Integrand size = 244, antiderivative size = 32 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\left (\frac {x}{8}+e^5 \left (-4+e^{\left (1+x^2\right )^2}-\frac {\log (3)}{1+x}\right )\right )^2 \] Output:
(exp(5)*(exp((x^2+1)^2)-ln(3)/(1+x)-4)+1/8*x)^2
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(32)=64\).
Time = 0.81 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.41 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=e^{2 \left (6+2 x^2+x^4\right )}+\frac {1}{4} e^{6+2 x^2+x^4} x+\frac {x^2}{64}-\frac {e^5 \left (4+8 x+4 x^2-\log (3)\right )}{4 (1+x)}-\frac {2 e^{11+2 x^2+x^4} (4+4 x+\log (3))}{1+x}+\frac {e^{10} \log (3) (8+8 x+\log (3))}{(1+x)^2} \] Input:
Integrate[(x + 3*x^2 + 3*x^3 + x^4 + E^5*(-32 - 96*x - 96*x^2 - 32*x^3) + E^(12 + 4*x^2 + 2*x^4)*(256*x + 768*x^2 + 1024*x^3 + 1024*x^4 + 768*x^5 + 256*x^6) + (E^10*(-256 - 256*x) + E^5*(-8 - 8*x))*Log[3] - 64*E^10*Log[3]^ 2 + E^(1 + 2*x^2 + x^4)*(E^10*(-1024*x - 3072*x^2 - 4096*x^3 - 4096*x^4 - 3072*x^5 - 1024*x^6) + E^5*(8 + 24*x + 56*x^2 + 104*x^3 + 128*x^4 + 128*x^ 5 + 96*x^6 + 32*x^7) + E^10*(64 - 192*x - 512*x^2 - 512*x^3 - 512*x^4 - 25 6*x^5)*Log[3]))/(32 + 96*x + 96*x^2 + 32*x^3),x]
Output:
E^(2*(6 + 2*x^2 + x^4)) + (E^(6 + 2*x^2 + x^4)*x)/4 + x^2/64 - (E^5*(4 + 8 *x + 4*x^2 - Log[3]))/(4*(1 + x)) - (2*E^(11 + 2*x^2 + x^4)*(4 + 4*x + Log [3]))/(1 + x) + (E^10*Log[3]*(8 + 8*x + Log[3]))/(1 + x)^2
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.
| \(\displaystyle \int \frac {x^4+3 x^3+3 x^2+e^5 \left (-32 x^3-96 x^2-96 x-32\right )+e^{2 x^4+4 x^2+12} \left (256 x^6+768 x^5+1024 x^4+1024 x^3+768 x^2+256 x\right )+e^{x^4+2 x^2+1} \left (e^{10} \left (-256 x^5-512 x^4-512 x^3-512 x^2-192 x+64\right ) \log (3)+e^{10} \left (-1024 x^6-3072 x^5-4096 x^4-4096 x^3-3072 x^2-1024 x\right )+e^5 \left (32 x^7+96 x^6+128 x^5+128 x^4+104 x^3+56 x^2+24 x+8\right )\right )+x+\left (e^{10} (-256 x-256)+e^5 (-8 x-8)\right ) \log (3)-64 e^{10} \log ^2(3)}{32 x^3+96 x^2+96 x+32} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {x^4+3 x^3+3 x^2+e^5 \left (-32 x^3-96 x^2-96 x-32\right )+e^{2 x^4+4 x^2+12} \left (256 x^6+768 x^5+1024 x^4+1024 x^3+768 