Integrand size = 210, antiderivative size = 24 \[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\frac {\left (16-\frac {5\ 2^{1-x}}{4+x+\log (5)}\right )^2}{x^2} \] Output:
(16-10/(x+ln(5)+4)/exp(x*ln(2)))^2/x^2
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\frac {4^{1-x} \left (-5+2^{5+x}+2^{3+x} x+2^{3+x} \log (5)\right )^2}{x^2 (4+x+\log (5))^2} \] Input:
Integrate[(-800 - 400*x + (-800*x - 200*x^2)*Log[2] + (-200 - 200*x*Log[2] )*Log[5] + 2^x*(10240 + 6400*x + 960*x^2 + (5120*x + 2560*x^2 + 320*x^3)*L og[2] + (5120 + 1600*x + (2560*x + 640*x^2)*Log[2])*Log[5] + (640 + 320*x* Log[2])*Log[5]^2) + 2^(2*x)*(-32768 - 24576*x - 6144*x^2 - 512*x^3 + (-245 76 - 12288*x - 1536*x^2)*Log[5] + (-6144 - 1536*x)*Log[5]^2 - 512*Log[5]^3 ))/(2^(2*x)*(64*x^3 + 48*x^4 + 12*x^5 + x^6 + (48*x^3 + 24*x^4 + 3*x^5)*Lo g[5] + (12*x^3 + 3*x^4)*Log[5]^2 + x^3*Log[5]^3)),x]
Output:
(4^(1 - x)*(-5 + 2^(5 + x) + 2^(3 + x)*x + 2^(3 + x)*Log[5])^2)/(x^2*(4 + x + Log[5])^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 14.58 (sec) , antiderivative size = 839, normalized size of antiderivative = 34.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6, 2026, 2007, 7292, 7293, 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.
| \(\displaystyle \int \frac {2^{-2 x} \left (\left (-200 x^2-800 x\right ) \log (2)+2^x \left (960 x^2+\log (5) \left (\left (640 x^2+2560 x\right ) \log (2)+1600 x+5120\right )+\left (320 x^3+2560 x^2+5120 x\right ) \log (2)+6400 x+\log ^2(5) (320 x \log (2)+640)+10240\right )+2^{2 x} \left (-512 x^3-6144 x^2+\left (-1536 x^2-12288 x-24576\right ) \log (5)-24576 x+(-1536 x-6144) \log ^2(5)-32768-512 \log ^3(5)\right )-400 x+\log (5) (-200 x \log (2)-200)-800\right )}{x^6+12 x^5+48 x^4+64 x^3+x^3 \log ^3(5)+\left (3 x^4+12 x^3\right ) \log ^2(5)+\left (3 x^5+24 x^4+48 x^3\right ) \log (5)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2^{-2 x} \left (\left (-200 x^2-800 x\right ) \log (2)+2^x \left (960 x^2+\log (5) \left (\left (640 x^2+2560 x\right ) \log (2)+1600 x+5120\right )+\left (320 x^3+2560 x^2+5120 x\right ) \log (2)+6400 x+\log ^2(5) (320 x \log (2)+640)+10240\right )+2^{2 x} \left (-512 x^3-6144 x^2+\left (-1536 x^2-12288 x-24576\right ) \log (5)-24576 x+(-1536 x-6144) \log ^2(5)-32768-512 \log ^3(5)\right )-400 x+\log (5) (-200 x \log (2)-200)-800\right )}{x^6+12 x^5+48 x^4+x^3 \left (64+\log ^3(5)\right )+\left (3 x^4+12 x^3\right ) \log ^2(5)+\left (3 x^5+24 x^4+48 x^3\right ) \log (5)}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2^{-2 x} \left (\left (-200 x^2-800 x\right ) \log (2)+2^x \left (960 x^2+\log (5) \left (\left (640 x^2+2560 x\right ) \log (2)+1600 x+5120\right )+\left (320 x^3+2560 x^2+5120 x\right ) \log (2)+6400 x+\log ^2(5) (320 x \log (2)+640)+10240\right )+2^{2 x} \left (-512 x^3-6144 x^2+\left (-1536 