Integrand size = 155, antiderivative size = 31 \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=x+\frac {x}{x+\frac {1}{8} \left (2 x-\log (4)+\frac {5 x}{5+x \log (x)}\right )} \] Output:
x+x/(5/4*x-1/4*ln(2)+5/8/(5+x*ln(x))*x)
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=x+\frac {\log (256)}{50 x-5 \log (4)}-\frac {40 x^2}{(10 x-\log (4)) (55 x-5 \log (4)+x (10 x-\log (4)) \log (x))} \] Input:
Integrate[(3065*x^2 + (-200 - 550*x)*Log[4] + 25*Log[4]^2 + (40*x^2 + 1100 *x^3 + (-80*x - 210*x^2)*Log[4] + 10*x*Log[4]^2)*Log[x] + (100*x^4 + (-8*x ^2 - 20*x^3)*Log[4] + x^2*Log[4]^2)*Log[x]^2)/(3025*x^2 - 550*x*Log[4] + 2 5*Log[4]^2 + (1100*x^3 - 210*x^2*Log[4] + 10*x*Log[4]^2)*Log[x] + (100*x^4 - 20*x^3*Log[4] + x^2*Log[4]^2)*Log[x]^2),x]
Output:
x + Log[256]/(50*x - 5*Log[4]) - (40*x^2)/((10*x - Log[4])*(55*x - 5*Log[4 ] + x*(10*x - Log[4])*Log[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.
| \(\displaystyle \int \frac {3065 x^2+\left (1100 x^3+40 x^2+\left (-210 x^2-80 x\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+x^2 \log ^2(4)+\left (-20 x^3-8 x^2\right ) \log (4)\right ) \log ^2(x)+(-550 x-200) \log (4)+25 \log ^2(4)}{3025 x^2+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)-550 x \log (4)+25 \log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3065 x^2+\left (1100 x^3+40 x^2+\left (-210 x^2-80 x\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+x^2 \log ^2(4)+\left (-20 x^3-8 x^2\right ) \log (4)\right ) \log ^2(x)+(-550 x-200) \log (4)+25 \log ^2(4)}{\left (10 x^2 \log (x)+55 x-x \log (4) \log (x)-5 \log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {40 x (10 x+\log (4))}{(10 x-\log (4))^2 \left (10 x^2 \log (x)+55 x-x \log (4) \log (x)-5 \log (4)\right )}+\frac {100 x^2-20 x \log (4)-(8-\log (4)) \log (4)}{(10 x-\log (4))^2}+\frac {40 x \left (100 x^3-10 x^2 (55+\log (16))+x \log (4) (100+\log (4))-5 \log ^2(4)\right )}{(10 x-\log (4))^2 \left (10 x^2 \log (x)+55 x-x \log (4) \log (x)-5 \log (4)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \log ^3(4) \int \frac {1}{(\log (4)-10 x)^2 \left (-10 \log (x) x^2+\log (4) \log (x) x-55 x+5 \log (4)\right )^2}dx-6 \log ^2(4) \int \frac {1}{(10 x-\log (4)) \left (10 \log (x) x^2-\log (4) \log (x) x+55 x-5 \log (4)\right )^2}dx-8 \log ^2(4) \int \frac {1}{(\log (4)-10 x)^2 \left (-10 \log (x) x^2+\log (4) \log (x) x-55 x+5 \log (4)\right )}dx-220 \int \frac {x}{\left (10 \log (x) x^2-\log (4) \log (x) x+55 x-5 \log (4)\right )^2}dx+40 \int \frac {x^2}{\left (10 \log (x) x^2-\log (4) \log (x) x+55 x-5 \log (4)\right )^2}dx+4 \int \frac {1}{10 \log (x) x^2-\log (4) \log (x) x+55 x-5 \log (4)}dx+4 \log (64) \int \frac {1}{(10 x-\log (4)) \left (10 \log (x) x^2-\log (4) \log (x) x+55 x-5 \log (4)\right )}dx-4 \log (4) \int \frac {1}{\left (-10 \log (x) x^2+\log (4) \log (x) x-55 x+5 \log (4)\right )^2}dx+x+\frac {4 \log (4)}{5 (10 x-\log (4))}\) |
Input:
Int[(3065*x^2 + (-200 - 550*x)*Log[4] + 25*Log[4]^2 + (40*x^2 + 1100*x^3 + (-80*x - 210*x^2)*Log[4] + 10*x*Log[4]^2)*Log[x] + (100*x^4 + (-8*x^2 - 2 0*x^3)*Log[4] + x^2*Log[4]^2)*Log[x]^2)/(3025*x^2 - 550*x*Log[4] + 25*Log[ 4]^2 + (1100*x^3 - 210*x^2*Log[4] + 10*x*Log[4]^2)*Log[x] + (100*x^4 - 20* x^3*Log[4] + x^2*Log[4]^2)*Log[x]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(25)=50\).