x^2+256 x\right )+e^{x^4+2 x^2+1} \left (e^{10} \left (-256 x^5-512 x^4-512 x^3-512 x^2-192 x+64\right ) \log (3)+e^{10} \left (-1024 x^6-3072 x^5-4096 x^4-4096 x^3-3072 x^2-1024 x\right )+e^5 \left (32 x^7+96 x^6+128 x^5+128 x^4+104 x^3+56 x^2+24 x+8\right )\right )+x+\left (e^{10} (-256 x-256)+e^5 (-8 x-8)\right ) \log (3)-64 e^{10} \log ^2(3)}{\left (2\ 2^{2/3} x+2\ 2^{2/3}\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^4}{32 (x+1)^3}+\frac {3 x^3}{32 (x+1)^3}+\frac {3 x^2}{32 (x+1)^3}+8 e^{2 x^4+4 x^2+12} \left (x^2+1\right ) x+\frac {e^{x^4+2 x^2+6} \left (4 x^6+8 \left (1-16 e^5\right ) x^5+8 x^4 \left (1-4 e^5 (8+\log (3))\right )+8 x^3 \left (1-4 e^5 (8+\log (3))\right )+5 x^2 \left (1-\frac {32}{5} e^5 (8+\log (3))\right )+2 x \left (1-16 e^5 (4+\log (3))\right )+1+8 e^5 \log (3)\right )}{4 (x+1)^2}+\frac {x}{32 (x+1)^3}-\frac {2 e^{10} \log ^2(3)}{(x+1)^3}-\frac {e^5 \left (1+32 e^5\right ) \log (3)}{4 (x+1)^2}-e^5\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int e^{x^4+2 x^2+6} x^4dx+\frac {1}{4} \left (1-64 e^5 \log (3)\right ) \int e^{x^4+2 x^2+6}dx+\left (1-8 e^5 \log (3)\right ) \int e^{x^4+2 x^2+6} x^2dx+2 \log (3) \int \frac {e^{x^4+2 x^2+11}}{(x+1)^2}dx+16 \log (3) \int \frac {e^{x^4+2 x^2+11}}{x+1}dx+8 e^{10} \sqrt {\pi } \text {erfi}\left (x^2+1\right )-2 e^{10} \sqrt {\pi } (4-\log (3)) \text {erfi}\left (x^2+1\right )+\frac {x^2}{64 (x+1)^2}+\frac {x^2}{64}-8 e^{x^4+2 x^2+11}+e^{2 x^4+4 x^2+12}-e^5 x+\frac {1}{32 (x+1)}-\frac {1}{64 (x+1)^2}+\frac {e^{10} \log ^2(3)}{(x+1)^2}+\frac {e^5 \left (1+32 e^5\right ) \log (3)}{4 (x+1)}\) |
Input:
Int[(x + 3*x^2 + 3*x^3 + x^4 + E^5*(-32 - 96*x - 96*x^2 - 32*x^3) + E^(12 + 4*x^2 + 2*x^4)*(256*x + 768*x^2 + 1024*x^3 + 1024*x^4 + 768*x^5 + 256*x^ 6) + (E^10*(-256 - 256*x) + E^5*(-8 - 8*x))*Log[3] - 64*E^10*Log[3]^2 + E^ (1 + 2*x^2 + x^4)*(E^10*(-1024*x - 3072*x^2 - 4096*x^3 - 4096*x^4 - 3072*x ^5 - 1024*x^6) + E^5*(8 + 24*x + 56*x^2 + 104*x^3 + 128*x^4 + 128*x^5 + 96 *x^6 + 32*x^7) + E^10*(64 - 192*x - 512*x^2 - 512*x^3 - 512*x^4 - 256*x^5) *Log[3]))/(32 + 96*x + 96*x^2 + 32*x^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(28)=56\).