x^2-12288 x-24576\right ) \log (5)-24576 x+(-1536 x-6144) \log ^2(5)-32768-512 \log ^3(5)\right )-400 x+\log (5) (-200 x \log (2)-200)-800\right )}{x^3 \left (x^3+3 x^2 (4+\log (5))+3 x (4+\log (5))^2+(4+\log (5))^3\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {2^{-2 x} \left (\left (-200 x^2-800 x\right ) \log (2)+2^x \left (960 x^2+\log (5) \left (\left (640 x^2+2560 x\right ) \log (2)+1600 x+5120\right )+\left (320 x^3+2560 x^2+5120 x\right ) \log (2)+6400 x+\log ^2(5) (320 x \log (2)+640)+10240\right )+2^{2 x} \left (-512 x^3-6144 x^2+\left (-1536 x^2-12288 x-24576\right ) \log (5)-24576 x+(-1536 x-6144) \log ^2(5)-32768-512 \log ^3(5)\right )-400 x+\log (5) (-200 x \log (2)-200)-800\right )}{x^3 (x+4+\log (5))^3}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2^{3-2 x} \left (-2^{x+3} x-2^{x+5} \left (1+\frac {\log (5)}{4}\right )+5\right ) \left (2^{x+3} x^2-5 x^2 \log (2)-10 x \left (1+\frac {1}{2} \log (2) (4+\log (5))\right )+2^{x+6} x \left (1+\frac {\log (5)}{4}\right )+2^{x+7} \left (1+\frac {1}{16} \log (5) (8+\log (5))\right )-20 \left (1+\frac {\log (5)}{4}\right )\right )}{x^3 (x+4+\log (5))^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {512}{x^3}-\frac {25\ 2^{3-2 x} (4+\log (5))}{x^3 (x+4+\log (5))^3}-\frac {25\ 2^{3-2 x} (2+\log (2) \log (5)+\log (16))}{x^2 (x+4+\log (5))^3}+\frac {5\ 2^{6-x} \left (x^2 \log (2)+x (3+\log (2) \log (5)+\log (16))+8+\log (25)\right )}{x^3 (x+4+\log (5))^2}-\frac {25\ 2^{3-2 x} \log (2)}{x (x+4+\log (5))^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {400 \log (2) (2+\log (2) \log (5)+\log (16)) \operatorname {ExpIntegralEi}(-2 x \log (2))}{(4+\log (5))^3}+\frac {600 (2+\log (2) \log (5)+\log (16)) \operatorname {ExpIntegralEi}(-2 x \log (2))}{(4+\log (5))^4}-\frac {400 \log ^2(2) \operatorname {ExpIntegralEi}(-2 x \log (2))}{(4+\log (5))^2}-\frac {1400 \log (2) \operatorname {ExpIntegralEi}(-2 x \log (2))}{(4+\log (5))^3}-\frac {1200 \operatorname {ExpIntegralEi}(-2 x \log (2))}{(4+\log (5))^4}+\frac {25\ 2^{3-2 x} \log ^2(2)}{(4+\log (5)) (x+\log (5)+4)}-\frac {5\ 2^{6-x}}{(4+\log (5))^2 (x+\log (5)+4)}-\frac {75\ 2^{3-2 x}}{(4+\log (5))^3 (x+\log (5)+4)}+\frac {256}{x^2}-\frac {25\ 2^{2-2 x} \log (2)}{(4+\log (5)) (x+\log (5)+4)^2}-\frac {25\ 2^{2-2 x}}{(4+\log (5))^2 (x+\log (5)+4)^2}+\frac {5\ 2^{6-x} \left (4-\log (2) \log ^2(5)-\log (16) \log (25)-\log \left (\frac {65536}{5}\right )\right )}{x (4+\log (5))^3}+\frac {320 \operatorname {ExpIntegralEi}(-x \log (2)) \log (2) \left (4-\log (2) \log ^2(5)-\log (16) \log (25)-\log \left (\frac {65536}{5}\right )\right )}{(4+\log (5))^3}-\frac {320 \operatorname {ExpIntegralEi}(-x \log (2)) \log (2) \left (16+\log ^2(5)+4 \log (25)\right )}{(4+\log (5))^4}+\frac {5\ 2^{10+\log (5)} \operatorname {ExpIntegralEi}(-\log (2) (x+\log (5)+4)) \log (2) \left (16+\log ^2(5)+4 \log (25)\right )}{(4+\log (5))^4}-\frac {5\ 2^{5-x} (8+\log (25))}{x^2 (4+\log (5))^2}+\frac {160 \operatorname {ExpIntegralEi}(-x \log (2)) \log ^2(2) (8+\log (25))}{(4+\log (5))^2}+\frac {5\ 2^{5-x} \log (2) (8+\log (25))}{x (4+\log (5))^2}-\frac {25\ 2^{3-2 x} \log (2) (2+\log (2) \log (5)+\log (16))}{(4+\log (5))^2 (x+\log (5)+4)}+\frac {25\ 2^{4-2 x} (2+\log (2) \log (5)+\log (16))}{(4+\log (5))^3 (x+\log (5)+4)}+\frac {25\ 2^{2-2 x} (2+\log (2) \log (5)+\log (16))}{(4+\log (5))^2 (x+\log (5)+4)^2}-\frac {25\ 2^{12+\log (25)} \operatorname {ExpIntegralEi}(-2 \log (2) (x+\log (5)+4)) \log ^2(2) (2+\log (2) \log (5)+\log (16))}{(4+\log (5))^2}+\frac {25\ 2^{3-2 x} (2+\log (2) \log (5)+\log (16))}{x (4+\log (5))^3}+\frac {25\ 2^{13+\log (25)} \operatorname {ExpIntegralEi}(-2 \log (2) (x+\log (5)+4)) \log (2) (2+\log (2) \log (5)+\log (16))}{(4+\log (5))^3}-\frac {75\ 2^{11+\log (25)} \operatorname {ExpIntegralEi}(-2 \log (2) (x+\log (5)+4)) (2+\log (2) \log (5)+\log (16))}{(4+\log (5))^4}+\frac {25\ 2^{12+\log (25)} \operatorname {ExpIntegralEi}(-2 \log (2) (x+\log (5)+4)) \log ^3(2)}{4+\log (5)}+\frac {25\ 2^{2-2 x}}{x^2 (4+\log (5))^2}-\frac {5\ 2^{10+\log (5)} \operatorname {ExpIntegralEi}(-\log (2) (x+\log (5)+4)) \log (2)}{(4+\log (5))^2}-\frac {25\ 2^{3-2 x} \log (2)}{x (4+\log (5))^2}-\frac {75\ 2^{3-2 x}}{x (4+\log (5))^3}-\frac {75\ 2^{12+\log (25)} \operatorname {ExpIntegralEi}(-2 \log (2) (x+\log (5)+4)) \log (2)}{(4+\log (5))^3}+\frac {25\ 2^{11+\log (25)} \operatorname {ExpIntegralEi}(-2 \log (2) (x+\log (5)+4)) \log (2)}{(4+\log (5))^3}+\frac {75\ 2^{12+\log (25)} \operatorname {ExpIntegralEi}(-2 \log (2) (x+\log (5)+4))}{(4+\log (5))^4}\) |
Input:
Int[(-800 - 400*x + (-800*x - 200*x^2)*Log[2] + (-200 - 200*x*Log[2])*Log[ 5] + 2^x*(10240 + 6400*x + 960*x^2 + (5120*x + 2560*x^2 + 320*x^3)*Log[2] + (5120 + 1600*x + (2560*x + 640*x^2)*Log[2])*Log[5] + (640 + 320*x*Log[2] )*Log[5]^2) + 2^(2*x)*(-32768 - 24576*x - 6144*x^2 - 512*x^3 + (-24576 - 1 2288*x - 1536*x^2)*Log[5] + (-6144 - 1536*x)*Log[5]^2 - 512*Log[5]^3))/(2^ (2*x)*(64*x^3 + 48*x^4 + 12*x^5 + x^6 + (48*x^3 + 24*x^4 + 3*x^5)*Log[5] + (12*x^3 + 3*x^4)*Log[5]^2 + x^3*Log[5]^3)),x]
Output:
256/x^2 - (1200*ExpIntegralEi[-2*x*Log[2]])/(4 + Log[5])^4 + (75*2^(12 + L og[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])])/(4 + Log[5])^4 - (75*2^ (3 - 2*x))/(x*(4 + Log[5])^3) - (1400*ExpIntegralEi[-2*x*Log[2]]*Log[2])/( 4 + Log[5])^3 + (25*2^(11 + Log[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[ 5])]*Log[2])/(4 + Log[5])^3 - (75*2^(12 + Log[25])*ExpIntegralEi[-2*Log[2] *(4 + x + Log[5])]*Log[2])/(4 + Log[5])^3 + (25*2^(2 - 2*x))/(x^2*(4 + Log [5])^2) - (25*2^(3 - 2*x)*Log[2])/(x*(4 + Log[5])^2) - (5*2^(10 + Log[5])* ExpIntegralEi[-(Log[2]*(4 + x + Log[5]))]*Log[2])/(4 + Log[5])^2 - (400*Ex pIntegralEi[-2*x*Log[2]]*Log[2]^2)/(4 + Log[5])^2 + (25*2^(12 + Log[25])*E xpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*Log[2]^3)/(4 + Log[5]) - (25*2^(2 - 2*x))/((4 + Log[5])^2*(4 + x + Log[5])^2) - (25*2^(2 - 2*x)*Log[2])/((4 + Log[5])*(4 + x + Log[5])^2) - (75*2^(3 - 2*x))/((4 + Log[5])^3*(4 + x + Log[5])) - (5*2^(6 - x))/((4 + Log[5])^2*(4 + x + Log[5])) + (25*2^(3 - 2 *x)*Log[2]^2)/((4 + Log[5])*(4 + x + Log[5])) + (600*ExpIntegralEi[-2*x*Lo g[2]]*(2 + Log[2]*Log[5] + Log[16]))/(4 + Log[5])^4 - (75*2^(11 + Log[25]) *ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*(2 + Log[2]*Log[5] + Log[16]))/ (4 + Log[5])^4 + (25*2^(3 - 2*x)*(2 + Log[2]*Log[5] + Log[16]))/(x*(4 + Lo g[5])^3) + (400*ExpIntegralEi[-2*x*Log[2]]*Log[2]*(2 + Log[2]*Log[5] + Log [16]))/(4 + Log[5])^3 + (25*2^(13 + Log[25])*ExpIntegralEi[-2*Log[2]*(4 + x + Log[5])]*Log[2]*(2 + Log[2]*Log[5] + Log[16]))/(4 + Log[5])^3 - (25...
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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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])
Time = 12.90 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71
| method | result | size |
| risch | \(\frac {256}{x^{2}}-\frac {320 \,2^{-x}}{x^{2} \left (x +\ln \left (5\right )+4\right )}+\frac {100 \,2^{-2 x}}{x^{2} \left (x +\ln \left (5\right )+4\right )^{2}}\) | \(41\) |
| norman | \(\frac {\left (100+\left (-320 \ln \left (5\right )-1280\right ) {\mathrm e}^{x \ln \left (2\right )}+\left (256 \ln \left (5\right )^{2}+2048 \ln \left (5\right )+4096\right ) {\mathrm e}^{2 x \ln \left (2\right )}+\left (512 \ln \left (5\right )+2048\right ) x \,{\mathrm e}^{2 x \ln \left (2\right )}+256 \,{\mathrm e}^{2 x \ln \left (2\right )} x^{2}-320 \,{\mathrm e}^{x \ln \left (2\right )} x \right ) {\mathrm e}^{-2 x \ln \left (2\right )}}{x^{2} \left (x +\ln \left (5\right )+4\right )^{2}}\) | \(88\) |
| parallelrisch | \(\frac {\left (100-1280 \,{\mathrm e}^{x \ln \left (2\right )}-320 \ln \left (5\right ) {\mathrm e}^{x \ln \left (2\right )}-320 \,{\mathrm e}^{x \ln \left (2\right )} x +256 \ln \left (5\right )^{2} {\mathrm e}^{2 x \ln \left (2\right )}+256 \,{\mathrm e}^{2 x \ln \left (2\right )} x^{2}+2048 \ln \left (5\right ) {\mathrm e}^{2 x \ln \left (2\right )}+2048 \,{\mathrm e}^{2 x \ln \left (2\right )} x +4096 \,{\mathrm e}^{2 x \ln \left (2\right )}+512 \ln \left (5\right ) {\mathrm e}^{2 x \ln \left (2\right )} x \right ) {\mathrm e}^{-2 x \ln \left (2\right )}}{x^{2} \left (\ln \left (5\right )^{2}+2 x \ln \left (5\right )+x^{2}+8 \ln \left (5\right )+8 x +16\right )}\) | \(128\) |
| parts | \(\text {Expression too large to display}\) | \(2805\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(6192\) |
| default | \(\text {Expression too large to display}\) | \(6192\) |
