begin {gather*} 4 x^2-16 \left (2+e^4\right ) x+16 e^4 x+24 x-\frac {4 e^x}{\left (2-e^4\right ) (x+2)}-\frac {32 e^8}{\left (2-e^4\right )^2 (x+2)}+\frac {136 e^4}{\left (2-e^4\right )^2 (x+2)}-\frac {144}{\left (2-e^4\right )^2 (x+2)}+\frac {8 e^4 \left (1+6 e^{12}-2 e^{16}\right )}{\left (2-e^4\right )^2 \left (x+e^4\right )}-\frac {8 e^{16} \left (3-e^4\right )}{\left (2-e^4\right )^2 \left (x+e^4\right )}+\frac {4 e^x}{\left (2-e^4\right ) \left (x+e^4\right )}+\frac {8 e^{20}}{\left (2-e^4\right )^2 \left (x+e^4\right )}-\frac {24 e^{16}}{\left (2-e^4\right )^2 \left (x+e^4\right )}-\frac {4 e^8}{\left (2-e^4\right )^2 \left (x+e^4\right )}-\frac {16 e^4 \left (9+4 e^4\right ) \log (x+2)}{\left (2-e^4\right )^3}-\frac {768 \left (1-e^4\right ) \log (x+2)}{\left (2-e^4\right )^3}+\frac {128 \left (6-5 e^4\right ) \log (x+2)}{\left (2-e^4\right )^3}+\frac {64 e^8 \log (x+2)}{\left (2-e^4\right )^3}+\frac {16 e^4 \log (x+2)}{\left (2-e^4\right )^3}+\frac {16 e^4 \left (1+18 e^8-11 e^{12}+2 e^{16}\right ) \log \left (x+e^4\right )}{\left (2-e^4\right )^3}-\frac {8 e^{12} \left (12-6 e^4+e^8\right ) \log \left (x+e^4\right )}{\left (2-e^4\right )^3}-\frac {48 e^{12} \left (4-e^4\right ) \log \left (x+e^4\right )}{\left (2-e^4\right )^3}+\frac {8 e^{16} \left (10-3 e^4\right ) \log \left (x+e^4\right )}{\left (2-e^4\right )^3}-\frac {16 e^4 \log \left (x+e^4\right )}{\left (2-e^4\right )^3} \end {gather*}

Int[(4*x^2 + 24*x^4 + 8*x^5 + E^8*(24*x^2 + 8*x^3) + E^4*(-8 + 48*x^3 + 16*x^4) + E^x*(-8 + 4*x^2 + E^4*(4 + 4

24*x + 16*E^4*x - 16*(2 + E^4)*x + 4*x^2 - 144/((2 - E^4)^2*(2 + x)) + (136*E^4)/((2 - E^4)^2*(2 + x)) - (32*E

^8)/((2 - E^4)^2*(2 + x)) - (4*E^x)/((2 - E^4)*(2 + x)) - (4*E^8)/((2 - E^4)^2*(E^4 + x)) - (24*E^16)/((2 - E^

4)^2*(E^4 + x)) + (8*E^20)/((2 - E^4)^2*(E^4 + x)) + (4*E^x)/((2 - E^4)*(E^4 + x)) - (8*E^16*(3 - E^4))/((2 -

E^4)^2*(E^4 + x)) + (8*E^4*(1 + 6*E^12 - 2*E^16))/((2 - E^4)^2*(E^4 + x)) + (16*E^4*Log[2 + x])/(2 - E^4)^3 +

(64*E^8*Log[2 + x])/(2 - E^4)^3 + (128*(6 - 5*E^4)*Log[2 + x])/(2 - E^4)^3 - (768*(1 - E^4)*Log[2 + x])/(2 - E

^4)^3 - (16*E^4*(9 + 4*E^4)*Log[2 + x])/(2 - E^4)^3 - (16*E^4*Log[E^4 + x])/(2 - E^4)^3 + (8*E^16*(10 - 3*E^4)