Time = 13.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.56
| method | result | size |
| risch | \(\frac {x^{2}}{64}-x \,{\mathrm e}^{5}+\frac {\frac {\left (256 \ln \left (3\right ) {\mathrm e}^{10}+8 \ln \left (3\right ) {\mathrm e}^{5}\right ) x}{32}+{\mathrm e}^{10} \ln \left (3\right )^{2}+8 \ln \left (3\right ) {\mathrm e}^{10}+\frac {\ln \left (3\right ) {\mathrm e}^{5}}{4}}{x^{2}+2 x +1}+{\mathrm e}^{2 x^{4}+4 x^{2}+12}-\frac {\left (8 \ln \left (3\right ) {\mathrm e}^{5}+32 x \,{\mathrm e}^{5}-x^{2}+32 \,{\mathrm e}^{5}-x \right ) {\mathrm e}^{x^{4}+2 x^{2}+6}}{4 \left (1+x \right )}\) | \(114\) |
| parts | \(\frac {x^{2}}{64}-x \,{\mathrm e}^{5}+\frac {{\mathrm e}^{10} \ln \left (3\right )^{2}}{\left (1+x \right )^{2}}+\frac {\ln \left (3\right ) \left (32 \,{\mathrm e}^{10}+{\mathrm e}^{5}\right )}{4+4 x}+\frac {\left (-2 \ln \left (3\right ) {\mathrm e}^{10}-8 \,{\mathrm e}^{10}\right ) {\mathrm e}^{x^{4}+2 x^{2}+1}+\left (\frac {{\mathrm e}^{5}}{4}-8 \,{\mathrm e}^{10}\right ) x \,{\mathrm e}^{x^{4}+2 x^{2}+1}+\frac {{\mathrm e}^{5} {\mathrm e}^{x^{4}+2 x^{2}+1} x^{2}}{4}}{1+x}+{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2}\) | \(134\) |
| norman | \(\frac {\left (\frac {1}{32}-{\mathrm e}^{5}\right ) x^{3}+\left (-2 \ln \left (3\right ) {\mathrm e}^{10}-8 \,{\mathrm e}^{10}\right ) {\mathrm e}^{x^{4}+2 x^{2}+1}+\left (-\frac {1}{32}+3 \,{\mathrm e}^{5}+8 \ln \left (3\right ) {\mathrm e}^{10}+\frac {\ln \left (3\right ) {\mathrm e}^{5}}{4}\right ) x +{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2}+\left (\frac {{\mathrm e}^{5}}{2}-8 \,{\mathrm e}^{10}\right ) x^{2} {\mathrm e}^{x^{4}+2 x^{2}+1}+\left (-2 \ln \left (3\right ) {\mathrm e}^{10}-16 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{5}}{4}\right ) x \,{\mathrm e}^{x^{4}+2 x^{2}+1}+{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2} x^{2}+\frac {x^{4}}{64}+\frac {{\mathrm e}^{5} {\mathrm e}^{x^{4}+2 x^{2}+1} x^{3}}{4}+2 \,{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2} x -\frac {1}{64}+2 \,{\mathrm e}^{5}+8 \ln \left (3\right ) {\mathrm e}^{10}+\frac {\ln \left (3\right ) {\mathrm e}^{5}}{4}+{\mathrm e}^{10} \ln \left (3\right )^{2}}{\left (1+x \right )^{2}}\) | \(235\) |
| parallelrisch | \(\frac {-1-2 x +64 \,{\mathrm e}^{10} \ln \left (3\right )^{2}+16 x \,{\mathrm e}^{5} \ln \left (3\right )-64 x^{3} {\mathrm e}^{5}+192 x \,{\mathrm e}^{5}-128 \,{\mathrm e}^{x^{4}+2 x^{2}+1} \ln \left (3\right ) {\mathrm e}^{10} x +128 \,{\mathrm e}^{5}+2 x^{3}+x^{4}+128 \,{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2} x -512 \,{\mathrm e}^{10} {\mathrm e}^{x^{4}+2 x^{2}+1} x^{2}+16 \,{\mathrm e}^{5} {\mathrm e}^{x^{4}+2 x^{2}+1} x^{3}-128 \,{\mathrm e}^{x^{4}+2 x^{2}+1} \ln \left (3\right ) {\mathrm