Integrand size = 152, antiderivative size = 32 \[ \int \frac {e^{\frac {-6544 x^5+6560 x^6-80 x^7+\left (4-405 x+5 x^2\right ) \log ^2(2)}{80 x^5-80 x^6+5 x \log ^2(2)}} \left (256 x^{10}-2560 x^{11}+1280 x^{12}+\left (-320 x^4+128 x^5+160 x^6-160 x^7\right ) \log ^2(2)+\left (-4+5 x^2\right ) \log ^4(2)\right )}{1280 x^{10}-2560 x^{11}+1280 x^{12}+\left (160 x^6-160 x^7\right ) \log ^2(2)+5 x^2 \log ^4(2)} \, dx=e^{-81+x-\frac {4}{5 \left (-x+\frac {x}{x-\frac {\log ^2(2)}{16 x^4}}\right )}} \]
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {-6544 x^5+6560 x^6-80 x^7+\left (4-405 x+5 x^2\right ) \log ^2(2)}{80 x^5-80 x^6+5 x \log ^2(2)}} \left (256 x^{10}-2560 x^{11}+1280 x^{12}+\left (-320 x^4+128 x^5+160 x^6-160 x^7\right ) \log ^2(2)+\left (-4+5 x^2\right ) \log ^4(2)\right )}{1280 x^{10}-2560 x^{11}+1280 x^{12}+\left (160 x^6-160 x^7\right ) \log ^2(2)+5 x^2 \log ^4(2)} \, dx=e^{-81+\frac {4}{5 x}+x-\frac {64 x^3}{80 x^4-80 x^5+5 \log ^2(2)}} \]
Integrate[(E^((-6544*x^5 + 6560*x^6 - 80*x^7 + (4 - 405*x + 5*x^2)*Log[2]^ 2)/(80*x^5 - 80*x^6 + 5*x*Log[2]^2))*(256*x^10 - 2560*x^11 + 1280*x^12 + ( -320*x^4 + 128*x^5 + 160*x^6 - 160*x^7)*Log[2]^2 + (-4 + 5*x^2)*Log[2]^4)) /(1280*x^10 - 2560*x^11 + 1280*x^12 + (160*x^6 - 160*x^7)*Log[2]^2 + 5*x^2 *Log[2]^4),x]
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.
| \(\displaystyle \int \frac {\left (1280 x^{12}-2560 x^{11}+256 x^{10}+\left (5 x^2-4\right ) \log ^4(2)+\left (-160 x^7+160 x^6+128 x^5-320 x^4\right ) \log ^2(2)\right ) \exp \left (\frac {-80 x^7+6560 x^6-6544 x^5+\left (5 x^2-405 x+4\right ) \log ^2(2)}{-80 x^6+80 x^5+5 x \log ^2(2)}\right )}{1280 x^{12}-2560 x^{11}+1280 x^{10}+5 x^2 \log ^4(2)+\left (160 x^6-160 x^7\right ) \log ^2(2)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (1280 x^{12}-2560 x^{11}+256 x^{10}+\left (5 x^2-4\right ) \log ^4(2)+\left (-160 x^7+160 x^6+128 x^5-320 x^4\right ) \log ^2(2)\right ) \exp \left (\frac {-80 x^7+6560 x^6-6544 x^5+\left (5 x^2-405 x+4\right ) \log ^2(2)}{-80 x^6+80 x^5+5 x \log ^2(2)}\right )}{x^2 \left (1280 x^{10}-2560 x^9+1280 x^8-160 x^5 \log ^2(2)+160 x^4 \log ^2(2)+5 \log ^4(2)\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {\left (1280 x^{12}-2560 x^{11}+256 x^{10}+\left (5 x^2-4\right ) \log ^4(2)+\left (-160 x^7+160 x^6+128 x^5-320 x^4\right ) \log ^2(2)\right ) \exp \left (\frac {-80 x^7+6560 x^6-6544 x^5+\left (5 x^2-405 x+4\right ) \log ^2(2)}{-80 x^6+80 x^5+5 x \log ^2(2)}\right )}{5 x^2 \left (16 x^5-16 x^4-\log ^2(2)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 \left (-16 x^6+16 x^5+\log ^2(2) x\right )}\right ) \left (1280 x^{12}-2560 x^{11}+256 x^{10}-\left (4-5 x^2\right ) \log ^4(2)-32 \left (5 x^7-5 x^6-4 x^5+10 x^4\right ) \log ^2(2)\right )}{x^2 \left (-16 x^5+16 x^4+\log ^2(2)\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) \left (1280 x^{12}-2560 x^{11}+256 x^{10}-\left (4-5 x^2\right ) \log ^4(2)-32 \left (5 x^7-5 x^6-4 x^5+10 x^4\right ) \log ^2(2)\right )}{x^2 \left (-16 x^5+16 x^4+\log ^2(2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (-\frac {64 \exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) \left (2 x^2+x+1\right )}{16 x^5-16 x^4-\log ^2(2)}+5 \exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right )-\frac {64 \exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) \left (16 x^4+5 \log ^2(2) x^2+\log ^2(2) x+\log ^2(2)\right )}{\left (16 x^5-16 x^4-\log ^2(2)\right )^2}-\frac {4 \exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (5 \int \exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right )dx-4 \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right )}{x^2}dx-64 \log ^2(2) \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) x}{\left (16 x^5-16 x^4-\log ^2(2)\right )^2}dx-320 \log ^2(2) \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) x^2}{\left (16 x^5-16 x^4-\log ^2(2)\right )^2}dx-1024 \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) x^4}{\left (16 x^5-16 x^4-\log ^2(2)\right )^2}dx-64 \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right )}{16 x^5-16 x^4-\log ^2(2)}dx-64 \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) x}{16 x^5-16 x^4-\log ^2(2)}dx-128 \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right ) x^2}{16 x^5-16 x^4-\log ^2(2)}dx-64 \log ^2(2) \int \frac {\exp \left (-\frac {80 x^7-6560 x^6+6544 x^5-\left (5 x^2-405 x+4\right ) \log ^2(2)}{5 x \left (-16 x^5+16 x^4+\log ^2(2)\right )}\right )}{\left (-16 x^5+16 x^4+\log ^2(2)\right )^2}dx\right )\) |
Int[(E^((-6544*x^5 + 6560*x^6 - 80*x^7 + (4 - 405*x + 5*x^2)*Log[2]^2)/(80 *x^5 - 80*x^6 + 5*x*Log[2]^2))*(256*x^10 - 2560*x^11 + 1280*x^12 + (-320*x ^4 + 128*x^5 + 160*x^6 - 160*x^7)*Log[2]^2 + (-4 + 5*x^2)*Log[2]^4))/(1280 *x^10 - 2560*x^11 + 1280*x^12 + (160*x^6 - 160*x^7)*Log[2]^2 + 5*x^2*Log[2 ]^4),x]
