Integrand size = 148, antiderivative size = 27 \[ \int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx=\frac {2}{x^2 \left (8+\frac {x}{\frac {1}{9}+x-\left (e^2+x\right )^2}\right )} \]
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx=-\frac {2 \left (1-9 e^4+9 x-18 e^2 x-9 x^2\right )}{x^2 \left (-8+72 e^4-81 x+144 e^2 x+72 x^2\right )} \]
Integrate[(-32 - 2592*E^8 - 630*x - 10368*E^6*x - 2340*x^2 + 5346*x^3 - 25 92*x^4 + E^4*(576 + 5670*x - 15552*x^2) + E^2*(1152*x + 11016*x^2 - 10368* x^3))/(64*x^3 + 5184*E^8*x^3 + 1296*x^4 + 20736*E^6*x^4 + 5409*x^5 - 11664 *x^6 + 5184*x^7 + E^4*(-1152*x^3 - 11664*x^4 + 31104*x^5) + E^2*(-2304*x^4 - 23328*x^5 + 20736*x^6)),x]
Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(27)=54\).
Time = 0.68 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.52, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6, 6, 6, 2026, 2462, 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.
| \(\displaystyle \int \frac {-2592 x^4+5346 x^3-2340 x^2+e^4 \left (-15552 x^2+5670 x+576\right )+e^2 \left (-10368 x^3+11016 x^2+1152 x\right )-10368 e^6 x-630 x-2592 e^8-32}{5184 x^7-11664 x^6+5409 x^5+20736 e^6 x^4+1296 x^4+5184 e^8 x^3+64 x^3+e^2 \left (20736 x^6-23328 x^5-2304 x^4\right )+e^4 \left (31104 x^5-11664 x^4-1152 x^3\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-2592 x^4+5346 x^3-2340 x^2+e^4 \left (-15552 x^2+5670 x+576\right )+e^2 \left (-10368 x^3+11016 x^2+1152 x\right )+\left (-630-10368 e^6\right ) x-2592 e^8-32}{5184 x^7-11664 x^6+5409 x^5+20736 e^6 x^4+1296 x^4+5184 e^8 x^3+64 x^3+e^2 \left (20736 x^6-23328 x^5-2304 x^4\right )+e^4 \left (31104 x^5-11664 x^4-1152 x^3\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-2592 x^4+5346 x^3-2340 x^2+e^4 \left (-15552 x^2+5670 x+576\right )+e^2 \left (-10368 x^3+11016 x^2+1152 x\right )+\left (-630-10368 e^6\right ) x-2592 e^8-32}{5184 x^7-11664 x^6+5409 x^5+20736 e^6 x^4+1296 x^4+\left (64+5184 e^8\right ) x^3+e^2 \left (20736 x^6-23328 x^5-2304 x^4\right )+e^4 \left (31104 x^5-11664 x^4-1152 x^3\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-2592 x^4+5346 x^3-2340 x^2+e^4 \left (-15552 x^2+5670 x+576\right )+e^2 \left (-10368 x^3+11016 x^2+1152 x\right )+\left (-630-10368 e^6\right ) x-2592 e^8-32}{5184 x^7-11664 x^6+5409 x^5+\left (1296+20736 e^6\right ) x^4+\left (64+5184 e^8\right ) x^3+e^2 \left (20736 x^6-23328 x^5-2304 x^4\right )+e^4 \left (31104 x^5-11664 x^4-1152 x^3\right )}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-2592 x^4+5346 x^3-2340 x^2+e^4 \left (-15552 x^2+5670 x+576\right )+e^2 \left (-10368 