*Log[E^4 + x])/(2 - E^4)^3 - (48*E^12*(4 - E^4)*Log[E^4 + x])/(2 - E^4)^3 - (8*E^12*(12 - 6*E^4 + E^8)*Log[E^4

+ x])/(2 - E^4)^3 + (16*E^4*(1 + 18*E^8 - 11*E^12 + 2*E^16)*Log[E^4 + x])/(2 - E^4)^3

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

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI

ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb

ol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)

^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&

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

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]

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (x^2+6 x^4+2 x^5+e^{4+x} (1+x)+2 e^8 x^2 (3+x)+e^x \left (-2+x^2\right )+2 e^4 \left (-1+6 x^3+2 x^4\right )\right )}{(2+x)^2 \left (e^4+x\right )^2} \, dx\\ &=4 \int \frac {x^2+6 x^4+2 x^5+e^{4+x} (1+x)+2 e^8 x^2 (3+x)+e^x \left (-2+x^2\right )+2 e^4 \left (-1+6 x^3+2 x^4\right )}{(2+x)^2 \left (e^4+x\right )^2} \, dx\\ &=4 \int \left (\frac {x^2}{(2+x)^2 \left (e^4+x\right )^2}+\frac {6 x^4}{(2+x)^2 \left (e^4+x\right )^2}+\frac {2 x^5}{(2+x)^2 \left (e^4+x\right )^2}+\frac {2 e^8 x^2 (3+x)}{(2+x)^2 \left (e^4+x\right )^2}+\frac {e^x \left (-2+e^4+e^4 x+x^2\right )}{(2+x)^2 \left (e^4+x\right )^2}+\frac {2 e^4 \left (-1+6 x^3+2 x^4\right )}{(2+x)^2 \left (e^4+x\right )^2}\right ) \, dx\\ &=4 \int \frac {x^2}{(2+x)^2 \left (e^4+x\right )^2} \, dx+4 \int \frac {e^x \left (-2+e^4+e^4 x+x^2\right )}{(2+x)^2 \left (e^4+x\right )^2} \, dx+8 \int \frac {x^5}{(2+x)^2 \left (e^4+x\right )^2} \, dx+24 \int \frac {x^4}{(2+x)^2 \left (e^4+x\right )^2} \, dx+\left (8 e^4\right ) \int \frac {-1+6 x^3+2 x^4}{(2+x)^2 \left (e^4+x\right )^2} \, dx+\left (8 e^8\right ) \int \frac {x^2 (3+x)}{(2+x)^2 \left (e^4+x\right )^2} \, dx\\ &=4 \int \left (\frac {4}{\left (-2+e^4\right )^2 (2+x)^2}-\frac {4 e^4}{\left (-2+e^4\right )^3 (2+x)}+\frac {e^8}{\left (-2+e^4\right )^2 \left (e^4+x\right )^2}+\frac {4 e^4}{\left (-2+e^4\right )^3 \left (e^4+x\right )}\right ) \, dx+4 \int \left (-\frac {e^x}{\left (-2+e^4\right ) (2+x)^2}+\frac {e^x}{\left (-2+e^4\right ) (2+x)}+\frac {e^x}{\left (-2+e^4\right ) \left (e^4+x\right )^2}-\frac {e^x}{\left (-2+e^4\right ) \left (e^4+x\right )}\right ) \, dx+8 \int \left (-2 \left (2+e^4\right )+x-\frac {32}{\left (-2+e^4\right )^2 (2+x)^2}+\frac {16 \left (-6+5 e^4\right )}{\left (-2+e^4\right )^3 (2+x)}-\frac {e^{20}}{\left (-2+e^4\right )^2 \left (e^4+x\right )^2}+\frac {e^{16} \left (-10+3 e^4\right )}{\left (-2+e^4\right )^3 \left (e^4+x\right )}\right ) \, dx+24 \int \left (1+\frac {16}{\left (-2+e^4\right )^2 (2+x)^2}-\frac {32 \left (-1+e^4\right )}{\left (-2+e^4\right )^3 (2+x)}+\frac {e^{16}}{\left (-2+e^4\right )^2 \left (e^4+x\right )^2}-\frac {2 e^{12} \left (-4+e^4\right )}{\left (-2+e^4\right )^3 \left (e^4+x\right )}\right ) \, dx+\left (8 e^4\right ) \int \left (2-\frac {17}{\left (-2+e^4\right )^2 (2+x)^2}+\frac {2 \left (9+4 e^4\right )}{\left (-2+e^4\right )^3 (2+x)}+\frac {-1-6 e^{12}+2 e^{16}}{\left (-2+e^4\right )^2 \left (e^4+x\right )^2}-\frac {2 \left (1+18 e^8-11 e^{12}+2 e^{16}\right )}{\left (-2+e^4\right )^3 \left (e^4+x\right )}\right ) \, dx+\left (8 e^8\right ) \int \left (\frac {4}{\left (-2+e^4\right )^2 (2+x)^2}-\frac {8}{\left (-2+e^4\right )^3 (2+x)}-\frac {e^8 \left (-3+e^4\right )}{\left (-2+e^4\right )^2 \left (e^4+x\right )^2}+\frac {e^4 \left (12-6 e^4+e^8\right )}{\left (-2+e^4\right )^3 \left (e^4+x\right )}\right ) \, dx\\ &=24 x+16 e^4 x-16 \left (2+e^4\right ) x+4 x^2-\frac {144}{\left (2-e^4\right )^2 (2+x)}+\frac {136 e^4}{\left (2-e^4\right )^2 (2+x)}-\frac {32 e^8}{\left (2-e^4\right )^2 (2+x)}-\frac {4 e^8}{\left (2-e^4\right )^2 \left (e^4+x\right )}-\frac {24 e^{16}}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {8 e^{20}}{\left (2-e^4\right )^2 \left (e^4+x\right )}-\frac {8 e^{16} \left (3-e^4\right )}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {8 e^4 \left (1+6 e^{12}-2 e^{16}\right )}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {16 e^4 \log (2+x)}{\left (2-e^4\right )^3}+\frac {64 e^8 \log (2+x)}{\left (2-e^4\right )^3}+\frac {128 \left (6-5 e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {768 \left (1-e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {16 e^4 \left (9+4 e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {16 e^4 \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {8 e^{16} \left (10-3 e^4\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}-\frac {48 e^{12} \left (4-e^4\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}-\frac {8 e^{12} \left (12-6 e^4+e^8\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {16 e^4 \left (1+18 e^8-11 e^{12}+2 e^{16}\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {4 \int \frac {e^x}{(2+x)^2} \, dx}{2-e^4}-\frac {4 \int \frac {e^x}{2+x} \, dx}{2-e^4}-\frac {4 \int \frac {e^x}{\left (e^4+x\right )^2} \, dx}{2-e^4}+\frac {4 \int \frac {e^x}{e^4+x} \, dx}{2-e^4}\\ &=24 x+16 e^4 x-16 \left (2+e^4\right ) x+4 x^2-\frac {144}{\left (2-e^4\right )^2 (2+x)}+\frac {136 e^4}{\left (2-e^4\right )^2 (2+x)}-\frac {32 e^8}{\left (2-e^4\right )^2 (2+x)}-\frac {4 e^x}{\left (2-e^4\right ) (2+x)}-\frac {4 e^8}{\left (2-e^4\right )^2 \left (e^4+x\right )}-\frac {24 e^{16}}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {8 e^{20}}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {4 e^x}{\left (2-e^4\right ) \left (e^4+x\right )}-\frac {8 e^{16} \left (3-e^4\right )}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {8 e^4 \left (1+6 e^{12}-2 e^{16}\right )}{\left (2-e^4\right )^2 \left (e^4+x\right )}-\frac {4 \text {Ei}(2+x)}{e^2 \left (2-e^4\right )}+\frac {4 e^{-e^4} \text {Ei}\left (e^4+x\right )}{2-e^4}+\frac {16 e^4 \log (2+x)}{\left (2-e^4\right )^3}+\frac {64 e^8 \log (2+x)}{\left (2-e^4\right )^3}+\frac {128 \left (6-5 e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {768 \left (1-e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {16 e^4 \left (9+4 e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {16 e^4 \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {8 e^{16} \left (10-3 e^4\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}-\frac {48 e^{12} \left (4-e^4\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}-\frac {8 e^{12} \left (12-6 e^4+e^8\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {16 e^4 \left (1+18 e^8-11 e^{12}+2 e^{16}\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {4 \int \frac {e^x}{2+x} \, dx}{2-e^4}-\frac {4 \int \frac {e^x}{e^4+x} \, dx}{2-e^4}\\ &=24 x+16 e^4 x-16 \left (2+e^4\right ) x+4 x^2-\frac {144}{\left (2-e^4\right )^2 (2+x)}+\frac {136 e^4}{\left (2-e^4\right )^2 (2+x)}-\frac {32 e^8}{\left (2-e^4\right )^2 (2+x)}-\frac {4 e^x}{\left (2-e^4\right ) (2+x)}-\frac {4 e^8}{\left (2-e^4\right )^2 \left (e^4+x\right )}-\frac {24 e^{16}}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {8 e^{20}}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {4 e^x}{\left (2-e^4\right ) \left (e^4+x\right )}-\frac {8 e^{16} \left (3-e^4\right )}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {8 e^4 \left (1+6 e^{12}-2 e^{16}\right )}{\left (2-e^4\right )^2 \left (e^4+x\right )}+\frac {16 e^4 \log (2+x)}{\left (2-e^4\right )^3}+\frac {64 e^8 \log (2+x)}{\left (2-e^4\right )^3}+\frac {128 \left (6-5 e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {768 \left (1-e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {16 e^4 \left (9+4 e^4\right ) \log (2+x)}{\left (2-e^4\right )^3}-\frac {16 e^4 \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {8 e^{16} \left (10-3 e^4\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}-\frac {48 e^{12} \left (4-e^4\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}-\frac {8 e^{12} \left (12-6 e^4+e^8\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}+\frac {16 e^4 \left (1+18 e^8-11 e^{12}+2 e^{16}\right ) \log \left (e^4+x\right )}{\left (2-e^4\right )^3}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 3.23, size = 40, normalized size = 1.38 \begin {gather*} \frac {4 \left (e^x+x \left (-9-4 x+x^3\right )+e^4 \left (-8-4 x+x^3\right )\right )}{(2+x) \left (e^4+x\right )} \end {gather*}

