Integrand size = 78, antiderivative size = 26 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=-e^{x^2}-x (5+x)^4+\log (2+2 x-\log (5)) \] Output:
ln(2-ln(5)+2*x)-exp(x^2)-(5+x)^4*x
Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(26)=52\).
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.92 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=-e^{x^2}-\frac {1}{2} x \left (300 x^2+40 x^3+2 x^4+5 x \left (200+4 \log ^2(5)-\log (5) \log (625)\right )+5 \left (250+4 \log ^3(5)+\log ^2(625)-\log ^2(5) (16+\log (625))\right )\right )-\frac {1}{4} \left (-4+20 \log ^4(5)+540 \log (625)-5 \log ^3(5) (24+\log (625))+10 \log ^2(5) (124+3 \log (625))-10 \log (5) (216+31 \log (625))\right ) \log (2+2 x-\log (5)) \] Input:
Integrate[(1248 + 3250*x + 2900*x^2 + 1060*x^3 + 170*x^4 + 10*x^5 + (-625 - 1000*x - 450*x^2 - 80*x^3 - 5*x^4)*Log[5] + E^x^2*(4*x + 4*x^2 - 2*x*Log [5]))/(-2 - 2*x + Log[5]),x]
Output:
-E^x^2 - (x*(300*x^2 + 40*x^3 + 2*x^4 + 5*x*(200 + 4*Log[5]^2 - Log[5]*Log [625]) + 5*(250 + 4*Log[5]^3 + Log[625]^2 - Log[5]^2*(16 + Log[625]))))/2 - ((-4 + 20*Log[5]^4 + 540*Log[625] - 5*Log[5]^3*(24 + Log[625]) + 10*Log[ 5]^2*(124 + 3*Log[625]) - 10*Log[5]*(216 + 31*Log[625]))*Log[2 + 2*x - Log [5]])/4
Leaf count is larger than twice the leaf count of optimal. \(367\) vs. \(2(26)=52\).
Time = 0.81 (sec) , antiderivative size = 367, normalized size of antiderivative = 14.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {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.
\(\displaystyle \int \frac {10 x^5+170 x^4+1060 x^3+2900 x^2+e^{x^2} \left (4 x^2+4 x-2 x \log (5)\right )+\left (-5 x^4-80 x^3-450 x^2-1000 x-625\right ) \log (5)+3250 x+1248}{-2 x-2+\log (5)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {10 x^5}{2 x+2-\log (5)}-\frac {170 x^4}{2 x+2-\log (5)}-\frac {1060 x^3}{2 x+2-\log (5)}-2 e^{x^2} x-\frac {2900 x^2}{2 x+2-\log (5)}-\frac {3250 x}{2 x+2-\log (5)}-\frac {1248}{2 x+2-\log (5)}+\frac {5 (x+1) (x+5)^3 \log (5)}{2 x+2-\log (5)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x^5-\frac {85 x^4}{4}+\frac {5}{8} x^4 (2-\log (5))-\frac {530 x^3}{3}-\frac {5}{12} x^3 (2-\log (5))^2+\frac {85}{6} x^3 (2-\log (5))-725 x^2-e^{x^2}+\frac {5}{16} x^2 (2-\log (5))^3-\frac {85}{8} x^2 (2-\log (5))^2+\frac {265}{2} x^2 (2-\log (5))-1625 x+\frac {5}{16} x \log ^2(5) (8+\log (5))^2+\frac {5}{32} \log ^2(5) (8+\log (5))^3 \log (2 x+2-\log (5))+\frac {5}{16} (x+5)^2 \log ^2(5) (8+\log (5))+\frac {5}{12} (x+5)^3 \log ^2(5)-\frac {5}{16} x (2-\log (5))^4+\frac {85}{8} x (2-\log (5))^3-\frac {265}{2} x (2-\log (5))^2+725 x (2-\log (5))+\frac {5}{32} (2-\log (5))^5 \log (2 x+2-\log (5))-\frac {85}{16} (2-\log (5))^4 \log (2 x+2-\log (5))+\frac {265}{4} (2-\log (5))^3 \log (2 x+2-\log (5))-\frac {725}{2} (2-\log (5))^2 \log (2 x+2-\log (5))+\frac {1625}{2} (2-\log (5)) \log (2 x+2-\log (5))-624 \log (2 x+2-\log (5))+\frac {5}{8} (x+5)^4 \log (5)\) |
Input:
Int[(1248 + 3250*x + 2900*x^2 + 1060*x^3 + 170*x^4 + 10*x^5 + (-625 - 1000 *x - 450*x^2 - 80*x^3 - 5*x^4)*Log[5] + E^x^2*(4*x + 4*x^2 - 2*x*Log[5]))/ (-2 - 2*x + Log[5]),x]
Output:
-E^x^2 - 1625*x - 725*x^2 - (530*x^3)/3 - (85*x^4)/4 - x^5 + 725*x*(2 - Lo g[5]) + (265*x^2*(2 - Log[5]))/2 + (85*x^3*(2 - Log[5]))/6 + (5*x^4*(2 - L og[5]))/8 - (265*x*(2 - Log[5])^2)/2 - (85*x^2*(2 - Log[5])^2)/8 - (5*x^3* (2 - Log[5])^2)/12 + (85*x*(2 - Log[5])^3)/8 + (5*x^2*(2 - Log[5])^3)/16 - (5*x*(2 - Log[5])^4)/16 + (5*(5 + x)^4*Log[5])/8 + (5*(5 + x)^3*Log[5]^2) /12 + (5*(5 + x)^2*Log[5]^2*(8 + Log[5]))/16 + (5*x*Log[5]^2*(8 + Log[5])^ 2)/16 - 624*Log[2 + 2*x - Log[5]] + (1625*(2 - Log[5])*Log[2 + 2*x - Log[5 ]])/2 - (725*(2 - Log[5])^2*Log[2 + 2*x - Log[5]])/2 + (265*(2 - Log[5])^3 *Log[2 + 2*x - Log[5]])/4 - (85*(2 - Log[5])^4*Log[2 + 2*x - Log[5]])/16 + (5*(2 - Log[5])^5*Log[2 + 2*x - Log[5]])/32 + (5*Log[5]^2*(8 + Log[5])^3* Log[2 + 2*x - Log[5]])/32
Time = 1.98 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50
method | result | size |
norman | \(-625 x -500 x^{2}-150 x^{3}-20 x^{4}-x^{5}-{\mathrm e}^{x^{2}}+\ln \left (\ln \left (5\right )-2 x -2\right )\) | \(39\) |
parallelrisch | \(-625 x -500 x^{2}-150 x^{3}-20 x^{4}-x^{5}-{\mathrm e}^{x^{2}}+\ln \left (-\frac {\ln \left (5\right )}{2}+x +1\right )\) | \(39\) |
risch | \(-x^{5}-20 x^{4}-150 x^{3}-500 x^{2}-625 x +\ln \left (2-\ln \left (5\right )+2 x \right )-{\mathrm e}^{x^{2}}\) | \(41\) |
parts | \(-x^{5}-20 x^{4}-150 x^{3}-500 x^{2}-625 x +\ln \left (2-\ln \left (5\right )+2 x \right )-{\mathrm e}^{x^{2}}\) | \(41\) |
Input:
int(((-2*x*ln(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x-625)*ln (5)+10*x^5+170*x^4+1060*x^3+2900*x^2+3250*x+1248)/(ln(5)-2*x-2),x,method=_ RETURNVERBOSE)
Output:
-625*x-500*x^2-150*x^3-20*x^4-x^5-exp(x^2)+ln(ln(5)-2*x-2)
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=-x^{5} - 20 \, x^{4} - 150 \, x^{3} - 500 \, x^{2} - 625 \, x - e^{\left (x^{2}\right )} + \log \left (2 \, x - \log \left (5\right ) + 2\right ) \] Input:
integrate(((-2*x*log(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x- 625)*log(5)+10*x^5+170*x^4+1060*x^3+2900*x^2+3250*x+1248)/(log(5)-2*x-2),x , algorithm="fricas")
Output:
-x^5 - 20*x^4 - 150*x^3 - 500*x^2 - 625*x - e^(x^2) + log(2*x - log(5) + 2 )
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=- x^{5} - 20 x^{4} - 150 x^{3} - 500 x^{2} - 625 x - e^{x^{2}} + \log {\left (2 x - \log {\left (5 \right )} + 2 \right )} \] Input:
integrate(((-2*x*ln(5)+4*x**2+4*x)*exp(x**2)+(-5*x**4-80*x**3-450*x**2-100 0*x-625)*ln(5)+10*x**5+170*x**4+1060*x**3+2900*x**2+3250*x+1248)/(ln(5)-2* x-2),x)
Output:
-x**5 - 20*x**4 - 150*x**3 - 500*x**2 - 625*x - exp(x**2) + log(2*x - log( 5) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 546, normalized size of antiderivative = 21.00 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=\text {Too large to display} \] Input:
integrate(((-2*x*log(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x- 625)*log(5)+10*x^5+170*x^4+1060*x^3+2900*x^2+3250*x+1248)/(log(5)-2*x-2),x , algorithm="maxima")
Output:
-x^5 - 5/8*x^4*(log(5) - 2) - 5/12*(log(5)^2 - 4*log(5) + 4)*x^3 - 85/4*x^ 4 - 85/6*x^3*(log(5) - 2) - 5/16*(log(5)^3 - 6*log(5)^2 + 12*log(5) - 8)*x ^2 - 85/8*(log(5)^2 - 4*log(5) + 4)*x^2 - 530/3*x^3 - 265/2*x^2*(log(5) - 2) - 5/16*(log(5)^4 - 8*log(5)^3 + 24*log(5)^2 - 32*log(5) + 16)*x - 85/8* (log(5)^3 - 6*log(5)^2 + 12*log(5) - 8)*x - 265/2*(log(5)^2 - 4*log(5) + 4 )*x - 725*x^2 - 725*x*(log(5) - 2) + 5/96*(12*x^4 + 8*x^3*(log(5) - 2) + 6 *(log(5)^2 - 4*log(5) + 4)*x^2 + 6*(log(5)^3 - 6*log(5)^2 + 12*log(5) - 8) *x + 3*(log(5)^4 - 8*log(5)^3 + 24*log(5)^2 - 32*log(5) + 16)*log(2*x - lo g(5) + 2))*log(5) + 5/3*(8*x^3 + 6*x^2*(log(5) - 2) + 6*(log(5)^2 - 4*log( 5) + 4)*x + 3*(log(5)^3 - 6*log(5)^2 + 12*log(5) - 8)*log(2*x - log(5) + 2 ))*log(5) + 225/4*(2*x^2 + 2*x*(log(5) - 2) + (log(5)^2 - 4*log(5) + 4)*lo g(2*x - log(5) + 2))*log(5) + 250*((log(5) - 2)*log(2*x - log(5) + 2) + 2* x)*log(5) - 5/32*(log(5)^5 - 10*log(5)^4 + 40*log(5)^3 - 80*log(5)^2 + 80* log(5) - 32)*log(2*x - log(5) + 2) - 85/16*(log(5)^4 - 8*log(5)^3 + 24*log (5)^2 - 32*log(5) + 16)*log(2*x - log(5) + 2) - 265/4*(log(5)^3 - 6*log(5) ^2 + 12*log(5) - 8)*log(2*x - log(5) + 2) - 725/2*(log(5)^2 - 4*log(5) + 4 )*log(2*x - log(5) + 2) - 1625/2*(log(5) - 2)*log(2*x - log(5) + 2) + 625/ 2*log(5)*log(2*x - log(5) + 2) - 1625*x - e^(x^2) - 624*log(2*x - log(5) + 2)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=-x^{5} - 20 \, x^{4} - 150 \, x^{3} - 500 \, x^{2} - 625 \, x - e^{\left (x^{2}\right )} + \log \left (2 \, x - \log \left (5\right ) + 2\right ) \] Input:
integrate(((-2*x*log(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x- 625)*log(5)+10*x^5+170*x^4+1060*x^3+2900*x^2+3250*x+1248)/(log(5)-2*x-2),x , algorithm="giac")
Output:
-x^5 - 20*x^4 - 150*x^3 - 500*x^2 - 625*x - e^(x^2) + log(2*x - log(5) + 2 )
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=\ln \left (2\,x-\ln \left (5\right )+2\right )-{\mathrm {e}}^{x^2}-625\,x-500\,x^2-150\,x^3-20\,x^4-x^5 \] Input:
int(-(3250*x + exp(x^2)*(4*x - 2*x*log(5) + 4*x^2) - log(5)*(1000*x + 450* x^2 + 80*x^3 + 5*x^4 + 625) + 2900*x^2 + 1060*x^3 + 170*x^4 + 10*x^5 + 124 8)/(2*x - log(5) + 2),x)
Output:
log(2*x - log(5) + 2) - exp(x^2) - 625*x - 500*x^2 - 150*x^3 - 20*x^4 - x^ 5
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {1248+3250 x+2900 x^2+1060 x^3+170 x^4+10 x^5+\left (-625-1000 x-450 x^2-80 x^3-5 x^4\right ) \log (5)+e^{x^2} \left (4 x+4 x^2-2 x \log (5)\right )}{-2-2 x+\log (5)} \, dx=-e^{x^{2}}+\mathrm {log}\left (\mathrm {log}\left (5\right )-2 x -2\right )-x^{5}-20 x^{4}-150 x^{3}-500 x^{2}-625 x \] Input:
int(((-2*x*log(5)+4*x^2+4*x)*exp(x^2)+(-5*x^4-80*x^3-450*x^2-1000*x-625)*l og(5)+10*x^5+170*x^4+1060*x^3+2900*x^2+3250*x+1248)/(log(5)-2*x-2),x)
Output:
- e**(x**2) + log(log(5) - 2*x - 2) - x**5 - 20*x**4 - 150*x**3 - 500*x** 2 - 625*x