e}^{10}+512 \ln \left (3\right ) {\mathrm e}^{10} x -1024 \,{\mathrm e}^{x^{4}+2 x^{2}+1} x \,{\mathrm e}^{10}+32 \,{\mathrm e}^{5} {\mathrm e}^{x^{4}+2 x^{2}+1} x^{2}+64 \,{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2}+512 \ln \left (3\right ) {\mathrm e}^{10}-512 \,{\mathrm e}^{10} {\mathrm e}^{x^{4}+2 x^{2}+1}+16 \ln \left (3\right ) {\mathrm e}^{5}+16 \,{\mathrm e}^{5} {\mathrm e}^{x^{4}+2 x^{2}+1} x +64 \,{\mathrm e}^{10} {\mathrm e}^{2 x^{4}+4 x^{2}+2} x^{2}}{64 x^{2}+128 x +64}\) | \(289\) |
Input:
int(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp(x^4+2* x^2+1)^2+((-256*x^5-512*x^4-512*x^3-512*x^2-192*x+64)*exp(5)^2*ln(3)+(-102 4*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^7+96*x^6+ 128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64*exp(5)^ 2*ln(3)^2+((-256*x-256)*exp(5)^2+(-8*x-8)*exp(5))*ln(3)+(-32*x^3-96*x^2-96 *x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x,method=_RETURNV ERBOSE)
Output:
1/64*x^2-x*exp(5)+(1/32*(256*ln(3)*exp(10)+8*ln(3)*exp(5))*x+exp(10)*ln(3) ^2+8*ln(3)*exp(10)+1/4*ln(3)*exp(5))/(x^2+2*x+1)+exp(2*x^4+4*x^2+12)-1/4/( 1+x)*(8*ln(3)*exp(5)+32*x*exp(5)-x^2+32*exp(5)-x)*exp(x^4+2*x^2+6)
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (32) = 64\).
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.25 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {x^{4} + 2 \, x^{3} + 64 \, e^{10} \log \left (3\right )^{2} + x^{2} - 64 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{5} + 64 \, {\left (x^{2} + 2 \, x + 1\right )} e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 16 \, {\left (8 \, {\left (x + 1\right )} e^{10} \log \left (3\right ) + 32 \, {\left (x^{2} + 2 \, x + 1\right )} e^{10} - {\left (x^{3} + 2 \, x^{2} + x\right )} e^{5}\right )} e^{\left (x^{4} + 2 \, x^{2} + 1\right )} + 16 \, {\left (32 \, {\left (x + 1\right )} e^{10} + {\left (x + 1\right )} e^{5}\right )} \log \left (3\right )}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} \] Input:
integrate(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp( x^4+2*x^2+1)^2+((-256*x^5-512*x^4-512*x^3-512*x^2-192*x+64)*exp(5)^2*log(3 )+(-1024*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^7+ 96*x^6+128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64* exp(5)^2*log(3)^2+((-256*x-256)*exp(5)^2+(-8*x-8)*exp(5))*log(3)+(-32*x^3- 96*x^2-96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x, algor ithm="fricas")
Output:
1/64*(x^4 + 2*x^3 + 64*e^10*log(3)^2 + x^2 - 64*(x^3 + 2*x^2 + x)*e^5 + 64 *(x^2 + 2*x + 1)*e^(2*x^4 + 4*x^2 + 12) - 16*(8*(x + 1)*e^10*log(3) + 32*( x^2 + 2*x + 1)*e^10 - (x^3 + 2*x^2 + x)*e^5)*e^(x^4 + 2*x^2 + 1) + 16*(32* (x + 1)*e^10 + (x + 1)*e^5)*log(3))/(x^2 + 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (24) = 48\).
Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.25 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {x^{2}}{64} - x e^{5} + \frac {x \left (e^{5} \log {\left (3 \right )} + 32 e^{10} \log {\left (3 \right )}\right ) + e^{5} \log {\left (3 \right )} + 4 e^{10} \log {\left (3 \right )}^{2} + 32 e^{10} \log {\left (3 \right )}}{4 x^{2} + 8 x + 4} + \frac {\left (4 x e^{10} + 4 e^{10}\right ) e^{2 x^{4} + 4 x^{2} + 2} + \left (x^{2} e^{5} - 32 x e^{10} + x e^{5} - 32 e^{10} - 8 e^{10} \log {\left (3 \right )}\right ) e^{x^{4} + 2 x^{2} + 1}}{4 x + 4} \] Input:
integrate(((256*x**6+768*x**5+1024*x**4+1024*x**3+768*x**2+256*x)*exp(5)** 2*exp(x**4+2*x**2+1)**2+((-256*x**5-512*x**4-512*x**3-512*x**2-192*x+64)*e xp(5)**2*ln(3)+(-1024*x**6-3072*x**5-4096*x**4-4096*x**3-3072*x**2-1024*x) *exp(5)**2+(32*x**7+96*x**6+128*x**5+128*x**4+104*x**3+56*x**2+24*x+8)*exp (5))*exp(x**4+2*x**2+1)-64*exp(5)**2*ln(3)**2+((-256*x-256)*exp(5)**2+(-8* x-8)*exp(5))*ln(3)+(-32*x**3-96*x**2-96*x-32)*exp(5)+x**4+3*x**3+3*x**2+x) /(32*x**3+96*x**2+96*x+32),x)
Output:
x**2/64 - x*exp(5) + (x*(exp(5)*log(3) + 32*exp(10)*log(3)) + exp(5)*log(3 ) + 4*exp(10)*log(3)**2 + 32*exp(10)*log(3))/(4*x**2 + 8*x + 4) + ((4*x*ex p(10) + 4*exp(10))*exp(2*x**4 + 4*x**2 + 2) + (x**2*exp(5) - 32*x*exp(10) + x*exp(5) - 32*exp(10) - 8*exp(10)*log(3))*exp(x**4 + 2*x**2 + 1))/(4*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (32) = 64\).
Time = 0.18 (sec) , antiderivative size = 323, normalized size of antiderivative = 10.09 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {1}{64} \, x^{2} - \frac {1}{2} \, {\left (2 \, x - \frac {6 \, x + 5}{x^{2} + 2 \, x + 1} - 6 \, \log \left (x + 1\right )\right )} e^{5} - \frac {3}{2} \, {\left (\frac {4 \, x + 3}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right )\right )} e^{5} + \frac {4 \, {\left (2 \, x + 1\right )} e^{10} \log \left (3\right )}{x^{2} + 2 \, x + 1} + \frac {{\left (2 \, x + 1\right )} e^{5} \log \left (3\right )}{8 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {e^{10} \log \left (3\right )^{2}}{x^{2} + 2 \, x + 1} + \frac {3 \, {\left (2 \, x + 1\right )} e^{5}}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {4 \, e^{10} \log \left (3\right )}{x^{2} + 2 \, x + 1} + \frac {e^{5} \log \left (3\right )}{8 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {8 \, x + 7}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {3 \, {\left (6 \, x + 5\right )}}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {3 \, {\left (4 \, x + 3\right )}}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {2 \, x + 1}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {4 \, {\left (x e^{12} + e^{12}\right )} e^{\left (2 \, x^{4} + 4 \, x^{2}\right )} + {\left (x^{2} e^{6} - x {\left (32 \, e^{11} - e^{6}\right )} - 8 \, {\left (\log \left (3\right ) + 4\right )} e^{11}\right )} e^{\left (x^{4} + 2 \, x^{2}\right )}}{4 \, {\left (x + 1\right )}} + \frac {e^{5}}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} \] Input:
integrate(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp( x^4+2*x^2+1)^2+((-256*x^5-512*x^4-512*x^3-512*x^2-192*x+64)*exp(5)^2*log(3 )+(-1024*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^7+ 96*x^6+128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64* exp(5)^2*log(3)^2+((-256*x-256)*exp(5)^2+(-8*x-8)*exp(5))*log(3)+(-32*x^3- 96*x^2-96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x, algor ithm="maxima")
Output:
1/64*x^2 - 1/2*(2*x - (6*x + 5)/(x^2 + 2*x + 1) - 6*log(x + 1))*e^5 - 3/2* ((4*x + 3)/(x^2 + 2*x + 1) + 2*log(x + 1))*e^5 + 4*(2*x + 1)*e^10*log(3)/( x^2 + 2*x + 1) + 1/8*(2*x + 1)*e^5*log(3)/(x^2 + 2*x + 1) + e^10*log(3)^2/ (x^2 + 2*x + 1) + 3/2*(2*x + 1)*e^5/(x^2 + 2*x + 1) + 4*e^10*log(3)/(x^2 + 2*x + 1) + 1/8*e^5*log(3)/(x^2 + 2*x + 1) + 1/64*(8*x + 7)/(x^2 + 2*x + 1 ) - 3/64*(6*x + 5)/(x^2 + 2*x + 1) + 3/64*(4*x + 3)/(x^2 + 2*x + 1) - 1/64 *(2*x + 1)/(x^2 + 2*x + 1) + 1/4*(4*(x*e^12 + e^12)*e^(2*x^4 + 4*x^2) + (x ^2*e^6 - x*(32*e^11 - e^6) - 8*(log(3) + 4)*e^11)*e^(x^4 + 2*x^2))/(x + 1) + 1/2*e^5/(x^2 + 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (32) = 64\).
