Integrand size = 114, antiderivative size = 29 \[ \int \frac {4 x^2+24 x^4+8 x^5+e^8 \left (24 x^2+8 x^3\right )+e^4 \left (-8+48 x^3+16 x^4\right )+e^x \left (-8+4 x^2+e^4 (4+4 x)\right )}{4 x^2+4 x^3+x^4+e^8 \left (4+4 x+x^2\right )+e^4 \left (8 x+8 x^2+2 x^3\right )} \, dx=\frac {4 x \left (x^2+\frac {-1+\frac {e^x}{x}}{e^4+x}\right )}{2+x} \]
Time = 3.87 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {4 x^2+24 x^4+8 x^5+e^8 \left (24 x^2+8 x^3\right )+e^4 \left (-8+48 x^3+16 x^4\right )+e^x \left (-8+4 x^2+e^4 (4+4 x)\right )}{4 x^2+4 x^3+x^4+e^8 \left (4+4 x+x^2\right )+e^4 \left (8 x+8 x^2+2 x^3\right )} \, dx=\frac {4 \left (e^x+x \left (-9-4 x+x^3\right )+e^4 \left (-8-4 x+x^3\right )\right )}{(2+x) \left (e^4+x\right )} \]
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]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.90 (sec) , antiderivative size = 849, normalized size of antiderivative = 29.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2463, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {8 x^5+24 x^4+4 x^2+e^x \left (4 x^2+e^4 (4 x+4)-8\right )+e^4 \left (16 x^4+48 x^3-8\right )+e^8 \left (8 x^3+24 x^2\right )}{x^4+4 x^3+4 x^2+e^8 \left (x^2+4 x+4\right )+e^4 \left (2 x^3+8 x^2+8 x\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {2 \left (8 x^5+24 x^4+4 x^2+e^x \left (4 x^2+e^4 (4 x+4)-8\right )+e^4 \left (16 x^4+48 x^3-8\right )+e^8 \left (8 x^3+24 x^2\right )\right )}{\left (e^4-2\right )^3 (x+2)}+\frac {2 \left (8 x^5+24 x^4+4 x^2+e^x \left (4 x^2+e^4 (4 x+4)-8\right )+e^4 \left (16 x^4+48 x^3-8\right )+e^8 \left (8 x^3+24 x^2\right )\right )}{\left (e^4-2\right )^3 \left (x+e^4\right )}+\frac {8 x^5+24 x^4+4 x^2+e^x \left (4 x^2+e^4 (4 x+4)-8\right )+e^4 \left (16 x^4+48 x^3-8\right )+e^8 \left (8 x^3+24 x^2\right )}{\left (e^4-2\right )^2 (x+2)^2}+\frac {8 x^5+24 x^4+4 x^2+e^x \left (4 x^2+e^4 (4 x+4)-8\right )+e^4 \left (16 x^4+48 x^3-8\right )+e^8 \left (8 x^3+24 x^2\right )}{\left (e^4-2\right )^2 \left (x+e^4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (1+2 e^4\right ) x^4}{\left (2-e^4\right )^3}-\frac {4 \left (3+e^4\right ) x^4}{\left (2-e^4\right )^3}+\frac {4 x^4}{\left (2-e^4\right )^2}+\frac {4 e^8 x^3}{\left (2-e^4\right )^2 (x+2)}-\frac {16 \left (2-2 e^4-e^8\right ) x^3}{3 \left (2-e^4\right )^3}+\frac {16 e^4 x^3}{3 \left (2-e^4\right )^2}+\frac {16 x^3}{3 \left (2-e^4\right )^2}-\frac {16 e^4 x^3}{\left (2-e^4\right )^3}+\frac {4 \left (9-8 e^4+2 e^8\right ) x^2}{\left (2-e^4\right )^3}+\frac {8 e^4 \left (3-2 e^4\right ) x^2}{\left (2-e^4\right )^2}+\frac {16 e^8 x^2}{\left (2-e^4\right )^2}-\frac {32 e^4 x^2}{\left (2-e^4\right )^2}-\frac {4 x^2}{\left (2-e^4\right )^3}-\frac {16 \left (9-8 e^4+2 e^8\right ) x}{\left (2-e^4\right )^3}-\frac {48 e^8 x}{2-e^4}+\frac {8 e^8 \left (3-2 e^4\right ) x}{\left (2-e^4\right )^2}-\frac {32 e^{12} x}{\left (2-e^4\right )^2}+\frac {72 e^8 x}{\left (2-e^4\right )^2}+\frac {40 x}{\left (2-e^4\right )^2}+\frac {8 e^4 x}{\left (2-e^4\right )^3}-\frac {4 \left (4-e^4\right ) \operatorname {ExpIntegralEi}(x+2)}{e^2 \left (2-e^4\right )^2}+\frac {4 \operatorname {ExpIntegralEi}(x+2)}{e^2 \left (2-e^4\right )}+\frac {8 \operatorname {ExpIntegralEi}(x+2)}{e^2 \left (2-e^4\right )^2}-\frac {4 e^{-e^4} \operatorname {ExpIntegralEi}\left (x+e^4\right )}{2-e^4}-\frac {4 e^{4-e^4} \operatorname {ExpIntegralEi}\left (x+e^4\right )}{\left (2-e^4\right )^2}+\frac {8 e^{-e^4} \operatorname {ExpIntegralEi}\left (x+e^4\right )}{\left (2-e^4\right )^2}+\frac {16 \left (9-4 e^4\right ) \log (x+2)}{\left (2-e^4\right )^2}+\frac {64 e^4 \log (x+2)}{\left (2-e^4\right )^2}-\frac {144 \log (x+2)}{\left (2-e^4\right )^2}-\frac {24 e^{12} \log \left (x+e^4\right )}{2-e^4}+\frac {16 e^{12} \left (9-4 e^4\right ) \log \left (x+e^4\right )}{\left (2-e^4\right )^2}+\frac {40 e^{16} \log \left (x+e^4\right )}{\left (2-e^4\right )^2}-\frac {96 e^{12} \log \left (x+e^4\right )}{\left (2-e^4\right )^2}-\frac {4 e^x}{\left (2-e^4\right ) (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 )}\) |
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]
(8*E^4*x)/(2 - E^4)^3 + (40*x)/(2 - E^4)^2 + (72*E^8*x)/(2 - E^4)^2 - (32* E^12*x)/(2 - E^4)^2 + (8*E^8*(3 - 2*E^4)*x)/(2 - E^4)^2 - (48*E^8*x)/(2 - E^4) - (16*(9 - 8*E^4 + 2*E^8)*x)/(2 - E^4)^3 - (4*x^2)/(2 - E^4)^3 - (32* E^4*x^2)/(2 - E^4)^2 + (16*E^8*x^2)/(2 - E^4)^2 + (8*E^4*(3 - 2*E^4)*x^2)/ (2 - E^4)^2 + (4*(9 - 8*E^4 + 2*E^8)*x^2)/(2 - E^4)^3 - (16*E^4*x^3)/(2 - E^4)^3 + (16*x^3)/(3*(2 - E^4)^2) + (16*E^4*x^3)/(3*(2 - E^4)^2) - (16*(2 - 2*E^4 - E^8)*x^3)/(3*(2 - E^4)^3) + (4*x^4)/(2 - E^4)^2 - (4*(3 + E^4)*x ^4)/(2 - E^4)^3 + (4*(1 + 2*E^4)*x^4)/(2 - E^4)^3 - 144/((2 - E^4)^2*(2 + x)) + (136*E^4)/((2 - E^4)^2*(2 + x)) - (4*E^x)/((2 - E^4)*(2 + x)) + (4*E ^8*x^3)/((2 - E^4)^2*(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)) + (8*ExpIntegralEi[2 + x])/( E^2*(2 - E^4)^2) + (4*ExpIntegralEi[2 + x])/(E^2*(2 - E^4)) - (4*(4 - E^4) *ExpIntegralEi[2 + x])/(E^2*(2 - E^4)^2) + (8*ExpIntegralEi[E^4 + x])/(E^E ^4*(2 - E^4)^2) - (4*E^(4 - E^4)*ExpIntegralEi[E^4 + x])/(2 - E^4)^2 - (4* ExpIntegralEi[E^4 + x])/(E^E^4*(2 - E^4)) - (144*Log[2 + x])/(2 - E^4)^2 + (64*E^4*Log[2 + x])/(2 - E^4)^2 + (16*(9 - 4*E^4)*Log[2 + x])/(2 - E^4)^2 - (96*E^12*Log[E^4 + x])/(2 - E^4)^2 + (40*E^16*Log[E^4 + x])/(2 - E^4)^2 + (16*E^12*(9 - 4*E^4)*Log[E^4 + x])/(2 - E^4)^2 - (24*E^12*Log[E^4 + ...
3.25.97.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.93 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14
| method | result | size |
| norman | \(\frac {-4 x +4 x^{4}+4 x^{3} {\mathrm e}^{4}+4 \,{\mathrm e}^{x}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}\) | \(33\) |
| parallelrisch | \(\frac {-4 x +4 x^{4}+4 x^{3} {\mathrm e}^{4}+4 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x}\) | \(39\) |