x^3+11016 x^2+1152 x\right )+\left (-630-10368 e^6\right ) x-2592 e^8-32}{x^3 \left (5184 x^4-1296 \left (9-16 e^2\right ) x^3+9 \left (601-2592 e^2+3456 e^4\right ) x^2+144 \left (9-16 e^2-81 e^4+144 e^6\right ) x+64 \left (1-9 e^4\right )^2\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (-\frac {1}{2 x^3}+\frac {81 \left (72 \left (9-16 e^2\right ) x-1152 e^4+2592 e^2-857\right )}{32 \left (1-9 e^4\right ) \left (-72 x^2+9 \left (9-16 e^2\right ) x+8 \left (1-9 e^4\right )\right )^2}+\frac {81}{4 \left (1-3 e^2\right ) \left (1+3 e^2\right ) \left (-72 x^2+9 \left (9-16 e^2\right ) x+8 \left (1-9 e^4\right )\right )}-\frac {9}{32 \left (9 e^4-1\right ) x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {81 \left (-8 \left (985-2592 e^2\right ) x+41472 e^4-39088 e^2+8865\right )}{32 \left (985-2592 e^2\right ) \left (1-9 e^4\right ) \left (-72 x^2+9 \left (9-16 e^2\right ) x+8 \left (1-9 e^4\right )\right )}+\frac {1}{4 x^2}-\frac {9}{32 \left (1-9 e^4\right ) x}\) |
Int[(-32 - 2592*E^8 - 630*x - 10368*E^6*x - 2340*x^2 + 5346*x^3 - 2592*x^4 + E^4*(576 + 5670*x - 15552*x^2) + E^2*(1152*x + 11016*x^2 - 10368*x^3))/ (64*x^3 + 5184*E^8*x^3 + 1296*x^4 + 20736*E^6*x^4 + 5409*x^5 - 11664*x^6 + 5184*x^7 + E^4*(-1152*x^3 - 11664*x^4 + 31104*x^5) + E^2*(-2304*x^4 - 233 28*x^5 + 20736*x^6)),x]
1/(4*x^2) - 9/(32*(1 - 9*E^4)*x) + (81*(8865 - 39088*E^2 + 41472*E^4 - 8*( 985 - 2592*E^2)*x))/(32*(985 - 2592*E^2)*(1 - 9*E^4)*(8*(1 - 9*E^4) + 9*(9 - 16*E^2)*x - 72*x^2))
3.12.74.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(\frac {-2+18 x^{2}+72 \left (\frac {{\mathrm e}^{2}}{2}-\frac {1}{4}\right ) x +18 \,{\mathrm e}^{4}}{x^{2} \left (72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{2} x +72 x^{2}-81 x -8\right )}\) | \(46\) |
| norman | \(\frac {\left (36 \,{\mathrm e}^{2}-18\right ) x +18 x^{2}-2+18 \,{\mathrm e}^{4}}{x^{2} \left (72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{2} x +72 x^{2}-81 x -8\right )}\) | \(49\) |
| gosper | \(\frac {18 \,{\mathrm e}^{4}+36 \,{\mathrm e}^{2} x +18 x^{2}-18 x -2}{x^{2} \left (72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{2} x +72 x^{2}-81 x -8\right )}\) | \(50\) |
| parallelrisch | \(\frac {-144+1296 \,{\mathrm e}^{4}+2592 \,{\mathrm e}^{2} x +1296 x^{2}-1296 x}{72 x^{2} \left (72 \,{\mathrm e}^{4}+144 \,{\mathrm e}^{2} x +72 x^{2}-81 x -8\right )}\) | \(50\) |