Integrate[(4*x^2 + 24*x^4 + 8*x^5 + E^8*(24*x^2 + 8*x^3) + E^4*(-8 + 48*x^3 + 16*x^4) + E^x*(-8 + 4*x^2 + E^4*

(4 + 4*x)))/(4*x^2 + 4*x^3 + x^4 + E^8*(4 + 4*x + x^2) + E^4*(8*x + 8*x^2 + 2*x^3)),x]

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 2.53, size = 1090, normalized size = 37.59


method	result	size

norman	$\frac {-4 x +4 x^{4}+4 x^{3} {\mathrm e}^{4}+4 \,{\mathrm e}^{x}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}$	$33$
risch	$4 x^{2}-8 x +\frac {-32 \,{\mathrm e}^{4}-36 x}{x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x}+\frac {4 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x}$	$57$
default	$\text {Expression too large to display}$	$1090$

int((((4*x+4)*exp(4)+4*x^2-8)*exp(x)+(8*x^3+24*x^2)*exp(4)^2+(16*x^4+48*x^3-8)*exp(4)+8*x^5+24*x^4+4*x^2)/((x^

2+4*x+4)*exp(4)^2+(2*x^3+8*x^2+8*x)*exp(4)+x^4+4*x^3+4*x^2),x,method=_RETURNVERBOSE)

((-16-4*exp(4)^2)/(exp(4)^2-4*exp(4)+4)*x+2*(-8-4*exp(4))*exp(4)/(exp(4)^2-4*exp(4)+4))/(2+x)/(x+exp(4))+(24*x