3.7.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 4.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (5 x^{2}-405 x +4\right ) \ln \left (2\right )^{2}-80 x^{7}+6560 x^{6}-6544 x^{5}}{5 x \left (-16 x^{5}+16 x^{4}+\ln \left (2\right )^{2}\right )}}\) | \(55\) |
| gosper | \({\mathrm e}^{\frac {-80 x^{7}+6560 x^{6}-6544 x^{5}+5 x^{2} \ln \left (2\right )^{2}-405 x \ln \left (2\right )^{2}+4 \ln \left (2\right )^{2}}{5 x \left (-16 x^{5}+16 x^{4}+\ln \left (2\right )^{2}\right )}}\) | \(62\) |
| risch | \({\mathrm e}^{\frac {-80 x^{7}+6560 x^{6}-6544 x^{5}+5 x^{2} \ln \left (2\right )^{2}-405 x \ln \left (2\right )^{2}+4 \ln \left (2\right )^{2}}{5 x \left (-16 x^{5}+16 x^{4}+\ln \left (2\right )^{2}\right )}}\) | \(62\) |
| norman | \(\frac {x \ln \left (2\right )^{2} {\mathrm e}^{\frac {\left (5 x^{2}-405 x +4\right ) \ln \left (2\right )^{2}-80 x^{7}+6560 x^{6}-6544 x^{5}}{5 x \ln \left (2\right )^{2}-80 x^{6}+80 x^{5}}}+16 x^{5} {\mathrm e}^{\frac {\left (5 x^{2}-405 x +4\right ) \ln \left (2\right )^{2}-80 x^{7}+6560 x^{6}-6544 x^{5}}{5 x \ln \left (2\right )^{2}-80 x^{6}+80 x^{5}}}-16 x^{6} {\mathrm e}^{\frac {\left (5 x^{2}-405 x +4\right ) \ln \left (2\right )^{2}-80 x^{7}+6560 x^{6}-6544 x^{5}}{5 x \ln \left (2\right )^{2}-80 x^{6}+80 x^{5}}}}{x \left (-16 x^{5}+16 x^{4}+\ln \left (2\right )^{2}\right )}\) | \(198\) |
int(((5*x^2-4)*ln(2)^4+(-160*x^7+160*x^6+128*x^5-320*x^4)*ln(2)^2+1280*x^1 2-2560*x^11+256*x^10)*exp(((5*x^2-405*x+4)*ln(2)^2-80*x^7+6560*x^6-6544*x^ 5)/(5*x*ln(2)^2-80*x^6+80*x^5))/(5*x^2*ln(2)^4+(-160*x^7+160*x^6)*ln(2)^2+ 1280*x^12-2560*x^11+1280*x^10),x,method=_RETURNVERBOSE)
Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {e^{\frac {-6544 x^5+6560 x^6-80 x^7+\left (4-405 x+5 x^2\right ) \log ^2(2)}{80 x^5-80 x^6+5 x \log ^2(2)}} \left (256 x^{10}-2560 x^{11}+1280 x^{12}+\left (-320 x^4+128 x^5+160 x^6-160 x^7\right ) \log ^2(2)+\left (-4+5 x^2\right ) \log ^4(2)\right )}{1280 x^{10}-2560 x^{11}+1280 x^{12}+\left (160 x^6-160 x^7\right ) \log ^2(2)+5 x^2 \log ^4(2)} \, dx=e^{\left (\frac {80 \, x^{7} - 6560 \, x^{6} + 6544 \, x^{5} - {\left (5 \, x^{2} - 405 \, x + 4\right )} \log \left (2\right )^{2}}{5 \, {\left (16 \, x^{6} - 16 \, x^{5} - x \log \left (2\right )^{2}\right )}}\right )} \]