Time = 0.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 7.69 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {x^{4} - 64 \, x^{3} e^{5} + 16 \, x^{3} e^{\left (x^{4} + 2 \, x^{2} + 6\right )} + 2 \, x^{3} - 128 \, x^{2} e^{5} + 64 \, x^{2} e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 512 \, x^{2} e^{\left (x^{4} + 2 \, x^{2} + 11\right )} + 32 \, x^{2} e^{\left (x^{4} + 2 \, x^{2} + 6\right )} + 512 \, x e^{10} \log \left (3\right ) + 16 \, x e^{5} \log \left (3\right ) - 128 \, x e^{\left (x^{4} + 2 \, x^{2} + 11\right )} \log \left (3\right ) + 64 \, e^{10} \log \left (3\right )^{2} + x^{2} - 64 \, x e^{5} + 128 \, x e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 1024 \, x e^{\left (x^{4} + 2 \, x^{2} + 11\right )} + 16 \, x e^{\left (x^{4} + 2 \, x^{2} + 6\right )} + 512 \, e^{10} \log \left (3\right ) + 16 \, e^{5} \log \left (3\right ) - 128 \, e^{\left (x^{4} + 2 \, x^{2} + 11\right )} \log \left (3\right ) + 64 \, e^{\left (2 \, x^{4} + 4 \, x^{2} + 12\right )} - 512 \, e^{\left (x^{4} + 2 \, x^{2} + 11\right )}}{64 \, {\left (x^{2} + 2 \, x + 1\right )}} \] Input:
integrate(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp( x^4+2*x^2+1)^2+((-256*x^5-512*x^4-512*x^3-512*x^2-192*x+64)*exp(5)^2*log(3 )+(-1024*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^7+ 96*x^6+128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64* exp(5)^2*log(3)^2+((-256*x-256)*exp(5)^2+(-8*x-8)*exp(5))*log(3)+(-32*x^3- 96*x^2-96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x, algor ithm="giac")
Output:
1/64*(x^4 - 64*x^3*e^5 + 16*x^3*e^(x^4 + 2*x^2 + 6) + 2*x^3 - 128*x^2*e^5 + 64*x^2*e^(2*x^4 + 4*x^2 + 12) - 512*x^2*e^(x^4 + 2*x^2 + 11) + 32*x^2*e^ (x^4 + 2*x^2 + 6) + 512*x*e^10*log(3) + 16*x*e^5*log(3) - 128*x*e^(x^4 + 2 *x^2 + 11)*log(3) + 64*e^10*log(3)^2 + x^2 - 64*x*e^5 + 128*x*e^(2*x^4 + 4 *x^2 + 12) - 1024*x*e^(x^4 + 2*x^2 + 11) + 16*x*e^(x^4 + 2*x^2 + 6) + 512* e^10*log(3) + 16*e^5*log(3) - 128*e^(x^4 + 2*x^2 + 11)*log(3) + 64*e^(2*x^ 4 + 4*x^2 + 12) - 512*e^(x^4 + 2*x^2 + 11))/(x^2 + 2*x + 1)
