Integrand size = 184, antiderivative size = 30 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {5}{\frac {4}{3}-\left (\frac {e^3}{3}-\frac {1}{e^2 (256+x)}\right )^2} \] Output:
5/(4/3-(1/3*exp(3)-1/exp(1)^2/(256+x))^2)
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {135 \left (3-2 e^5 (256+x)\right )}{\left (-12+e^6\right ) \left (9-6 e^5 (256+x)-12 e^4 (256+x)^2+e^{10} (256+x)^2\right )} \] Input:
Integrate[(E^4*(-207360 - 810*x) + E^9*(17694720 + 138240*x + 270*x^2))/(8 1 + E^4*(-14155776 - 110592*x - 216*x^2) + E^15*(-201326592 - 2359296*x - 9216*x^2 - 12*x^3) + E^20*(4294967296 + 67108864*x + 393216*x^2 + 1024*x^3 + x^4) + E^8*(618475290624 + 9663676416*x + 56623104*x^2 + 147456*x^3 + 1 44*x^4) + E^3*(E^2*(-27648 - 108*x) + E^6*(2415919104 + 28311552*x + 11059 2*x^2 + 144*x^3)) + E^6*(E^4*(3538944 + 27648*x + 54*x^2) + E^8*(-10307921 5104 - 1610612736*x - 9437184*x^2 - 24576*x^3 - 24*x^4))),x]
Output:
(135*(3 - 2*E^5*(256 + x)))/((-12 + E^6)*(9 - 6*E^5*(256 + x) - 12*E^4*(25 6 + x)^2 + E^10*(256 + x)^2))
Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(30)=60\).
Time = 0.95 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.67, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2459, 1380, 27, 2345, 24}
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.
| \(\displaystyle \int \frac {e^9 \left (270 x^2+138240 x+17694720\right )+e^4 (-810 x-207360)}{e^4 \left (-216 x^2-110592 x-14155776\right )+e^{15} \left (-12 x^3-9216 x^2-2359296 x-201326592\right )+e^3 \left (e^6 \left (144 x^3+110592 x^2+28311552 x+2415919104\right )+e^2 (-108 x-27648)\right )+e^{20} \left (x^4+1024 x^3+393216 x^2+67108864 x+4294967296\right )+e^8 \left (144 x^4+147456 x^3+56623104 x^2+9663676416 x+618475290624\right )+e^6 \left (e^4 \left (54 x^2+27648 x+3538944\right )+e^8 \left (-24 x^4-24576 x^3-9437184 x^2-1610612736 x-103079215104\right )\right )+81} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {270 e^9 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2-\frac {810 e^4 \left (12+e^6\right ) \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )}{12-e^6}+\frac {29160 e^5}{\left (12-e^6\right )^2}}{e^8 \left (12-e^6\right )^2 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^4-216 e^4 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2+\frac {11664}{\left (12-e^6\right )^2}}d\left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle e^8 \left (12-e^6\right )^2 \int \frac {270 \left (e^9 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2-\frac {3 e^4 \left (12+e^6\right ) \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )}{12-e^6}+\frac {108 e^5}{\left (12-e^6\right )^2}\right )}{e^8 \left (108-e^4 \left (12-e^6\right )^2 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2\right )^2}d\left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 270 \left (12-e^6\right )^2 \int \frac {e^9 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2-\frac {3 e^4 \left (12+e^6\right ) \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )}{12-e^6}+\frac {108 e^5}{\left (12-e^6\right )^2}}{\left (108-e^4 \left (12-e^6\right )^2 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2\right )^2}d\left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle 270 \left (12-e^6\right )^2 \left (-\frac {1}{216} \int 0d\left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )-\frac {\frac {3 \left (12+e^6\right )}{12-e^6}-2 e^5 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )}{2 \left (12-e^6\right )^2 \left (108-e^4 \left (12-e^6\right )^2 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2\right )}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {135 \left (\frac {3 \left (12+e^6\right )}{12-e^6}-2 e^5 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )\right )}{108-e^4 \left (12-e^6\right )^2 \left (x+\frac {147456 e^8+144 e^9-24576 e^{14}-12 e^{15}+1024 e^{20}}{4 \left (144 e^8-24 e^{14}+e^{20}\right )}\right )^2}\) |
Input:
Int[(E^4*(-207360 - 810*x) + E^9*(17694720 + 138240*x + 270*x^2))/(81 + E^ 4*(-14155776 - 110592*x - 216*x^2) + E^15*(-201326592 - 2359296*x - 9216*x ^2 - 12*x^3) + E^20*(4294967296 + 67108864*x + 393216*x^2 + 1024*x^3 + x^4 ) + E^8*(618475290624 + 9663676416*x + 56623104*x^2 + 147456*x^3 + 144*x^4 ) + E^3*(E^2*(-27648 - 108*x) + E^6*(2415919104 + 28311552*x + 110592*x^2 + 144*x^3)) + E^6*(E^4*(3538944 + 27648*x + 54*x^2) + E^8*(-103079215104 - 1610612736*x - 9437184*x^2 - 24576*x^3 - 24*x^4))),x]
Output:
(-135*((3*(12 + E^6))/(12 - E^6) - 2*E^5*((147456*E^8 + 144*E^9 - 24576*E^ 14 - 12*E^15 + 1024*E^20)/(4*(144*E^8 - 24*E^14 + E^20)) + x)))/(108 - E^4 *(12 - E^6)^2*((147456*E^8 + 144*E^9 - 24576*E^14 - 12*E^15 + 1024*E^20)/( 4*(144*E^8 - 24*E^14 + E^20)) + x)^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(26)=52\).