Input:
int(((-512*ln(5)^3+(-1536*x-6144)*ln(5)^2+(-1536*x^2-12288*x-24576)*ln(5)- 512*x^3-6144*x^2-24576*x-32768)*exp(x*ln(2))^2+((320*x*ln(2)+640)*ln(5)^2+ ((640*x^2+2560*x)*ln(2)+1600*x+5120)*ln(5)+(320*x^3+2560*x^2+5120*x)*ln(2) +960*x^2+6400*x+10240)*exp(x*ln(2))+(-200*x*ln(2)-200)*ln(5)+(-200*x^2-800 *x)*ln(2)-400*x-800)/(x^3*ln(5)^3+(3*x^4+12*x^3)*ln(5)^2+(3*x^5+24*x^4+48* x^3)*ln(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*ln(2))^2,x,method=_RETURNVERBOS E)
Output:
256/x^2-320/x^2/(x+ln(5)+4)/(2^x)+100/x^2/(x+ln(5)+4)^2/(2^x)^2
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (23) = 46\).
Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.50 \[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\frac {4 \, {\left (64 \, {\left (x^{2} + 2 \, {\left (x + 4\right )} \log \left (5\right ) + \log \left (5\right )^{2} + 8 \, x + 16\right )} 2^{2 \, x} - 80 \cdot 2^{x} {\left (x + \log \left (5\right ) + 4\right )} + 25\right )}}{{\left (x^{4} + x^{2} \log \left (5\right )^{2} + 8 \, x^{3} + 16 \, x^{2} + 2 \, {\left (x^{3} + 4 \, x^{2}\right )} \log \left (5\right )\right )} 2^{2 \, x}} \] Input:
integrate(((-512*log(5)^3+(-1536*x-6144)*log(5)^2+(-1536*x^2-12288*x-24576 )*log(5)-512*x^3-6144*x^2-24576*x-32768)*exp(x*log(2))^2+((320*x*log(2)+64 0)*log(5)^2+((640*x^2+2560*x)*log(2)+1600*x+5120)*log(5)+(320*x^3+2560*x^2 +5120*x)*log(2)+960*x^2+6400*x+10240)*exp(x*log(2))+(-200*x*log(2)-200)*lo g(5)+(-200*x^2-800*x)*log(2)-400*x-800)/(x^3*log(5)^3+(3*x^4+12*x^3)*log(5 )^2+(3*x^5+24*x^4+48*x^3)*log(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*log(2))^2 ,x, algorithm="fricas")
Output:
4*(64*(x^2 + 2*(x + 4)*log(5) + log(5)^2 + 8*x + 16)*2^(2*x) - 80*2^x*(x + log(5) + 4) + 25)/((x^4 + x^2*log(5)^2 + 8*x^3 + 16*x^2 + 2*(x^3 + 4*x^2) *log(5))*2^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 6.46 \[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\frac {\left (100 x^{3} + 100 x^{2} \log {\left (5 \right )} + 400 x^{2}\right ) e^{- 2 x \log {\left (2 \right )}} + \left (- 320 x^{4} - 2560 x^{3} - 640 x^{3} \log {\left (5 \right )} - 5120 x^{2} - 2560 x^{2} \log {\left (5 \right )} - 320 x^{2} \log {\left (5 \right )}^{2}\right ) e^{- x \log {\left (2 \right )}}}{x^{7} + 3 x^{6} \log {\left (5 \right )} + 12 x^{6} + 3 x^{5} \log {\left (5 \right )}^{2} + 24 x^{5} \log {\left (5 \right )} + 48 x^{5} + x^{4} \log {\left (5 \right )}^{3} + 12 x^{4} \log {\left (5 \right )}^{2} + 64 x^{4} + 48 x^{4} \log {\left (5 \right )}} + \frac {256}{x^{2}} \] Input:
integrate(((-512*ln(5)**3+(-1536*x-6144)*ln(5)**2+(-1536*x**2-12288*x-2457 6)*ln(5)-512*x**3-6144*x**2-24576*x-32768)*exp(x*ln(2))**2+((320*x*ln(2)+6 40)*ln(5)**2+((640*x**2+2560*x)*ln(2)+1600*x+5120)*ln(5)+(320*x**3+2560*x* *2+5120*x)*ln(2)+960*x**2+6400*x+10240)*exp(x*ln(2))+(-200*x*ln(2)-200)*ln (5)+(-200*x**2-800*x)*ln(2)-400*x-800)/(x**3*ln(5)**3+(3*x**4+12*x**3)*ln( 5)**2+(3*x**5+24*x**4+48*x**3)*ln(5)+x**6+12*x**5+48*x**4+64*x**3)/exp(x*l n(2))**2,x)