| risch | \(4 x^{2}-8 x +\frac {-32 \,{\mathrm e}^{4}-36 x}{x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x}+\frac {4 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{4}+x^{2}+2 \,{\mathrm e}^{4}+2 x}\) | \(57\) |
| parts | \(\frac {-4 x +4 x^{4}+4 x^{3} {\mathrm e}^{4}}{\left (2+x \right ) \left (x +{\mathrm e}^{4}\right )}+4 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{x} \left (2 x +{\mathrm e}^{4}+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 {\left ({\mathrm e}^{4}-4\right ) {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2-x \right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}+\frac {{\mathrm e}^{4} {\mathrm e}^{-{\mathrm e}^{4}} \operatorname {Ei}_{1}\left (-{\mathrm e}^{4}-x \right )}{\left ({\mathrm e}^{4}-2\right ) \left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\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 {32 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2-x \right )}{\left ({\mathrm e}^{4}-2\right ) \left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right )}+\frac {4 \left ({\mathrm e}^{8} {\mathrm e}^{4}-2 \,{\mathrm e}^{8}+4 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-{\mathrm e}^{4}} \operatorname {Ei}_{1}\left (-{\mathrm e}^{4}-x \right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\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}^{4}-6\right ) {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2-x \right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}-\frac {\left ({\mathrm e}^{8}-{\mathrm e}^{4}+2\right ) {\mathrm e}^{-{\mathrm e}^{4}} \operatorname {Ei}_{1}\left (-{\mathrm e}^{4}-x \right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}\right )+\frac {8 \,{\mathrm e}^{x} \left (2 x +{\mathrm e}^{4}+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 \left ({\mathrm e}^{4}-4\right ) {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2-x \right )}{\left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right ) \left ({\mathrm e}^{4}-2\right )}-\frac {8 \,{\mathrm e}^{4} {\mathrm e}^{-{\mathrm e}^{4}} \operatorname {Ei}_{1}\left (-{\mathrm e}^{4}-x \right )}{\left ({\mathrm e}^{4}-2\right ) \left (4 \,{\mathrm e}^{4}-4-{\mathrm e}^{8}\right )}\) | \(511\) |
| default | \(\text {Expression too large to display}\) | \(1468\) |
int((((4+4*x)*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)*ex p(4)+x^4+4*x^3+4*x^2),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {4 x^2+24 x^4+8 x^5+e^8 \left (24 x^2+8 x^3\right )+e^4 \left (-8+48 x^3+16 x^4\right )+e^x \left (-8+4 x^2+e^4 (4+4 x)\right )}{4 x^2+4 x^3+x^4+e^8 \left (4+4 x+x^2\right )+e^4 \left (8 x+8 x^2+2 x^3\right )} \, dx=\frac {4 \, {\left (x^{4} - 4 \, x^{2} + {\left (x^{3} - 4 \, x - 8\right )} e^{4} - 9 \, x + e^{x}\right )}}{x^{2} + {\left (x + 2\right )} e^{4} + 2 \, x} \]
integrate((((4+4*x)*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=\
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).