Time = 1.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40
| method | result | size |
| risch | \(\frac {-\frac {270 \,{\mathrm e}^{5} x}{{\mathrm e}^{6}-12}-\frac {135 \left (512 \,{\mathrm e}^{5}-3\right )}{{\mathrm e}^{6}-12}}{{\mathrm e}^{10} x^{2}+512 x \,{\mathrm e}^{10}+65536 \,{\mathrm e}^{10}-6 x \,{\mathrm e}^{5}-12 x^{2} {\mathrm e}^{4}-1536 \,{\mathrm e}^{5}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}+9}\) | \(72\) |
| gosper | \(-\frac {135 \left (2 x \,{\mathrm e}^{2} {\mathrm e}^{3}+512 \,{\mathrm e}^{2} {\mathrm e}^{3}-3\right )}{\left ({\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9\right ) \left ({\mathrm e}^{6}-12\right )}\) | \(106\) |
| parallelrisch | \(-\frac {\left (270 \,{\mathrm e}^{6} {\mathrm e}^{3} x +69120 \,{\mathrm e}^{6} {\mathrm e}^{3}-405 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-4}}{\left ({\mathrm e}^{6}-12\right ) \left ({\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9\right )}\) | \(115\) |
| norman | \(\frac {-\frac {270 \,{\mathrm e}^{2} {\mathrm e}^{3} x}{{\mathrm e}^{6}-12}-\frac {135 \left (512 \,{\mathrm e}^{2} {\mathrm e}^{3}-3\right )}{{\mathrm e}^{6}-12}}{{\mathrm e}^{4} {\mathrm e}^{6} x^{2}+512 \,{\mathrm e}^{4} {\mathrm e}^{6} x +65536 \,{\mathrm e}^{4} {\mathrm e}^{6}-12 x^{2} {\mathrm e}^{4}-6144 x \,{\mathrm e}^{4}-786432 \,{\mathrm e}^{4}-6 x \,{\mathrm e}^{2} {\mathrm e}^{3}-1536 \,{\mathrm e}^{2} {\mathrm e}^{3}+9}\) | \(116\) |
| default | \(-\frac {135 \,{\mathrm e}^{4} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (-{\mathrm e}^{20}+24 \,{\mathrm e}^{14}-144 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{4}+\left (-147456 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{15}-1024 \,{\mathrm e}^{20}+24576 \,{\mathrm e}^{14}-144 \,{\mathrm e}^{9}\right ) \textit {\_Z}^{3}+\left (-56623104 \,{\mathrm e}^{8}+9216 \,{\mathrm e}^{15}-393216 \,{\mathrm e}^{20}+9437184 \,{\mathrm e}^{14}-110592 \,{\mathrm e}^{9}-54 \,{\mathrm e}^{10}+216 \,{\mathrm e}^{4}\right ) \textit {\_Z}^{2}+\left (108 \,{\mathrm e}^{5}-9663676416 \,{\mathrm e}^{8}+2359296 \,{\mathrm e}^{15}-67108864 \,{\mathrm e}^{20}+1610612736 \,{\mathrm e}^{14}-28311552 \,{\mathrm e}^{9}-27648 \,{\mathrm e}^{10}+110592 \,{\mathrm e}^{4}\right ) \textit {\_Z} -81+103079215104 \,{\mathrm e}^{14}+201326592 \,{\mathrm e}^{15}-618475290624 \,{\mathrm e}^{8}-4294967296 \,{\mathrm e}^{20}-3538944 \,{\mathrm e}^{10}+27648 \,{\mathrm e}^{5}+14155776 \,{\mathrm e}^{4}-2415919104 \,{\mathrm e}^{9}\right )}{\sum }\frac {\left (\textit {\_R}^{2} {\mathrm e}^{5}+\left (512 \,{\mathrm e}^{5}-3\right ) \textit {\_R} -768+65536 \,{\mathrm e}^{5}\right ) \ln \left (x -\textit {\_R} \right )}{-2415919104 \,{\mathrm e}^{8}+589824 \,{\mathrm e}^{15}-196608 \textit {\_R} \,{\mathrm e}^{20}-6912 \,{\mathrm e}^{10}+27 \,{\mathrm e}^{5}-55296 \textit {\_R} \,{\mathrm e}^{9}-16777216 \,{\mathrm e}^{20}-110592 \textit {\_R}^{2} {\mathrm e}^{8}-768 \textit {\_R}^{2} {\mathrm e}^{20}-28311552 \textit {\_R} \,{\mathrm e}^{8}-27 \textit {\_R} \,{\mathrm e}^{10}-{\mathrm e}^{20} \textit {\_R}^{3}+108 \textit {\_R} \,{\mathrm e}^{4}+27648 \,{\mathrm e}^{4}+24 \textit {\_R}^{3} {\mathrm e}^{14}+402653184 \,{\mathrm e}^{14}-7077888 \,{\mathrm e}^{9}-144 \textit {\_R}^{3} {\mathrm e}^{8}-108 \textit {\_R}^{2} {\mathrm e}^{9}+18432 \textit {\_R}^{2} {\mathrm e}^{14}+4718592 \textit {\_R} \,{\mathrm e}^{14}+9 \textit {\_R}^{2} {\mathrm e}^{15}+4608 \textit {\_R} \,{\mathrm e}^{15}}\right )}{2}\) | \(308\) |
Input:
int(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp(1)^4) /((x^4+1024*x^3+393216*x^2+67108864*x+4294967296)*exp(1)^8*exp(3)^4+(-12*x ^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576*x^3-943 7184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944)*exp( 1)^4)*exp(3)^2+((144*x^3+110592*x^2+28311552*x+2415919104)*exp(1)^6+(-108* x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+9663676416*x+61 8475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x,method=_R ETURNVERBOSE)
Output:
(-270*exp(5)/(exp(6)-12)*x-135*(512*exp(5)-3)/(exp(6)-12))/(exp(10)*x^2+51 2*x*exp(10)+65536*exp(10)-6*x*exp(5)-12*x^2*exp(4)-1536*exp(5)-6144*x*exp( 4)-786432*exp(4)+9)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=-\frac {135 \, {\left (2 \, {\left (x + 256\right )} e^{5} - 3\right )}}{{\left (x^{2} + 512 \, x + 65536\right )} e^{16} - 6 \, {\left (x + 256\right )} e^{11} - 24 \, {\left (x^{2} + 512 \, x + 65536\right )} e^{10} + 72 \, {\left (x + 256\right )} e^{5} + 144 \, {\left (x^{2} + 512 \, x + 65536\right )} e^{4} + 9 \, e^{6} - 108} \] Input:
integrate(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp (1)^4)/((x^4+1024*x^3+393216*x^2+67108864*x+4294967296)*exp(1)^8*exp(3)^4+ (-12*x^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576*x ^3-9437184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944 )*exp(1)^4)*exp(3)^2+((144*x^3+110592*x^2+28311552*x+2415919104)*exp(1)^6+ (-108*x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+966367641 6*x+618475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x, al gorithm="fricas")
Output:
-135*(2*(x + 256)*e^5 - 3)/((x^2 + 512*x + 65536)*e^16 - 6*(x + 256)*e^11 - 24*(x^2 + 512*x + 65536)*e^10 + 72*(x + 256)*e^5 + 144*(x^2 + 512*x + 65 536)*e^4 + 9*e^6 - 108)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
Time = 1.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {- 270 x e^{5} - 69120 e^{5} + 405}{x^{2} \left (- 24 e^{10} + 144 e^{4} + e^{16}\right ) + x \left (- 12288 e^{10} - 6 e^{11} + 72 e^{5} + 73728 e^{4} + 512 e^{16}\right ) - 1572864 e^{10} - 1536 e^{11} - 108 + 9 e^{6} + 18432 e^{5} + 9437184 e^{4} + 65536 e^{16}} \] Input:
integrate(((270*x**2+138240*x+17694720)*exp(1)**6*exp(3)+(-810*x-207360)*e xp(1)**4)/((x**4+1024*x**3+393216*x**2+67108864*x+4294967296)*exp(1)**8*ex p(3)**4+(-12*x**3-9216*x**2-2359296*x-201326592)*exp(1)**6*exp(3)**3+((-24 *x**4-24576*x**3-9437184*x**2-1610612736*x-103079215104)*exp(1)**8+(54*x** 2+27648*x+3538944)*exp(1)**4)*exp(3)**2+((144*x**3+110592*x**2+28311552*x+ 2415919104)*exp(1)**6+(-108*x-27648)*exp(1)**2)*exp(3)+(144*x**4+147456*x* *3+56623104*x**2+9663676416*x+618475290624)*exp(1)**8+(-216*x**2-110592*x- 14155776)*exp(1)**4+81),x)
Output:
(-270*x*exp(5) - 69120*exp(5) + 405)/(x**2*(-24*exp(10) + 144*exp(4) + exp (16)) + x*(-12288*exp(10) - 6*exp(11) + 72*exp(5) + 73728*exp(4) + 512*exp (16)) - 1572864*exp(10) - 1536*exp(11) - 108 + 9*exp(6) + 18432*exp(5) + 9 437184*exp(4) + 65536*exp(16))
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=-\frac {135 \, {\left (2 \, x e^{5} + 512 \, e^{5} - 3\right )}}{x^{2} {\left (e^{16} - 24 \, e^{10} + 144 \, e^{4}\right )} + 2 \, x {\left (256 \, e^{16} - 3 \, e^{11} - 6144 \, e^{10} + 36 \, e^{5} + 36864 \, e^{4}\right )} + 65536 \, e^{16} - 1536 \, e^{11} - 1572864 \, e^{10} + 9 \, e^{6} + 18432 \, e^{5} + 9437184 \, e^{4} - 108} \] Input:
integrate(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp (1)^4)/((x^4+1024*x^3+393216*x^2+67108864*x+4294967296)*exp(1)^8*exp(3)^4+ (-12*x^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576*x ^3-9437184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944 )*exp(1)^4)*exp(3)^2+((144*x^3+110592*x^2+28311552*x+2415919104)*exp(1)^6+ (-108*x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+966367641 6*x+618475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x, al gorithm="maxima")
Output:
-135*(2*x*e^5 + 512*e^5 - 3)/(x^2*(e^16 - 24*e^10 + 144*e^4) + 2*x*(256*e^ 16 - 3*e^11 - 6144*e^10 + 36*e^5 + 36864*e^4) + 65536*e^16 - 1536*e^11 - 1 572864*e^10 + 9*e^6 + 18432*e^5 + 9437184*e^4 - 108)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=-\frac {135 \, {\left (2 \, x e^{5} + 512 \, e^{5} - 3\right )}}{{\left (x^{2} e^{10} - 12 \, x^{2} e^{4} + 512 \, x e^{10} - 6 \, x e^{5} - 6144 \, x e^{4} + 65536 \, e^{10} - 1536 \, e^{5} - 786432 \, e^{4} + 9\right )} {\left (e^{6} - 12\right )}} \] Input:
integrate(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp (1)^4)/((x^4+1024*x^3+393216*x^2+67108864*x+4294967296)*exp(1)^8*exp(3)^4+ (-12*x^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576*x ^3-9437184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944 )*exp(1)^4)*exp(3)^2+((144*x^3+110592*x^2+28311552*x+2415919104)*exp(1)^6+ (-108*x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+966367641 6*x+618475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x, al gorithm="giac")
Output:
-135*(2*x*e^5 + 512*e^5 - 3)/((x^2*e^10 - 12*x^2*e^4 + 512*x*e^10 - 6*x*e^ 5 - 6144*x*e^4 + 65536*e^10 - 1536*e^5 - 786432*e^4 + 9)*(e^6 - 12))