Time = 3.19 (sec) , antiderivative size = 218, normalized size of antiderivative = 6.81 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=-\frac {x^2\,\left (64\,{\mathrm {e}}^5+8\,{\mathrm {e}}^5\,\ln \left (3\right )+256\,{\mathrm {e}}^{10}\,\ln \left (3\right )+32\,{\mathrm {e}}^{10}\,{\ln \left (3\right )}^2-\frac {1}{2}\right )-64\,x\,{\mathrm {e}}^{2\,x^4+4\,x^2+12}-8\,x^3\,{\mathrm {e}}^{x^4+2\,x^2+6}-32\,{\mathrm {e}}^{2\,x^4+4\,x^2+12}+x^3\,\left (32\,{\mathrm {e}}^5-1\right )+{\mathrm {e}}^{x^4+2\,x^2+11}\,\left (64\,\ln \left (3\right )+256\right )-32\,x^2\,{\mathrm {e}}^{2\,x^4+4\,x^2+12}-\frac {x^4}{2}+x\,\left (32\,{\mathrm {e}}^5+8\,{\mathrm {e}}^5\,\ln \left (3\right )+256\,{\mathrm {e}}^{10}\,\ln \left (3\right )+64\,{\mathrm {e}}^{10}\,{\ln \left (3\right )}^2\right )+8\,x\,{\mathrm {e}}^{x^4+2\,x^2+6}\,\left (64\,{\mathrm {e}}^5+8\,{\mathrm {e}}^5\,\ln \left (3\right )-1\right )+16\,x^2\,{\mathrm {e}}^{x^4+2\,x^2+6}\,\left (16\,{\mathrm {e}}^5-1\right )}{32\,x^2+64\,x+32} \] Input:
int((x - log(3)*(exp(5)*(8*x + 8) + exp(10)*(256*x + 256)) - exp(2*x^2 + x ^4 + 1)*(exp(10)*(1024*x + 3072*x^2 + 4096*x^3 + 4096*x^4 + 3072*x^5 + 102 4*x^6) - exp(5)*(24*x + 56*x^2 + 104*x^3 + 128*x^4 + 128*x^5 + 96*x^6 + 32 *x^7 + 8) + exp(10)*log(3)*(192*x + 512*x^2 + 512*x^3 + 512*x^4 + 256*x^5 - 64)) - exp(5)*(96*x + 96*x^2 + 32*x^3 + 32) - 64*exp(10)*log(3)^2 + 3*x^ 2 + 3*x^3 + x^4 + exp(4*x^2 + 2*x^4 + 2)*exp(10)*(256*x + 768*x^2 + 1024*x ^3 + 1024*x^4 + 768*x^5 + 256*x^6))/(96*x + 96*x^2 + 32*x^3 + 32),x)
Output:
-(x^2*(64*exp(5) + 8*exp(5)*log(3) + 256*exp(10)*log(3) + 32*exp(10)*log(3 )^2 - 1/2) - 64*x*exp(4*x^2 + 2*x^4 + 12) - 8*x^3*exp(2*x^2 + x^4 + 6) - 3 2*exp(4*x^2 + 2*x^4 + 12) + x^3*(32*exp(5) - 1) + exp(2*x^2 + x^4 + 11)*(6 4*log(3) + 256) - 32*x^2*exp(4*x^2 + 2*x^4 + 12) - x^4/2 + x*(32*exp(5) + 8*exp(5)*log(3) + 256*exp(10)*log(3) + 64*exp(10)*log(3)^2) + 8*x*exp(2*x^ 2 + x^4 + 6)*(64*exp(5) + 8*exp(5)*log(3) - 1) + 16*x^2*exp(2*x^2 + x^4 + 6)*(16*exp(5) - 1))/(64*x + 32*x^2 + 32)