integrate(((5*x^2-4)*log(2)^4+(-160*x^7+160*x^6+128*x^5-320*x^4)*log(2)^2+ 1280*x^12-2560*x^11+256*x^10)*exp(((5*x^2-405*x+4)*log(2)^2-80*x^7+6560*x^ 6-6544*x^5)/(5*x*log(2)^2-80*x^6+80*x^5))/(5*x^2*log(2)^4+(-160*x^7+160*x^ 6)*log(2)^2+1280*x^12-2560*x^11+1280*x^10),x, algorithm=\
e^(1/5*(80*x^7 - 6560*x^6 + 6544*x^5 - (5*x^2 - 405*x + 4)*log(2)^2)/(16*x ^6 - 16*x^5 - x*log(2)^2))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 1.53 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^{\frac {-6544 x^5+6560 x^6-80 x^7+\left (4-405 x+5 x^2\right ) \log ^2(2)}{80 x^5-80 x^6+5 x \log ^2(2)}} \left (256 x^{10}-2560 x^{11}+1280 x^{12}+\left (-320 x^4+128 x^5+160 x^6-160 x^7\right ) \log ^2(2)+\left (-4+5 x^2\right ) \log ^4(2)\right )}{1280 x^{10}-2560 x^{11}+1280 x^{12}+\left (160 x^6-160 x^7\right ) \log ^2(2)+5 x^2 \log ^4(2)} \, dx=e^{\frac {- 80 x^{7} + 6560 x^{6} - 6544 x^{5} + \left (5 x^{2} - 405 x + 4\right ) \log {\left (2 \right )}^{2}}{- 80 x^{6} + 80 x^{5} + 5 x \log {\left (2 \right )}^{2}}} \]
integrate(((5*x**2-4)*ln(2)**4+(-160*x**7+160*x**6+128*x**5-320*x**4)*ln(2 )**2+1280*x**12-2560*x**11+256*x**10)*exp(((5*x**2-405*x+4)*ln(2)**2-80*x* *7+6560*x**6-6544*x**5)/(5*x*ln(2)**2-80*x**6+80*x**5))/(5*x**2*ln(2)**4+( -160*x**7+160*x**6)*ln(2)**2+1280*x**12-2560*x**11+1280*x**10),x)
exp((-80*x**7 + 6560*x**6 - 6544*x**5 + (5*x**2 - 405*x + 4)*log(2)**2)/(- 80*x**6 + 80*x**5 + 5*x*log(2)**2))
Time = 97.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {-6544 x^5+6560 x^6-80 x^7+\left (4-405 x+5 x^2\right ) \log ^2(2)}{80 x^5-80 x^6+5 x \log ^2(2)}} \left (256 x^{10}-2560 x^{11}+1280 x^{12}+\left (-320 x^4+128 x^5+160 x^6-160 x^7\right ) \log ^2(2)+\left (-4+5 x^2\right ) \log ^4(2)\right )}{1280 x^{10}-2560 x^{11}+1280 x^{12}+\left (160 x^6-160 x^7\right ) \log ^2(2)+5 x^2 \log ^4(2)} \, dx=e^{\left (\frac {64 \, x^{3}}{5 \, {\left (16 \, x^{5} - 16 \, x^{4} - \log \left (2\right )^{2}\right )}} + x + \frac {4}{5 \, x} - 81\right )} \]
integrate(((5*x^2-4)*log(2)^4+(-160*x^7+160*x^6+128*x^5-320*x^4)*log(2)^2+ 1280*x^12-2560*x^11+256*x^10)*exp(((5*x^2-405*x+4)*log(2)^2-80*x^7+6560*x^ 6-6544*x^5)/(5*x*log(2)^2-80*x^6+80*x^5))/(5*x^2*log(2)^4+(-160*x^7+160*x^ 6)*log(2)^2+1280*x^12-2560*x^11+1280*x^10),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (28) = 56\).
