\(\int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 (16-8 x+x^2)+e^2 (384 x^2-192 x^3+24 x^4)} \, dx\) [1236]

Optimal result

Integrand size = 65, antiderivative size = 32 \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=1-\frac {1}{(-4+x) \left (x^2+\frac {1}{2} x \left (\frac {e^2}{4 x}+x\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {2099, 653, 209} \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=\frac {96 (x+4)}{\left (192+e^2\right ) \left (12 x^2+e^2\right )}+\frac {8}{\left (192+e^2\right ) (4-x)} \]

[In]

Int[(8*E^2 - 768*x + 288*x^2)/(2304*x^4 - 1152*x^5 + 144*x^6 + E^4*(16 - 8*x + x^2) + E^2*(384*x^2 - 192*x^3 +
24*x^4)),x]

[Out]

8/((192 + E^2)*(4 - x)) + (96*(4 + x))/((192 + E^2)*(E^2 + 12*x^2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {8}{\left (192+e^2\right ) (-4+x)^2}+\frac {192 \left (e^2-48 x\right )}{\left (192+e^2\right ) \left (e^2+12 x^2\right )^2}-\frac {96}{\left (192+e^2\right ) \left (e^2+12 x^2\right )}\right ) \, dx \\ & = \frac {8}{\left (192+e^2\right ) (4-x)}-\frac {96 \int \frac {1}{e^2+12 x^2} \, dx}{192+e^2}+\frac {192 \int \frac {e^2-48 x}{\left (e^2+12 x^2\right )^2} \, dx}{192+e^2} \\ & = \frac {8}{\left (192+e^2\right ) (4-x)}+\frac {96 (4+x)}{\left (192+e^2\right ) \left (e^2+12 x^2\right )}-\frac {16 \sqrt {3} \arctan \left (\frac {2 \sqrt {3} x}{e}\right )}{e \left (192+e^2\right )}+\frac {96 \int \frac {1}{e^2+12 x^2} \, dx}{192+e^2} \\ & = \frac {8}{\left (192+e^2\right ) (4-x)}+\frac {96 (4+x)}{\left (192+e^2\right ) \left (e^2+12 x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56 \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=-\frac {8}{(-4+x) \left (e^2+12 x^2\right )} \]

[In]

Integrate[(8*E^2 - 768*x + 288*x^2)/(2304*x^4 - 1152*x^5 + 144*x^6 + E^4*(16 - 8*x + x^2) + E^2*(384*x^2 - 192
*x^3 + 24*x^4)),x]

[Out]

-8/((-4 + x)*(E^2 + 12*x^2))

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56

method	result	size
norman	\(-\frac {8}{\left (x -4\right ) \left (12 x^{2}+{\mathrm e}^{2}\right )}\)	\(18\)
gosper	\(-\frac {8}{12 x^{3}+{\mathrm e}^{2} x -48 x^{2}-4 \,{\mathrm e}^{2}}\)	\(24\)
risch	\(-\frac {8}{12 x^{3}+{\mathrm e}^{2} x -48 x^{2}-4 \,{\mathrm e}^{2}}\)	\(24\)
parallelrisch	\(-\frac {8}{12 x^{3}+{\mathrm e}^{2} x -48 x^{2}-4 \,{\mathrm e}^{2}}\)	\(24\)
default	\(\frac {-\frac {96 \left (-\frac {{\mathrm e}^{-2} \left (221184 \,{\mathrm e}^{4}+576 \,{\mathrm e}^{2} {\mathrm e}^{4}+{\mathrm e}^{2} {\mathrm e}^{6}[phone] \,{\mathrm e}^{2}+576 \,{\mathrm e}^{6}+{\mathrm e}^{8}\right ) x}{2}-442368 \,{\mathrm e}^{2}-2304 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{6}-28311552\right )}{12 x^{2}+{\mathrm e}^{2}}-8 \left (576 \,{\mathrm e}^{2} {\mathrm e}^{4}+{\mathrm e}^{2} {\mathrm e}^{6}-576 \,{\mathrm e}^{6}-{\mathrm e}^{8}\right ) {\mathrm e}^{-2} \sqrt {3}\, {\mathrm e}^{-1} \arctan \left (2 x \sqrt {3}\, {\mathrm e}^{-1}\right )}{\left (36864+384 \,{\mathrm e}^{2}+{\mathrm e}^{4}\right )^{2}}-\frac {8 \left (110592 \,{\mathrm e}^{2}+576 \,{\mathrm e}^{4}+{\mathrm e}^{6}+7077888\right )}{\left (36864+384 \,{\mathrm e}^{2}+{\mathrm e}^{4}\right )^{2} \left (x -4\right )}\)	\(163\)

[In]

int((8*exp(2)+288*x^2-768*x)/((x^2-8*x+16)*exp(2)^2+(24*x^4-192*x^3+384*x^2)*exp(2)+144*x^6-1152*x^5+2304*x^4)
,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=-\frac {8}{12 \, x^{3} - 48 \, x^{2} + {\left (x - 4\right )} e^{2}} \]

[In]

integrate((8*exp(2)+288*x^2-768*x)/((x^2-8*x+16)*exp(2)^2+(24*x^4-192*x^3+384*x^2)*exp(2)+144*x^6-1152*x^5+230
4*x^4),x, algorithm="fricas")

[Out]

-8/(12*x^3 - 48*x^2 + (x - 4)*e^2)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=- \frac {8}{12 x^{3} - 48 x^{2} + x e^{2} - 4 e^{2}} \]

[In]

integrate((8*exp(2)+288*x**2-768*x)/((x**2-8*x+16)*exp(2)**2+(24*x**4-192*x**3+384*x**2)*exp(2)+144*x**6-1152*
x**5+2304*x**4),x)

[Out]

-8/(12*x**3 - 48*x**2 + x*exp(2) - 4*exp(2))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=-\frac {8}{12 \, x^{3} - 48 \, x^{2} + x e^{2} - 4 \, e^{2}} \]

[In]

integrate((8*exp(2)+288*x^2-768*x)/((x^2-8*x+16)*exp(2)^2+(24*x^4-192*x^3+384*x^2)*exp(2)+144*x^6-1152*x^5+230
4*x^4),x, algorithm="maxima")

[Out]

-8/(12*x^3 - 48*x^2 + x*e^2 - 4*e^2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=-\frac {8}{12 \, x^{3} - 48 \, x^{2} + x e^{2} - 4 \, e^{2}} \]

[In]

integrate((8*exp(2)+288*x^2-768*x)/((x^2-8*x+16)*exp(2)^2+(24*x^4-192*x^3+384*x^2)*exp(2)+144*x^6-1152*x^5+230
4*x^4),x, algorithm="giac")

[Out]

-8/(12*x^3 - 48*x^2 + x*e^2 - 4*e^2)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int \frac {8 e^2-768 x+288 x^2}{2304 x^4-1152 x^5+144 x^6+e^4 \left (16-8 x+x^2\right )+e^2 \left (384 x^2-192 x^3+24 x^4\right )} \, dx=-\frac {8}{\left (x-4\right )\,\left (12\,x^2+{\mathrm {e}}^2\right )} \]

[In]

int((8*exp(2) - 768*x + 288*x^2)/(exp(2)*(384*x^2 - 192*x^3 + 24*x^4) + exp(4)*(x^2 - 8*x + 16) + 2304*x^4 - 1
152*x^5 + 144*x^6),x)

[Out]

-8/((x - 4)*(exp(2) + 12*x^2))