^3-2*(48*exp(4)^3+384-48*exp(4)^2-96*exp(4))*exp(4)/(exp(4)^2-4*exp(4)+4)-(48*exp(4)^4+768-48*exp(4)^3-192*exp

(4))/(exp(4)^2-4*exp(4)+4)*x)/(2+x)/(x+exp(4))+((-24-12*exp(4))*x^3+(24*exp(4)^5-8*exp(4)^4+768-48*exp(4)^3-96

*exp(4)^2-64*exp(4))/(exp(4)^2-4*exp(4)+4)*x+4*x^4+2*(24*exp(4)^4-8*exp(4)^3+384-96*exp(4)^2-32*exp(4))*exp(4)

/(exp(4)^2-4*exp(4)+4))/(2+x)/(x+exp(4))+(16*exp(4)/(exp(4)^2-4*exp(4)+4)*x+2*(4*exp(4)+8)*exp(4)/(exp(4)^2-4*

exp(4)+4))/(2+x)/(x+exp(4))+8*exp(x)*(exp(4)+2*x+2)/(exp(4)^2-4*exp(4)+4)/(x*exp(4)+x^2+2*exp(4)+2*x)-8*exp(4)

/(exp(4)-2)/(4*exp(4)-4-exp(8))*exp(-exp(4))*Ei(1,-exp(4)-x)-8/(4*exp(4)-4-exp(8))*(exp(4)-4)/(exp(4)-2)*exp(-

2)*Ei(1,-x-2)+(-2*(24*exp(4)^3+48*exp(4)^2)*exp(4)/(exp(4)^2-4*exp(4)+4)-(24*exp(4)^4+96*exp(4)^2)/(exp(4)^2-4

*exp(4)+4)*x)/(2+x)/(x+exp(4))+((48*exp(4)^4+384*exp(4))/(exp(4)^2-4*exp(4)+4)*x+2*(48*exp(4)^3+192*exp(4))*ex

p(4)/(exp(4)^2-4*exp(4)+4))/(2+x)/(x+exp(4))+((8*exp(4)^5+64*exp(4)^2)/(exp(4)^2-4*exp(4)+4)*x+2*(8*exp(4)^4+3

2*exp(4)^2)*exp(4)/(exp(4)^2-4*exp(4)+4))/(2+x)/(x+exp(4))+(16*x^3*exp(4)-2*(32*exp(4)^4-32*exp(4)^3-64*exp(4)

^2+256*exp(4))*exp(4)/(exp(4)^2-4*exp(4)+4)-(32*exp(4)^5-32*exp(4)^4-128*exp(4)^2+512*exp(4))/(exp(4)^2-4*exp(

4)+4)*x)/(2+x)/(x+exp(4))+4*exp(4)*(-exp(x)*(exp(4)+2*x+2)/(exp(4)^2-4*exp(4)+4)/(x*exp(4)+x^2+2*exp(4)+2*x)+e

xp(4)/(exp(4)-2)/(4*exp(4)-4-exp(8))*exp(-exp(4))*Ei(1,-exp(4)-x)+1/(4*exp(4)-4-exp(8))*(exp(4)-4)/(exp(4)-2)*

exp(-2)*Ei(1,-x-2))-4*exp(x)*(x*exp(4)^2+2*exp(4)^2+4*exp(4)+4*x)/(exp(4)^2-4*exp(4)+4)/(x*exp(4)+x^2+2*exp(4)

+2*x)+4/(4*exp(4)-4-exp(8))*(exp(4)*exp(8)-2*exp(8)+4*exp(4))/(exp(4)-2)*exp(-exp(4))*Ei(1,-exp(4)-x)-32/(exp(

4)-2)/(4*exp(4)-4-exp(8))*exp(-2)*Ei(1,-x-2)+4*exp(4)*(exp(x)*(x*exp(4)+4*exp(4)+2*x)/(exp(4)^2-4*exp(4)+4)/(x

*exp(4)+x^2+2*exp(4)+2*x)-1/(4*exp(4)-4-exp(8))*(exp(4)^2-exp(4)+2)/(exp(4)-2)*exp(-exp(4))*Ei(1,-exp(4)-x)-1/