Output:
((100*x**3 + 100*x**2*log(5) + 400*x**2)*exp(-2*x*log(2)) + (-320*x**4 - 2 560*x**3 - 640*x**3*log(5) - 5120*x**2 - 2560*x**2*log(5) - 320*x**2*log(5 )**2)*exp(-x*log(2)))/(x**7 + 3*x**6*log(5) + 12*x**6 + 3*x**5*log(5)**2 + 24*x**5*log(5) + 48*x**5 + x**4*log(5)**3 + 12*x**4*log(5)**2 + 64*x**4 + 48*x**4*log(5)) + 256/x**2
Leaf count of result is larger than twice the leaf count of optimal. 1714 vs. \(2 (23) = 46\).
Time = 0.21 (sec) , antiderivative size = 1714, normalized size of antiderivative = 71.42 \[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\text {Too large to display} \] Input:
integrate(((-512*log(5)^3+(-1536*x-6144)*log(5)^2+(-1536*x^2-12288*x-24576 )*log(5)-512*x^3-6144*x^2-24576*x-32768)*exp(x*log(2))^2+((320*x*log(2)+64 0)*log(5)^2+((640*x^2+2560*x)*log(2)+1600*x+5120)*log(5)+(320*x^3+2560*x^2 +5120*x)*log(2)+960*x^2+6400*x+10240)*exp(x*log(2))+(-200*x*log(2)-200)*lo g(5)+(-200*x^2-800*x)*log(2)-400*x-800)/(x^3*log(5)^3+(3*x^4+12*x^3)*log(5 )^2+(3*x^5+24*x^4+48*x^3)*log(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*log(2))^2 ,x, algorithm="maxima")
Output:
-256*((12*x^3 + 18*x^2*(log(5) + 4) - log(5)^3 + 4*(log(5)^2 + 8*log(5) + 16)*x - 12*log(5)^2 - 48*log(5) - 64)/((log(5)^4 + 16*log(5)^3 + 96*log(5) ^2 + 256*log(5) + 256)*x^4 + 2*(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 64 0*log(5)^2 + 1280*log(5) + 1024)*x^3 + (log(5)^6 + 24*log(5)^5 + 240*log(5 )^4 + 1280*log(5)^3 + 3840*log(5)^2 + 6144*log(5) + 4096)*x^2) - 12*log(x + log(5) + 4)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280 *log(5) + 1024) + 12*log(x)/(log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*l og(5)^2 + 1280*log(5) + 1024))*log(5)^3 - 3072*((12*x^3 + 18*x^2*(log(5) + 4) - log(5)^3 + 4*(log(5)^2 + 8*log(5) + 16)*x - 12*log(5)^2 - 48*log(5) - 64)/((log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^4 + 2*( log(5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024) *x^3 + (log(5)^6 + 24*log(5)^5 + 240*log(5)^4 + 1280*log(5)^3 + 3840*log(5 )^2 + 6144*log(5) + 4096)*x^2) - 12*log(x + log(5) + 4)/(log(5)^5 + 20*log (5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024) + 12*log(x)/(log (5)^5 + 20*log(5)^4 + 160*log(5)^3 + 640*log(5)^2 + 1280*log(5) + 1024))*l og(5)^2 + 768*((6*x^2 + 9*x*(log(5) + 4) + 2*log(5)^2 + 16*log(5) + 32)/(( log(5)^3 + 12*log(5)^2 + 48*log(5) + 64)*x^3 + 2*(log(5)^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256)*x^2 + (log(5)^5 + 20*log(5)^4 + 160*log(5 )^3 + 640*log(5)^2 + 1280*log(5) + 1024)*x) - 6*log(x + log(5) + 4)/(log(5 )^4 + 16*log(5)^3 + 96*log(5)^2 + 256*log(5) + 256) + 6*log(x)/(log(5)^...