Time = 0.33 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.97 \[ \int \frac {e^{\frac {-6544 x^5+6560 x^6-80 x^7+\left (4-405 x+5 x^2\right ) \log ^2(2)}{80 x^5-80 x^6+5 x \log ^2(2)}} \left (256 x^{10}-2560 x^{11}+1280 x^{12}+\left (-320 x^4+128 x^5+160 x^6-160 x^7\right ) \log ^2(2)+\left (-4+5 x^2\right ) \log ^4(2)\right )}{1280 x^{10}-2560 x^{11}+1280 x^{12}+\left (160 x^6-160 x^7\right ) \log ^2(2)+5 x^2 \log ^4(2)} \, dx=e^{\left (\frac {16 \, x^{7}}{16 \, x^{6} - 16 \, x^{5} - x \log \left (2\right )^{2}} - \frac {1312 \, x^{6}}{16 \, x^{6} - 16 \, x^{5} - x \log \left (2\right )^{2}} + \frac {6544 \, x^{5}}{5 \, {\left (16 \, x^{6} - 16 \, x^{5} - x \log \left (2\right )^{2}\right )}} - \frac {x^{2} \log \left (2\right )^{2}}{16 \, x^{6} - 16 \, x^{5} - x \log \left (2\right )^{2}} + \frac {81 \, x \log \left (2\right )^{2}}{16 \, x^{6} - 16 \, x^{5} - x \log \left (2\right )^{2}} - \frac {4 \, \log \left (2\right )^{2}}{5 \, {\left (16 \, x^{6} - 16 \, x^{5} - x \log \left (2\right )^{2}\right )}}\right )} \]
integrate(((5*x^2-4)*log(2)^4+(-160*x^7+160*x^6+128*x^5-320*x^4)*log(2)^2+ 1280*x^12-2560*x^11+256*x^10)*exp(((5*x^2-405*x+4)*log(2)^2-80*x^7+6560*x^ 6-6544*x^5)/(5*x*log(2)^2-80*x^6+80*x^5))/(5*x^2*log(2)^4+(-160*x^7+160*x^ 6)*log(2)^2+1280*x^12-2560*x^11+1280*x^10),x, algorithm=\
e^(16*x^7/(16*x^6 - 16*x^5 - x*log(2)^2) - 1312*x^6/(16*x^6 - 16*x^5 - x*l og(2)^2) + 6544/5*x^5/(16*x^6 - 16*x^5 - x*log(2)^2) - x^2*log(2)^2/(16*x^ 6 - 16*x^5 - x*log(2)^2) + 81*x*log(2)^2/(16*x^6 - 16*x^5 - x*log(2)^2) - 4/5*log(2)^2/(16*x^6 - 16*x^5 - x*log(2)^2))
Time = 13.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.59 \[ \int \frac {e^{\frac {-6544 x^5+6560 x^6-80 x^7+\left (4-405 x+5 x^2\right ) \log ^2(2)}{80 x^5-80 x^6+5 x \log ^2(2)}} \left (256 x^{10}-2560 x^{11}+1280 x^{12}+\left (-320 x^4+128 x^5+160 x^6-160 x^7\right ) \log ^2(2)+\left (-4+5 x^2\right ) \log ^4(2)\right )}{1280 x^{10}-2560 x^{11}+1280 x^{12}+\left (160 x^6-160 x^7\right ) \log ^2(2)+5 x^2 \log ^4(2)} \, dx={\mathrm {e}}^{\frac {x\,{\ln \left (2\right )}^2}{-16\,x^5+16\,x^4+{\ln \left (2\right )}^2}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (2\right )}^2}{-80\,x^6+80\,x^5+5\,{\ln \left (2\right )}^2\,x}}\,{\mathrm {e}}^{-\frac {81\,{\ln \left (2\right )}^2}{-16\,x^5+16\,x^4+{\ln \left (2\right )}^2}}\,{\mathrm {e}}^{-\frac {16\,x^6}{-16\,x^5+16\,x^4+{\ln \left (2\right )}^2}}\,{\mathrm {e}}^{\frac {1312\,x^5}{-16\,x^5+16\,x^4+{\ln \left (2\right )}^2}}\,{\mathrm {e}}^{-\frac {6544\,x^4}{-80\,x^5+80\,x^4+5\,{\ln \left (2\right )}^2}} \]
int((exp((log(2)^2*(5*x^2 - 405*x + 4) - 6544*x^5 + 6560*x^6 - 80*x^7)/(5* x*log(2)^2 + 80*x^5 - 80*x^6))*(log(2)^4*(5*x^2 - 4) - log(2)^2*(320*x^4 - 128*x^5 - 160*x^6 + 160*x^7) + 256*x^10 - 2560*x^11 + 1280*x^12))/(5*x^2* log(2)^4 + 1280*x^10 - 2560*x^11 + 1280*x^12 + log(2)^2*(160*x^6 - 160*x^7 )),x)