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

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

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Rubi [B]  time = 1.44, 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 = {6688, 12, 6742, 88, 148, 2177, 2178, 1620} \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*}

Antiderivative was successfully verified.

[In]

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
*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]

[Out]

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

Rule 12

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

Rule 88

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] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 148

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
, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))

Rule 1620

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]) &&
GtQ[Expon[Px, x], 2]

Rule 2177

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] &&  !$UseGamma ===
True

Rule 2178

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

Rule 6688

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

Rule 6742

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

Rubi steps

\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*}

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Mathematica [B]  time = 0.52, size = 124, normalized size = 4.28 \begin {gather*} \frac {4 \left (4 e^x-4 e^{4+x}+e^{8+x}-2 e^{20} (2+x)-2 e^{16} \left (-6-x+x^2\right )+e^{12} x \left (12+6 x+x^2\right )+4 x \left (-9-4 x+x^3\right )+e^8 \left (32+15 x-4 x^3+x^4\right )-4 e^4 \left (8-5 x-4 x^2-x^3+x^4\right )\right )}{\left (-2+e^4\right )^2 (2+x) \left (e^4+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

(4*(4*E^x - 4*E^(4 + x) + E^(8 + x) - 2*E^20*(2 + x) - 2*E^16*(-6 - x + x^2) + E^12*x*(12 + 6*x + x^2) + 4*x*(
-9 - 4*x + x^3) + E^8*(32 + 15*x - 4*x^3 + x^4) - 4*E^4*(8 - 5*x - 4*x^2 - x^3 + x^4)))/((-2 + E^4)^2*(2 + x)*
(E^4 + x))

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fricas [A]  time = 1.08, 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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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

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giac [A]  time = 0.37, 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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)

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maple [A]  time = 0.57, size = 33, normalized size = 1.14

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	\(\frac {\frac {\left (-16-4 \,{\mathrm e}^{8}\right ) x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}+\frac {2 \left (-8-4 \,{\mathrm e}^{4}\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+\frac {24 x^{3}-\frac {2 \left (48 \,{\mathrm e}^{12}+384-48 \,{\mathrm e}^{8}-96 \,{\mathrm e}^{4}\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}-\frac {\left (48 \,{\mathrm e}^{16}+768-48 \,{\mathrm e}^{12}-192 \,{\mathrm e}^{4}\right ) x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+\frac {\left (-24-12 \,{\mathrm e}^{4}\right ) x^{3}+\frac {\left (24 \,{\mathrm e}^{20}-8 \,{\mathrm e}^{16}+768-48 \,{\mathrm e}^{12}-96 \,{\mathrm e}^{8}-64 \,{\mathrm e}^{4}\right ) x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}+4 x^{4}+\frac {2 \left (24 \,{\mathrm e}^{16}-8 \,{\mathrm e}^{12}+384-96 \,{\mathrm e}^{8}-32 \,{\mathrm e}^{4}\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+\frac {\frac {16 \,{\mathrm e}^{4} x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}+\frac {2 \left (4 \,{\mathrm e}^{4}+8\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+\frac {8 \,{\mathrm e}^{x} \left ({\mathrm e}^{4}+2 x +2\right )}{\left (-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}\right ) \left (x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x \right )}-\frac {8 \,{\mathrm e}^{4} {\mathrm e}^{-{\mathrm e}^{4}} \expIntegralEi \left (1, -{\mathrm e}^{4}-x \right )}{\left ({\mathrm e}^{4}-2\right ) \left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right )}-\frac {8 \left ({\mathrm e}^{4}-4\right ) {\mathrm e}^{-2} \expIntegralEi \left (1, -x -2\right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}+\frac {-\frac {2 \left (24 \,{\mathrm e}^{12}+48 \,{\mathrm e}^{8}\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}-\frac {\left (24 \,{\mathrm e}^{16}+96 \,{\mathrm e}^{8}\right ) x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+\frac {\frac {\left (48 \,{\mathrm e}^{16}+384 \,{\mathrm e}^{4}\right ) x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}+\frac {2 \left (48 \,{\mathrm e}^{12}+192 \,{\mathrm e}^{4}\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+\frac {\frac {\left (8 \,{\mathrm e}^{20}+64 \,{\mathrm e}^{8}\right ) x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}+\frac {2 \left (8 \,{\mathrm e}^{16}+32 \,{\mathrm e}^{8}\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+\frac {16 x^{3} {\mathrm e}^{4}-\frac {2 \left (32 \,{\mathrm e}^{16}-32 \,{\mathrm e}^{12}-64 \,{\mathrm e}^{8}+256 \,{\mathrm e}^{4}\right ) {\mathrm e}^{4}}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}-\frac {\left (32 \,{\mathrm e}^{20}-32 \,{\mathrm e}^{16}-128 \,{\mathrm e}^{8}+512 \,{\mathrm e}^{4}\right ) x}{-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+4 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{x} \left ({\mathrm e}^{4}+2 x +2\right )}{\left (-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}\right ) \left (x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x \right )}+\frac {{\mathrm e}^{4} {\mathrm e}^{-{\mathrm e}^{4}} \expIntegralEi \left (1, -{\mathrm e}^{4}-x \right )}{\left ({\mathrm e}^{4}-2\right ) \left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right )}+\frac {\left ({\mathrm e}^{4}-4\right ) {\mathrm e}^{-2} \expIntegralEi \left (1, -x -2\right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}\right )-\frac {4 \,{\mathrm e}^{x} \left (x \,{\mathrm e}^{8}+2 \,{\mathrm e}^{8}+4 \,{\mathrm e}^{4}+4 x \right )}{\left (-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}\right ) \left (x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x \right )}+\frac {4 \left ({\mathrm e}^{8} {\mathrm e}^{4}-2 \,{\mathrm e}^{8}+4 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-{\mathrm e}^{4}} \expIntegralEi \left (1, -{\mathrm e}^{4}-x \right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}-\frac {32 \,{\mathrm e}^{-2} \expIntegralEi \left (1, -x -2\right )}{\left ({\mathrm e}^{4}-2\right ) \left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right )}+4 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{x} \left (x \,{\mathrm e}^{4}+4 \,{\mathrm e}^{4}+2 x \right )}{\left (-4 \,{\mathrm e}^{4}+4+{\mathrm e}^{8}\right ) \left (x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x \right )}-\frac {\left ({\mathrm e}^{8}-{\mathrm e}^{4}+2\right ) {\mathrm e}^{-{\mathrm e}^{4}} \expIntegralEi \left (1, -{\mathrm e}^{4}-x \right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}-\frac {\left ({\mathrm e}^{4}-6\right ) {\mathrm e}^{-2} \expIntegralEi \left (1, -x -2\right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}\right )\)	\(1090\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

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maxima [B]  time = 0.44, size = 912, normalized size = 31.45 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)
+ 2*e^4)

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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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

(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*
x^2

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sympy [B]  time = 0.77, 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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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
p(4))

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