\[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\int { -\frac {8 \, {\left (64 \, {\left (x^{3} + 3 \, {\left (x + 4\right )} \log \left (5\right )^{2} + \log \left (5\right )^{3} + 12 \, x^{2} + 3 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (5\right ) + 48 \, x + 64\right )} 2^{2 \, x} - 40 \, {\left ({\left (x \log \left (2\right ) + 2\right )} \log \left (5\right )^{2} + 3 \, x^{2} + {\left (2 \, {\left (x^{2} + 4 \, x\right )} \log \left (2\right ) + 5 \, x + 16\right )} \log \left (5\right ) + {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (2\right ) + 20 \, x + 32\right )} 2^{x} + 25 \, {\left (x \log \left (2\right ) + 1\right )} \log \left (5\right ) + 25 \, {\left (x^{2} + 4 \, x\right )} \log \left (2\right ) + 50 \, x + 100\right )}}{{\left (x^{6} + x^{3} \log \left (5\right )^{3} + 12 \, x^{5} + 48 \, x^{4} + 64 \, x^{3} + 3 \, {\left (x^{4} + 4 \, x^{3}\right )} \log \left (5\right )^{2} + 3 \, {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3}\right )} \log \left (5\right )\right )} 2^{2 \, x}} \,d x } \] Input:
integrate(((-512*log(5)^3+(-1536*x-6144)*log(5)^2+(-1536*x^2-12288*x-24576 )*log(5)-512*x^3-6144*x^2-24576*x-32768)*exp(x*log(2))^2+((320*x*log(2)+64 0)*log(5)^2+((640*x^2+2560*x)*log(2)+1600*x+5120)*log(5)+(320*x^3+2560*x^2 +5120*x)*log(2)+960*x^2+6400*x+10240)*exp(x*log(2))+(-200*x*log(2)-200)*lo g(5)+(-200*x^2-800*x)*log(2)-400*x-800)/(x^3*log(5)^3+(3*x^4+12*x^3)*log(5 )^2+(3*x^5+24*x^4+48*x^3)*log(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*log(2))^2 ,x, algorithm="giac")
Output:
integrate(-8*(64*(x^3 + 3*(x + 4)*log(5)^2 + log(5)^3 + 12*x^2 + 3*(x^2 + 8*x + 16)*log(5) + 48*x + 64)*2^(2*x) - 40*((x*log(2) + 2)*log(5)^2 + 3*x^ 2 + (2*(x^2 + 4*x)*log(2) + 5*x + 16)*log(5) + (x^3 + 8*x^2 + 16*x)*log(2) + 20*x + 32)*2^x + 25*(x*log(2) + 1)*log(5) + 25*(x^2 + 4*x)*log(2) + 50* x + 100)/((x^6 + x^3*log(5)^3 + 12*x^5 + 48*x^4 + 64*x^3 + 3*(x^4 + 4*x^3) *log(5)^2 + 3*(x^5 + 8*x^4 + 16*x^3)*log(5))*2^(2*x)), x)
Timed out. \[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\text {Hanged} \] Input:
int(-(exp(-2*x*log(2))*(400*x + log(2)*(800*x + 200*x^2) + log(5)*(200*x*l og(2) + 200) + exp(2*x*log(2))*(24576*x + log(5)*(12288*x + 1536*x^2 + 245 76) + log(5)^2*(1536*x + 6144) + 512*log(5)^3 + 6144*x^2 + 512*x^3 + 32768 ) - exp(x*log(2))*(6400*x + log(5)*(1600*x + log(2)*(2560*x + 640*x^2) + 5 120) + log(2)*(5120*x + 2560*x^2 + 320*x^3) + log(5)^2*(320*x*log(2) + 640 ) + 960*x^2 + 10240) + 800))/(x^3*log(5)^3 + log(5)*(48*x^3 + 24*x^4 + 3*x ^5) + 64*x^3 + 48*x^4 + 12*x^5 + x^6 + log(5)^2*(12*x^3 + 3*x^4)),x)
Output:
\text{Hanged}
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.58 \[ \int \frac {2^{-2 x} \left (-800-400 x+\left (-800 x-200 x^2\right ) \log (2)+(-200-200 x \log (2)) \log (5)+2^x \left (10240+6400 x+960 x^2+\left (5120 x+2560 x^2+320 x^3\right ) \log (2)+\left (5120+1600 x+\left (2560 x+640 x^2\right ) \log (2)\right ) \log (5)+(640+320 x \log (2)) \log ^2(5)\right )+2^{2 x} \left (-32768-24576 x-6144 x^2-512 x^3+\left (-24576-12288 x-1536 x^2\right ) \log (5)+(-6144-1536 x) \log ^2(5)-512 \log ^3(5)\right )\right )}{64 x^3+48 x^4+12 x^5+x^6+\left (48 x^3+24 x^4+3 x^5\right ) \log (5)+\left (12 x^3+3 x^4\right ) \log ^2(5)+x^3 \log ^3(5)} \, dx=\frac {256 \,2^{2 x} \mathrm {log}\left (5\right )^{2}+512 \,2^{2 x} \mathrm {log}\left (5\right ) x +2048 \,2^{2 x} \mathrm {log}\left (5\right )+256 \,2^{2 x} x^{2}+2048 \,2^{2 x} x +4096 \,2^{2 x}-320 \,2^{x} \mathrm {log}\left (5\right )-320 \,2^{x} x -1280 \,2^{x}+100}{2^{2 x} x^{2} \left (\mathrm {log}\left (5\right )^{2}+2 \,\mathrm {log}\left (5\right ) x +8 \,\mathrm {log}\left (5\right )+x^{2}+8 x +16\right )} \] Input:
int(((-512*log(5)^3+(-1536*x-6144)*log(5)^2+(-1536*x^2-12288*x-24576)*log( 5)-512*x^3-6144*x^2-24576*x-32768)*exp(x*log(2))^2+((320*x*log(2)+640)*log (5)^2+((640*x^2+2560*x)*log(2)+1600*x+5120)*log(5)+(320*x^3+2560*x^2+5120* x)*log(2)+960*x^2+6400*x+10240)*exp(x*log(2))+(-200*x*log(2)-200)*log(5)+( -200*x^2-800*x)*log(2)-400*x-800)/(x^3*log(5)^3+(3*x^4+12*x^3)*log(5)^2+(3 *x^5+24*x^4+48*x^3)*log(5)+x^6+12*x^5+48*x^4+64*x^3)/exp(x*log(2))^2,x)
Output:
(4*(64*2**(2*x)*log(5)**2 + 128*2**(2*x)*log(5)*x + 512*2**(2*x)*log(5) + 64*2**(2*x)*x**2 + 512*2**(2*x)*x + 1024*2**(2*x) - 80*2**x*log(5) - 80*2* *x*x - 320*2**x + 25))/(2**(2*x)*x**2*(log(5)**2 + 2*log(5)*x + 8*log(5) + x**2 + 8*x + 16))