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 912 vs. $2 (27) = 54$.
time = 0.34, size = 912, normalized size = 31.45 \begin {gather*} 4 \, x^{2} - 16 \, x {\left (e^{4} + 2\right )} + 8 \, {\left (\frac {{\left (e^{12} - 6 \, e^{8}\right )} \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {4 \, {\left (3 \, e^{4} - 2\right )} \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {x {\left (e^{12} + 8\right )} + 2 \, e^{12} + 8 \, e^{4}}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}}\right )} e^{8} + 24 \, {\left (\frac {4 \, e^{4} \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} - \frac {4 \, e^{4} \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} - \frac {x {\left (e^{8} + 4\right )} + 2 \, e^{8} + 4 \, e^{4}}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}}\right )} e^{8} + 16 \, {\left (x - \frac {2 \, {\left (e^{16} - 4 \, e^{12}\right )} \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} - \frac {32 \, {\left (e^{4} - 1\right )} \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} - \frac {x {\left (e^{16} + 16\right )} + 2 \, e^{16} + 16 \, e^{4}}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}}\right )} e^{4} + 48 \, {\left (\frac {{\left (e^{12} - 6 \, e^{8}\right )} \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {4 \, {\left (3 \, e^{4} - 2\right )} \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {x {\left (e^{12} + 8\right )} + 2 \, e^{12} + 8 \, e^{4}}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}}\right )} e^{4} + 8 \, {\left (\frac {2 \, x + e^{4} + 2}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}} - \frac {2 \, \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {2 \, \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8}\right )} e^{4} + 24 \, x + \frac {8 \, {\left (3 \, e^{20} - 10 \, e^{16}\right )} \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} - \frac {48 \, {\left (e^{16} - 4 \, e^{12}\right )} \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {16 \, e^{4} \log \left (x + e^{4}\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {128 \, {\left (5 \, e^{4} - 6\right )} \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} - \frac {768 \, {\left (e^{4} - 1\right )} \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} - \frac {16 \, e^{4} \log \left (x + 2\right )}{e^{12} - 6 \, e^{8} + 12 \, e^{4} - 8} + \frac {8 \, {\left (x {\left (e^{20} + 32\right )} + 2 \, e^{20} + 32 \, e^{4}\right )}}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}} - \frac {24 \, {\left (x {\left (e^{16} + 16\right )} + 2 \, e^{16} + 16 \, e^{4}\right )}}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}} - \frac {4 \, {\left (x {\left (e^{8} + 4\right )} + 2 \, e^{8} + 4 \, e^{4}\right )}}{x^{2} {\left (e^{8} - 4 \, e^{4} + 4\right )} + x {\left (e^{12} - 2 \, e^{8} - 4 \, e^{4} + 8\right )} + 2 \, e^{12} - 8 \, e^{8} + 8 \, e^{4}} + \frac {4 \, e^{x}}{x^{2} + x {\left (e^{4} + 2\right )} + 2 \, e^{4}} \end {gather*}

integrate((((4*x+4)*exp(4)+4*x^2-8)*exp(x)+(8*x^3+24*x^2)*exp(4)^2+(16*x^4+48*x^3-8)*exp(4)+8*x^5+24*x^4+4*x^2

)/((x^2+4*x+4)*exp(4)^2+(2*x^3+8*x^2+8*x)*exp(4)+x^4+4*x^3+4*x^2),x, algorithm="maxima")

4*x^2 - 16*x*(e^4 + 2) + 8*((e^12 - 6*e^8)*log(x + e^4)/(e^12 - 6*e^8 + 12*e^4 - 8) + 4*(3*e^4 - 2)*log(x + 2)

/(e^12 - 6*e^8 + 12*e^4 - 8) + (x*(e^12 + 8) + 2*e^12 + 8*e^4)/(x^2*(e^8 - 4*e^4 + 4) + x*(e^12 - 2*e^8 - 4*e^

4 + 8) + 2*e^12 - 8*e^8 + 8*e^4))*e^8 + 24*(4*e^4*log(x + e^4)/(e^12 - 6*e^8 + 12*e^4 - 8) - 4*e^4*log(x + 2)/