| default | \(-\frac {-243 \,{\mathrm e}^{4}-6561 \,{\mathrm e}^{12}+2187 \,{\mathrm e}^{8}+9}{32 \left (18 \,{\mathrm e}^{4}-81 \,{\mathrm e}^{8}-1\right )^{2} x}-\frac {36 \,{\mathrm e}^{4}+2916 \,{\mathrm e}^{12}-6561 \,{\mathrm e}^{16}-486 \,{\mathrm e}^{8}-1}{4 \left (18 \,{\mathrm e}^{4}-81 \,{\mathrm e}^{8}-1\right )^{2} x^{2}}+\frac {9 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (5184 \textit {\_Z}^{4}+\left (20736 \,{\mathrm e}^{2}-11664\right ) \textit {\_Z}^{3}+\left (-23328 \,{\mathrm e}^{2}+31104 \,{\mathrm e}^{4}+5409\right ) \textit {\_Z}^{2}+\left (-2304 \,{\mathrm e}^{2}-11664 \,{\mathrm e}^{4}+20736 \,{\mathrm e}^{6}+1296\right ) \textit {\_Z} -1152 \,{\mathrm e}^{4}+5184 \,{\mathrm e}^{8}+64\right )}{\sum }\frac {\left (-793+576 \left (27 \,{\mathrm e}^{4}+729 \,{\mathrm e}^{12}-243 \,{\mathrm e}^{8}-1\right ) \textit {\_R}^{2}+144 \left (9-16 \,{\mathrm e}^{2}-243 \,{\mathrm e}^{4}-6561 \,{\mathrm e}^{12}-3888 \,{\mathrm e}^{10}+11664 \,{\mathrm e}^{14}+2187 \,{\mathrm e}^{8}+432 \,{\mathrm e}^{6}\right ) \textit {\_R} +2592 \,{\mathrm e}^{2}+19683 \,{\mathrm e}^{4}+158193 \,{\mathrm e}^{12}+629856 \,{\mathrm e}^{10}-1889568 \,{\mathrm e}^{14}+1259712 \,{\mathrm e}^{16}-146043 \,{\mathrm e}^{8}-69984 \,{\mathrm e}^{6}\right ) \ln \left (x -\textit {\_R} \right )}{72+1152 \,{\mathrm e}^{6}+3456 \textit {\_R} \,{\mathrm e}^{4}+3456 \textit {\_R}^{2} {\mathrm e}^{2}+1152 \textit {\_R}^{3}-648 \,{\mathrm e}^{4}-2592 \,{\mathrm e}^{2} \textit {\_R} -1944 \textit {\_R}^{2}-128 \,{\mathrm e}^{2}+601 \textit {\_R}}\right )}{64 \left (18 \,{\mathrm e}^{4}-81 \,{\mathrm e}^{8}-1\right )^{2}}\) | \(280\) |
int((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^2+(-10 368*x^3+11016*x^2+1152*x)*exp(2)-2592*x^4+5346*x^3-2340*x^2-630*x-32)/(518 4*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*exp(2)^2+ (20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+1296*x^4 +64*x^3),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx=\frac {2 \, {\left (9 \, x^{2} + 18 \, x e^{2} - 9 \, x + 9 \, e^{4} - 1\right )}}{72 \, x^{4} + 144 \, x^{3} e^{2} - 81 \, x^{3} + 72 \, x^{2} e^{4} - 8 \, x^{2}} \]
integrate((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^ 2+(-10368*x^3+11016*x^2+1152*x)*exp(2)-2592*x^4+5346*x^3-2340*x^2-630*x-32 )/(5184*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*exp (2)^2+(20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+12 96*x^4+64*x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 1.75 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx=- \frac {- 18 x^{2} + x \left (18 - 36 e^{2}\right ) - 18 e^{4} + 2}{72 x^{4} + x^{3} \left (-81 + 144 e^{2}\right ) + x^{2} \left (-8 + 72 e^{4}\right )} \]