Time = 3.77 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {135\,\left (512\,{\mathrm {e}}^5+2\,x\,{\mathrm {e}}^5-3\right )}{\left ({\mathrm {e}}^6-12\right )\,\left (\left (12\,{\mathrm {e}}^4-{\mathrm {e}}^{10}\right )\,x^2+\left (6144\,{\mathrm {e}}^4+6\,{\mathrm {e}}^5-512\,{\mathrm {e}}^{10}\right )\,x+786432\,{\mathrm {e}}^4+1536\,{\mathrm {e}}^5-65536\,{\mathrm {e}}^{10}-9\right )} \] Input:
int((exp(9)*(138240*x + 270*x^2 + 17694720) - exp(4)*(810*x + 207360))/(ex p(3)*(exp(6)*(28311552*x + 110592*x^2 + 144*x^3 + 2415919104) - exp(2)*(10 8*x + 27648)) - exp(4)*(110592*x + 216*x^2 + 14155776) + exp(20)*(67108864 *x + 393216*x^2 + 1024*x^3 + x^4 + 4294967296) - exp(6)*(exp(8)*(161061273 6*x + 9437184*x^2 + 24576*x^3 + 24*x^4 + 103079215104) - exp(4)*(27648*x + 54*x^2 + 3538944)) - exp(15)*(2359296*x + 9216*x^2 + 12*x^3 + 201326592) + exp(8)*(9663676416*x + 56623104*x^2 + 147456*x^3 + 144*x^4 + 61847529062 4) + 81),x)
Output:
(135*(512*exp(5) + 2*x*exp(5) - 3))/((exp(6) - 12)*(786432*exp(4) + 1536*e xp(5) - 65536*exp(10) + x^2*(12*exp(4) - exp(10)) + x*(6144*exp(4) + 6*exp (5) - 512*exp(10)) - 9))
Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.30 \[ \int \frac {e^4 (-207360-810 x)+e^9 \left (17694720+138240 x+270 x^2\right )}{81+e^4 \left (-14155776-110592 x-216 x^2\right )+e^{15} \left (-201326592-2359296 x-9216 x^2-12 x^3\right )+e^{20} \left (4294967296+67108864 x+393216 x^2+1024 x^3+x^4\right )+e^8 \left (618475290624+9663676416 x+56623104 x^2+147456 x^3+144 x^4\right )+e^3 \left (e^2 (-27648-108 x)+e^6 \left (2415919104+28311552 x+110592 x^2+144 x^3\right )\right )+e^6 \left (e^4 \left (3538944+27648 x+54 x^2\right )+e^8 \left (-103079215104-1610612736 x-9437184 x^2-24576 x^3-24 x^4\right )\right )} \, dx=\frac {135 e^{5} x^{2}-8847360 e^{5}+103680}{256 e^{16} x^{2}+131072 e^{16} x +16777216 e^{16}-3 e^{11} x^{2}-3072 e^{11} x -6144 e^{10} x^{2}-589824 e^{11}-3145728 e^{10} x -402653184 e^{10}+18 e^{6} x +36 e^{5} x^{2}+6912 e^{6}+36864 e^{5} x +36864 e^{4} x^{2}+7077888 e^{5}+18874368 e^{4} x +2415919104 e^{4}-27 e -27648} \] Input:
int(((270*x^2+138240*x+17694720)*exp(1)^6*exp(3)+(-810*x-207360)*exp(1)^4) /((x^4+1024*x^3+393216*x^2+67108864*x+4294967296)*exp(1)^8*exp(3)^4+(-12*x ^3-9216*x^2-2359296*x-201326592)*exp(1)^6*exp(3)^3+((-24*x^4-24576*x^3-943 7184*x^2-1610612736*x-103079215104)*exp(1)^8+(54*x^2+27648*x+3538944)*exp( 1)^4)*exp(3)^2+((144*x^3+110592*x^2+28311552*x+2415919104)*exp(1)^6+(-108* x-27648)*exp(1)^2)*exp(3)+(144*x^4+147456*x^3+56623104*x^2+9663676416*x+61 8475290624)*exp(1)^8+(-216*x^2-110592*x-14155776)*exp(1)^4+81),x)
Output:
(135*(e**5*x**2 - 65536*e**5 + 768))/(256*e**16*x**2 + 131072*e**16*x + 16 777216*e**16 - 3*e**11*x**2 - 3072*e**11*x - 589824*e**11 - 6144*e**10*x** 2 - 3145728*e**10*x - 402653184*e**10 + 18*e**6*x + 6912*e**6 + 36*e**5*x* *2 + 36864*e**5*x + 7077888*e**5 + 36864*e**4*x**2 + 18874368*e**4*x + 241 5919104*e**4 - 27*e - 27648)