(e^12 - 6*e^8 + 12*e^4 - 8) - (x*(e^8 + 4) + 2*e^8 + 4*e^4)/(x^2*(e^8 - 4*e^4 + 4) + x*(e^12 - 2*e^8 - 4*e^4 +

8) + 2*e^12 - 8*e^8 + 8*e^4))*e^8 + 16*(x - 2*(e^16 - 4*e^12)*log(x + e^4)/(e^12 - 6*e^8 + 12*e^4 - 8) - 32*(

e^4 - 1)*log(x + 2)/(e^12 - 6*e^8 + 12*e^4 - 8) - (x*(e^16 + 16) + 2*e^16 + 16*e^4)/(x^2*(e^8 - 4*e^4 + 4) + x

*(e^12 - 2*e^8 - 4*e^4 + 8) + 2*e^12 - 8*e^8 + 8*e^4))*e^4 + 48*((e^12 - 6*e^8)*log(x + e^4)/(e^12 - 6*e^8 + 1

2*e^4 - 8) + 4*(3*e^4 - 2)*log(x + 2)/(e^12 - 6*e^8 + 12*e^4 - 8) + (x*(e^12 + 8) + 2*e^12 + 8*e^4)/(x^2*(e^8

- 4*e^4 + 4) + x*(e^12 - 2*e^8 - 4*e^4 + 8) + 2*e^12 - 8*e^8 + 8*e^4))*e^4 + 8*((2*x + e^4 + 2)/(x^2*(e^8 - 4*

e^4 + 4) + x*(e^12 - 2*e^8 - 4*e^4 + 8) + 2*e^12 - 8*e^8 + 8*e^4) - 2*log(x + e^4)/(e^12 - 6*e^8 + 12*e^4 - 8)

+ 2*log(x + 2)/(e^12 - 6*e^8 + 12*e^4 - 8))*e^4 + 24*x + 8*(3*e^20 - 10*e^16)*log(x + e^4)/(e^12 - 6*e^8 + 12

*e^4 - 8) - 48*(e^16 - 4*e^12)*log(x + e^4)/(e^12 - 6*e^8 + 12*e^4 - 8) + 16*e^4*log(x + e^4)/(e^12 - 6*e^8 +

12*e^4 - 8) + 128*(5*e^4 - 6)*log(x + 2)/(e^12 - 6*e^8 + 12*e^4 - 8) - 768*(e^4 - 1)*log(x + 2)/(e^12 - 6*e^8

+ 12*e^4 - 8) - 16*e^4*log(x + 2)/(e^12 - 6*e^8 + 12*e^4 - 8) + 8*(x*(e^20 + 32) + 2*e^20 + 32*e^4)/(x^2*(e^8

- 4*e^4 + 4) + x*(e^12 - 2*e^8 - 4*e^4 + 8) + 2*e^12 - 8*e^8 + 8*e^4) - 24*(x*(e^16 + 16) + 2*e^16 + 16*e^4)/(

x^2*(e^8 - 4*e^4 + 4) + x*(e^12 - 2*e^8 - 4*e^4 + 8) + 2*e^12 - 8*e^8 + 8*e^4) - 4*(x*(e^8 + 4) + 2*e^8 + 4*e^

4)/(x^2*(e^8 - 4*e^4 + 4) + x*(e^12 - 2*e^8 - 4*e^4 + 8) + 2*e^12 - 8*e^8 + 8*e^4) + 4*e^x/(x^2 + x*(e^4 + 2)

________________________________________________________________________________________

Fricas [A]
time = 0.52, size = 42, normalized size = 1.45 \begin {gather*} \frac {4 \, {\left (x^{4} - 4 \, x^{2} + {\left (x^{3} - 4 \, x - 8\right )} e^{4} - 9 \, x + e^{x}\right )}}{x^{2} + {\left (x + 2\right )} e^{4} + 2 \, x} \end {gather*}

integrate((((4*x+4)*exp(4)+4*x^2-8)*exp(x)+(8*x^3+24*x^2)*exp(4)^2+(16*x^4+48*x^3-8)*exp(4)+8*x^5+24*x^4+4*x^2