integrate((-2592*exp(2)**4-10368*x*exp(2)**3+(-15552*x**2+5670*x+576)*exp( 2)**2+(-10368*x**3+11016*x**2+1152*x)*exp(2)-2592*x**4+5346*x**3-2340*x**2 -630*x-32)/(5184*x**3*exp(2)**4+20736*x**4*exp(2)**3+(31104*x**5-11664*x** 4-1152*x**3)*exp(2)**2+(20736*x**6-23328*x**5-2304*x**4)*exp(2)+5184*x**7- 11664*x**6+5409*x**5+1296*x**4+64*x**3),x)
-(-18*x**2 + x*(18 - 36*exp(2)) - 18*exp(4) + 2)/(72*x**4 + x**3*(-81 + 14 4*exp(2)) + x**2*(-8 + 72*exp(4)))
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx=\frac {2 \, {\left (9 \, x^{2} + 9 \, x {\left (2 \, e^{2} - 1\right )} + 9 \, e^{4} - 1\right )}}{72 \, x^{4} + 9 \, x^{3} {\left (16 \, e^{2} - 9\right )} + 8 \, x^{2} {\left (9 \, e^{4} - 1\right )}} \]
integrate((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^ 2+(-10368*x^3+11016*x^2+1152*x)*exp(2)-2592*x^4+5346*x^3-2340*x^2-630*x-32 )/(5184*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*exp (2)^2+(20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+12 96*x^4+64*x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx=-\frac {81 \, {\left (8 \, x + 16 \, e^{2} - 9\right )}}{32 \, {\left (72 \, x^{2} + 144 \, x e^{2} - 81 \, x + 72 \, e^{4} - 8\right )} {\left (9 \, e^{4} - 1\right )}} + \frac {9 \, x + 72 \, e^{4} - 8}{32 \, x^{2} {\left (9 \, e^{4} - 1\right )}} \]
integrate((-2592*exp(2)^4-10368*x*exp(2)^3+(-15552*x^2+5670*x+576)*exp(2)^ 2+(-10368*x^3+11016*x^2+1152*x)*exp(2)-2592*x^4+5346*x^3-2340*x^2-630*x-32 )/(5184*x^3*exp(2)^4+20736*x^4*exp(2)^3+(31104*x^5-11664*x^4-1152*x^3)*exp (2)^2+(20736*x^6-23328*x^5-2304*x^4)*exp(2)+5184*x^7-11664*x^6+5409*x^5+12 96*x^4+64*x^3),x, algorithm=\
-81/32*(8*x + 16*e^2 - 9)/((72*x^2 + 144*x*e^2 - 81*x + 72*e^4 - 8)*(9*e^4 - 1)) + 1/32*(9*x + 72*e^4 - 8)/(x^2*(9*e^4 - 1))
Time = 8.36 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {-32-2592 e^8-630 x-10368 e^6 x-2340 x^2+5346 x^3-2592 x^4+e^4 \left (576+5670 x-15552 x^2\right )+e^2 \left (1152 x+11016 x^2-10368 x^3\right )}{64 x^3+5184 e^8 x^3+1296 x^4+20736 e^6 x^4+5409 x^5-11664 x^6+5184 x^7+e^4 \left (-1152 x^3-11664 x^4+31104 x^5\right )+e^2 \left (-2304 x^4-23328 x^5+20736 x^6\right )} \, dx=\frac {18\,x^2+\left (36\,{\mathrm {e}}^2-18\right )\,x+18\,{\mathrm {e}}^4-2}{72\,x^4+\left (144\,{\mathrm {e}}^2-81\right )\,x^3+\left (72\,{\mathrm {e}}^4-8\right )\,x^2} \]
int(-(630*x + 2592*exp(8) - exp(4)*(5670*x - 15552*x^2 + 576) + 10368*x*ex p(6) - exp(2)*(1152*x + 11016*x^2 - 10368*x^3) + 2340*x^2 - 5346*x^3 + 259 2*x^4 + 32)/(20736*x^4*exp(6) + 5184*x^3*exp(8) - exp(4)*(1152*x^3 + 11664 *x^4 - 31104*x^5) - exp(2)*(2304*x^4 + 23328*x^5 - 20736*x^6) + 64*x^3 + 1 296*x^4 + 5409*x^5 - 11664*x^6 + 5184*x^7),x)