Time = 0.39 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {4 x^2+24 x^4+8 x^5+e^8 \left (24 x^2+8 x^3\right )+e^4 \left (-8+48 x^3+16 x^4\right )+e^x \left (-8+4 x^2+e^4 (4+4 x)\right )}{4 x^2+4 x^3+x^4+e^8 \left (4+4 x+x^2\right )+e^4 \left (8 x+8 x^2+2 x^3\right )} \, dx=4 x^{2} - 8 x + \frac {- 36 x - 32 e^{4}}{x^{2} + x \left (2 + e^{4}\right ) + 2 e^{4}} + \frac {4 e^{x}}{x^{2} + 2 x + x e^{4} + 2 e^{4}} \]
integrate((((4+4*x)*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*exp(4))
Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 912, normalized size of antiderivative = 31.45 \[ \int \frac {4 x^2+24 x^4+8 x^5+e^8 \left (24 x^2+8 x^3\right )+e^4 \left (-8+48 x^3+16 x^4\right )+e^x \left (-8+4 x^2+e^4 (4+4 x)\right )}{4 x^2+4 x^3+x^4+e^8 \left (4+4 x+x^2\right )+e^4 \left (8 x+8 x^2+2 x^3\right )} \, dx=\text {Too large to display} \]
integrate((((4+4*x)*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=\
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^1 2 + 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 + 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^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^1 2 - 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)*l og(x + 2)/(e^12 - 6*e^8 + 12*e^4 - 8) - 16*e^4*log(x + 2)/(e^12 - 6*e^8...
Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {4 x^2+24 x^4+8 x^5+e^8 \left (24 x^2+8 x^3\right )+e^4 \left (-8+48 x^3+16 x^4\right )+e^x \left (-8+4 x^2+e^4 (4+4 x)\right )}{4 x^2+4 x^3+x^4+e^8 \left (4+4 x+x^2\right )+e^4 \left (8 x+8 x^2+2 x^3\right )} \, dx=\frac {4 \, {\left (x^{4} + x^{3} e^{4} - 4 \, x^{2} - 4 \, x e^{4} - 9 \, x - 8 \, e^{4} + e^{x}\right )}}{x^{2} + x e^{4} + 2 \, x + 2 \, e^{4}} \]
integrate((((4+4*x)*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=\
Time = 12.68 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {4 x^2+24 x^4+8 x^5+e^8 \left (24 x^2+8 x^3\right )+e^4 \left (-8+48 x^3+16 x^4\right )+e^x \left (-8+4 x^2+e^4 (4+4 x)\right )}{4 x^2+4 x^3+x^4+e^8 \left (4+4 x+x^2\right )+e^4 \left (8 x+8 x^2+2 x^3\right )} \, dx=\frac {4\,{\mathrm {e}}^x}{x^2+\left ({\mathrm {e}}^4+2\right )\,x+2\,{\mathrm {e}}^4}-8\,x-\frac {36\,x+32\,{\mathrm {e}}^4}{x^2+\left ({\mathrm {e}}^4+2\right )\,x+2\,{\mathrm {e}}^4}+4\,x^2 \]
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)