Time = 6.52 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06
| method | result | size |
| risch | \(\frac {5 x \ln \left (2\right )-25 x^{2}-4 \ln \left (2\right )}{5 \ln \left (2\right )-25 x}-\frac {20 x^{2}}{\left (\ln \left (2\right )-5 x \right ) \left (2 x \ln \left (2\right ) \ln \left (x \right )-10 x^{2} \ln \left (x \right )+10 \ln \left (2\right )-55 x \right )}\) | \(64\) |
| default | \(\frac {\left (4-\ln \left (2\right )\right ) x +\left (\frac {2 \ln \left (2\right )^{2}}{5}-\frac {8 \ln \left (2\right )}{5}\right ) \ln \left (x \right ) x -55 x^{2}-10 x^{3} \ln \left (x \right )+2 \ln \left (2\right )^{2}-8 \ln \left (2\right )}{2 x \ln \left (2\right ) \ln \left (x \right )-10 x^{2} \ln \left (x \right )+10 \ln \left (2\right )-55 x}\) | \(72\) |
| norman | \(\frac {\left (4-\ln \left (2\right )\right ) x +\left (\frac {2 \ln \left (2\right )^{2}}{5}-\frac {8 \ln \left (2\right )}{5}\right ) \ln \left (x \right ) x -55 x^{2}-10 x^{3} \ln \left (x \right )+2 \ln \left (2\right )^{2}-8 \ln \left (2\right )}{2 x \ln \left (2\right ) \ln \left (x \right )-10 x^{2} \ln \left (x \right )+10 \ln \left (2\right )-55 x}\) | \(72\) |
| parallelrisch | \(\frac {4 \ln \left (2\right )^{2} \ln \left (x \right ) x -100 x^{3} \ln \left (x \right )-16 x \ln \left (2\right ) \ln \left (x \right )+20 \ln \left (2\right )^{2}-10 x \ln \left (2\right )-550 x^{2}-80 \ln \left (2\right )+40 x}{20 x \ln \left (2\right ) \ln \left (x \right )-100 x^{2} \ln \left (x \right )+100 \ln \left (2\right )-550 x}\) | \(74\) |
Input:
int(((4*x^2*ln(2)^2+2*(-20*x^3-8*x^2)*ln(2)+100*x^4)*ln(x)^2+(40*x*ln(2)^2 +2*(-210*x^2-80*x)*ln(2)+1100*x^3+40*x^2)*ln(x)+100*ln(2)^2+2*(-550*x-200) *ln(2)+3065*x^2)/((4*x^2*ln(2)^2-40*x^3*ln(2)+100*x^4)*ln(x)^2+(40*x*ln(2) ^2-420*x^2*ln(2)+1100*x^3)*ln(x)+100*ln(2)^2-1100*x*ln(2)+3025*x^2),x,meth od=_RETURNVERBOSE)
Output:
1/5*(5*x*ln(2)-25*x^2-4*ln(2))/(ln(2)-5*x)-20*x^2/(ln(2)-5*x)/(2*x*ln(2)*l n(x)-10*x^2*ln(x)+10*ln(2)-55*x)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=\frac {275 \, x^{2} - 10 \, {\left (5 \, x - 4\right )} \log \left (2\right ) + 2 \, {\left (25 \, x^{3} - {\left (5 \, x^{2} - 4 \, x\right )} \log \left (2\right )\right )} \log \left (x\right ) - 20 \, x}{5 \, {\left (2 \, {\left (5 \, x^{2} - x \log \left (2\right )\right )} \log \left (x\right ) + 55 \, x - 10 \, \log \left (2\right )\right )}} \] Input:
integrate(((4*x^2*log(2)^2+2*(-20*x^3-8*x^2)*log(2)+100*x^4)*log(x)^2+(40* x*log(2)^2+2*(-210*x^2-80*x)*log(2)+1100*x^3+40*x^2)*log(x)+100*log(2)^2+2 *(-550*x-200)*log(2)+3065*x^2)/((4*x^2*log(2)^2-40*x^3*log(2)+100*x^4)*log (x)^2+(40*x*log(2)^2-420*x^2*log(2)+1100*x^3)*log(x)+100*log(2)^2-1100*x*l og(2)+3025*x^2),x, algorithm="fricas")
Output:
1/5*(275*x^2 - 10*(5*x - 4)*log(2) + 2*(25*x^3 - (5*x^2 - 4*x)*log(2))*log (x) - 20*x)/(2*(5*x^2 - x*log(2))*log(x) + 55*x - 10*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=- \frac {20 x^{2}}{275 x^{2} - 105 x \log {\left (2 \right )} + \left (50 x^{3} - 20 x^{2} \log {\left (2 \right )} + 2 x \log {\left (2 \right )}^{2}\right ) \log {\left (x \right )} + 10 \log {\left (2 \right )}^{2}} + x + \frac {4 \log {\left (2 \right )}}{25 x - 5 \log {\left (2 \right )}} \] Input:
integrate(((4*x**2*ln(2)**2+2*(-20*x**3-8*x**2)*ln(2)+100*x**4)*ln(x)**2+( 40*x*ln(2)**2+2*(-210*x**2-80*x)*ln(2)+1100*x**3+40*x**2)*ln(x)+100*ln(2)* *2+2*(-550*x-200)*ln(2)+3065*x**2)/((4*x**2*ln(2)**2-40*x**3*ln(2)+100*x** 4)*ln(x)**2+(40*x*ln(2)**2-420*x**2*ln(2)+1100*x**3)*ln(x)+100*ln(2)**2-11 00*x*ln(2)+3025*x**2),x)
Output:
-20*x**2/(275*x**2 - 105*x*log(2) + (50*x**3 - 20*x**2*log(2) + 2*x*log(2) **2)*log(x) + 10*log(2)**2) + x + 4*log(2)/(25*x - 5*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=\frac {275 \, x^{2} - 10 \, x {\left (5 \, \log \left (2\right ) + 2\right )} + 2 \, {\left (25 \, x^{3} - 5 \, x^{2} \log \left (2\right ) + 4 \, x \log \left (2\right )\right )} \log \left (x\right ) + 40 \, \log \left (2\right )}{5 \, {\left (2 \, {\left (5 \, x^{2} - x \log \left (2\right )\right )} \log \left (x\right ) + 55 \, x - 10 \, \log \left (2\right )\right )}} \] Input:
integrate(((4*x^2*log(2)^2+2*(-20*x^3-8*x^2)*log(2)+100*x^4)*log(x)^2+(40* x*log(2)^2+2*(-210*x^2-80*x)*log(2)+1100*x^3+40*x^2)*log(x)+100*log(2)^2+2 *(-550*x-200)*log(2)+3065*x^2)/((4*x^2*log(2)^2-40*x^3*log(2)+100*x^4)*log (x)^2+(40*x*log(2)^2-420*x^2*log(2)+1100*x^3)*log(x)+100*log(2)^2-1100*x*l og(2)+3025*x^2),x, algorithm="maxima")
Output:
1/5*(275*x^2 - 10*x*(5*log(2) + 2) + 2*(25*x^3 - 5*x^2*log(2) + 4*x*log(2) )*log(x) + 40*log(2))/(2*(5*x^2 - x*log(2))*log(x) + 55*x - 10*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=x - \frac {20 \, x^{2}}{50 \, x^{3} \log \left (x\right ) - 20 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 2 \, x \log \left (2\right )^{2} \log \left (x\right ) + 275 \, x^{2} - 105 \, x \log \left (2\right ) + 10 \, \log \left (2\right )^{2}} + \frac {4 \, \log \left (2\right )}{5 \, {\left (5 \, x - \log \left (2\right )\right )}} \] Input:
integrate(((4*x^2*log(2)^2+2*(-20*x^3-8*x^2)*log(2)+100*x^4)*log(x)^2+(40* x*log(2)^2+2*(-210*x^2-80*x)*log(2)+1100*x^3+40*x^2)*log(x)+100*log(2)^2+2 *(-550*x-200)*log(2)+3065*x^2)/((4*x^2*log(2)^2-40*x^3*log(2)+100*x^4)*log (x)^2+(40*x*log(2)^2-420*x^2*log(2)+1100*x^3)*log(x)+100*log(2)^2-1100*x*l og(2)+3025*x^2),x, algorithm="giac")
Output:
x - 20*x^2/(50*x^3*log(x) - 20*x^2*log(2)*log(x) + 2*x*log(2)^2*log(x) + 2 75*x^2 - 105*x*log(2) + 10*log(2)^2) + 4/5*log(2)/(5*x - log(2))