)/((x^2+4*x+4)*exp(4)^2+(2*x^3+8*x^2+8*x)*exp(4)+x^4+4*x^3+4*x^2),x, algorithm="fricas")

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. $2 (22) = 44$.
time = 0.43, size = 54, normalized size = 1.86 \begin {gather*} 4 x^{2} - 8 x + \frac {- 36 x - 32 e^{4}}{x^{2} + x \left (2 + e^{4}\right ) + 2 e^{4}} + \frac {4 e^{x}}{x^{2} + 2 x + x e^{4} + 2 e^{4}} \end {gather*}

integrate((((4*x+4)*exp(4)+4*x**2-8)*exp(x)+(8*x**3+24*x**2)*exp(4)**2+(16*x**4+48*x**3-8)*exp(4)+8*x**5+24*x*

*4+4*x**2)/((x**2+4*x+4)*exp(4)**2+(2*x**3+8*x**2+8*x)*exp(4)+x**4+4*x**3+4*x**2),x)

4*x**2 - 8*x + (-36*x - 32*exp(4))/(x**2 + x*(2 + exp(4)) + 2*exp(4)) + 4*exp(x)/(x**2 + 2*x + x*exp(4) + 2*ex

________________________________________________________________________________________

Giac [A]
time = 0.47, size = 48, normalized size = 1.66 \begin {gather*} \frac {4 \, {\left (x^{4} + x^{3} e^{4} - 4 \, x^{2} - 4 \, x e^{4} - 9 \, x - 8 \, e^{4} + e^{x}\right )}}{x^{2} + x e^{4} + 2 \, x + 2 \, e^{4}} \end {gather*}

integrate((((4*x+4)*exp(4)+4*x^2-8)*exp(x)+(8*x^3+24*x^2)*exp(4)^2+(16*x^4+48*x^3-8)*exp(4)+8*x^5+24*x^4+4*x^2

)/((x^2+4*x+4)*exp(4)^2+(2*x^3+8*x^2+8*x)*exp(4)+x^4+4*x^3+4*x^2),x, algorithm="giac")

4*(x^4 + x^3*e^4 - 4*x^2 - 4*x*e^4 - 9*x - 8*e^4 + e^x)/(x^2 + x*e^4 + 2*x + 2*e^4)

________________________________________________________________________________________

Mupad [B]
time = 6.10, size = 55, normalized size = 1.90 \begin {gather*} \frac {4\,{\mathrm {e}}^x}{x^2+\left ({\mathrm {e}}^4+2\right )\,x+2\,{\mathrm {e}}^4}-8\,x-\frac {36\,x+32\,{\mathrm {e}}^4}{x^2+\left ({\mathrm {e}}^4+2\right )\,x+2\,{\mathrm {e}}^4}+4\,x^2 \end {gather*}

int((exp(8)*(24*x^2 + 8*x^3) + exp(4)*(48*x^3 + 16*x^4 - 8) + exp(x)*(4*x^2 + exp(4)*(4*x + 4) - 8) + 4*x^2 +

24*x^4 + 8*x^5)/(exp(4)*(8*x + 8*x^2 + 2*x^3) + exp(8)*(4*x + x^2 + 4) + 4*x^2 + 4*x^3 + x^4),x)

(4*exp(x))/(2*exp(4) + x*(exp(4) + 2) + x^2) - 8*x - (36*x + 32*exp(4))/(2*exp(4) + x*(exp(4) + 2) + x^2) + 4*

________________________________________________________________________________________

3.85.97 \(\int \frac {4 x^2+24 x^4+8 x^5+e^8 (24 x^2+8 x^3)+e^4 (-8+48 x^3+16 x^4)+e^x (-8+4 x^2+e^4 (4+4 x))}{4 x^2+4 x^3+x^4+e^8 (4+4 x+x^2)+e^4 (8 x+8 x^2+2 x^3)} \, dx\) [8497]