Time = 0.22 (sec) , antiderivative size = 291, normalized size of antiderivative = 9.09 \[ \int \frac {x+3 x^2+3 x^3+x^4+e^5 \left (-32-96 x-96 x^2-32 x^3\right )+e^{12+4 x^2+2 x^4} \left (256 x+768 x^2+1024 x^3+1024 x^4+768 x^5+256 x^6\right )+\left (e^{10} (-256-256 x)+e^5 (-8-8 x)\right ) \log (3)-64 e^{10} \log ^2(3)+e^{1+2 x^2+x^4} \left (e^{10} \left (-1024 x-3072 x^2-4096 x^3-4096 x^4-3072 x^5-1024 x^6\right )+e^5 \left (8+24 x+56 x^2+104 x^3+128 x^4+128 x^5+96 x^6+32 x^7\right )+e^{10} \left (64-192 x-512 x^2-512 x^3-512 x^4-256 x^5\right ) \log (3)\right )}{32+96 x+96 x^2+32 x^3} \, dx=\frac {64 e^{2 x^{4}+4 x^{2}} e^{12} x^{2}+128 e^{2 x^{4}+4 x^{2}} e^{12} x +64 e^{2 x^{4}+4 x^{2}} e^{12}-128 e^{x^{4}+2 x^{2}} \mathrm {log}\left (3\right ) e^{11} x -128 e^{x^{4}+2 x^{2}} \mathrm {log}\left (3\right ) e^{11}-512 e^{x^{4}+2 x^{2}} e^{11} x^{2}-1024 e^{x^{4}+2 x^{2}} e^{11} x -512 e^{x^{4}+2 x^{2}} e^{11}+16 e^{x^{4}+2 x^{2}} e^{6} x^{3}+32 e^{x^{4}+2 x^{2}} e^{6} x^{2}+16 e^{x^{4}+2 x^{2}} e^{6} x +64 \mathrm {log}\left (3\right )^{2} e^{10}-256 \,\mathrm {log}\left (3\right ) e^{10} x^{2}+256 \,\mathrm {log}\left (3\right ) e^{10}-8 \,\mathrm {log}\left (3\right ) e^{5} x^{2}+8 \,\mathrm {log}\left (3\right ) e^{5}-64 e^{5} x^{3}-96 e^{5} x^{2}+32 e^{5}+x^{4}+2 x^{3}+x^{2}}{64 x^{2}+128 x +64} \] Input:
int(((256*x^6+768*x^5+1024*x^4+1024*x^3+768*x^2+256*x)*exp(5)^2*exp(x^4+2* x^2+1)^2+((-256*x^5-512*x^4-512*x^3-512*x^2-192*x+64)*exp(5)^2*log(3)+(-10 24*x^6-3072*x^5-4096*x^4-4096*x^3-3072*x^2-1024*x)*exp(5)^2+(32*x^7+96*x^6 +128*x^5+128*x^4+104*x^3+56*x^2+24*x+8)*exp(5))*exp(x^4+2*x^2+1)-64*exp(5) ^2*log(3)^2+((-256*x-256)*exp(5)^2+(-8*x-8)*exp(5))*log(3)+(-32*x^3-96*x^2 -96*x-32)*exp(5)+x^4+3*x^3+3*x^2+x)/(32*x^3+96*x^2+96*x+32),x)
Output:
(64*e**(2*x**4 + 4*x**2)*e**12*x**2 + 128*e**(2*x**4 + 4*x**2)*e**12*x + 6 4*e**(2*x**4 + 4*x**2)*e**12 - 128*e**(x**4 + 2*x**2)*log(3)*e**11*x - 128 *e**(x**4 + 2*x**2)*log(3)*e**11 - 512*e**(x**4 + 2*x**2)*e**11*x**2 - 102 4*e**(x**4 + 2*x**2)*e**11*x - 512*e**(x**4 + 2*x**2)*e**11 + 16*e**(x**4 + 2*x**2)*e**6*x**3 + 32*e**(x**4 + 2*x**2)*e**6*x**2 + 16*e**(x**4 + 2*x* *2)*e**6*x + 64*log(3)**2*e**10 - 256*log(3)*e**10*x**2 + 256*log(3)*e**10 - 8*log(3)*e**5*x**2 + 8*log(3)*e**5 - 64*e**5*x**3 - 96*e**5*x**2 + 32*e **5 + x**4 + 2*x**3 + x**2)/(64*(x**2 + 2*x + 1))