Timed out. \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=\int \frac {100\,{\ln \left (2\right )}^2-2\,\ln \left (2\right )\,\left (550\,x+200\right )+{\ln \left (x\right )}^2\,\left (4\,x^2\,{\ln \left (2\right )}^2-2\,\ln \left (2\right )\,\left (20\,x^3+8\,x^2\right )+100\,x^4\right )+3065\,x^2+\ln \left (x\right )\,\left (40\,x\,{\ln \left (2\right )}^2-2\,\ln \left (2\right )\,\left (210\,x^2+80\,x\right )+40\,x^2+1100\,x^3\right )}{\ln \left (x\right )\,\left (1100\,x^3-420\,\ln \left (2\right )\,x^2+40\,{\ln \left (2\right )}^2\,x\right )-1100\,x\,\ln \left (2\right )+100\,{\ln \left (2\right )}^2+3025\,x^2+{\ln \left (x\right )}^2\,\left (100\,x^4-40\,\ln \left (2\right )\,x^3+4\,{\ln \left (2\right )}^2\,x^2\right )} \,d x \] Input:
int((100*log(2)^2 - 2*log(2)*(550*x + 200) + log(x)^2*(4*x^2*log(2)^2 - 2* log(2)*(8*x^2 + 20*x^3) + 100*x^4) + 3065*x^2 + log(x)*(40*x*log(2)^2 - 2* log(2)*(80*x + 210*x^2) + 40*x^2 + 1100*x^3))/(log(x)*(40*x*log(2)^2 - 420 *x^2*log(2) + 1100*x^3) - 1100*x*log(2) + 100*log(2)^2 + 3025*x^2 + log(x) ^2*(4*x^2*log(2)^2 - 40*x^3*log(2) + 100*x^4)),x)
Output:
int((100*log(2)^2 - 2*log(2)*(550*x + 200) + log(x)^2*(4*x^2*log(2)^2 - 2* log(2)*(8*x^2 + 20*x^3) + 100*x^4) + 3065*x^2 + log(x)*(40*x*log(2)^2 - 2* log(2)*(80*x + 210*x^2) + 40*x^2 + 1100*x^3))/(log(x)*(40*x*log(2)^2 - 420 *x^2*log(2) + 1100*x^3) - 1100*x*log(2) + 100*log(2)^2 + 3025*x^2 + log(x) ^2*(4*x^2*log(2)^2 - 40*x^3*log(2) + 100*x^4)), x)
Time = 0.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.58 \[ \int \frac {3065 x^2+(-200-550 x) \log (4)+25 \log ^2(4)+\left (40 x^2+1100 x^3+\left (-80 x-210 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4+\left (-8 x^2-20 x^3\right ) \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)}{3025 x^2-550 x \log (4)+25 \log ^2(4)+\left (1100 x^3-210 x^2 \log (4)+10 x \log ^2(4)\right ) \log (x)+\left (100 x^4-20 x^3 \log (4)+x^2 \log ^2(4)\right ) \log ^2(x)} \, dx=\frac {4 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )^{2} x +2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{2}-16 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x -110 \,\mathrm {log}\left (x \right ) x^{3}-8 \,\mathrm {log}\left (x \right ) x^{2}+20 \mathrm {log}\left (2\right )^{2}-80 \,\mathrm {log}\left (2\right )-605 x^{2}}{22 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x -110 \,\mathrm {log}\left (x \right ) x^{2}+110 \,\mathrm {log}\left (2\right )-605 x} \] Input:
int(((4*x^2*log(2)^2+2*(-20*x^3-8*x^2)*log(2)+100*x^4)*log(x)^2+(40*x*log( 2)^2+2*(-210*x^2-80*x)*log(2)+1100*x^3+40*x^2)*log(x)+100*log(2)^2+2*(-550 *x-200)*log(2)+3065*x^2)/((4*x^2*log(2)^2-40*x^3*log(2)+100*x^4)*log(x)^2+ (40*x*log(2)^2-420*x^2*log(2)+1100*x^3)*log(x)+100*log(2)^2-1100*x*log(2)+ 3025*x^2),x)
Output:
(4*log(x)*log(2)**2*x + 2*log(x)*log(2)*x**2 - 16*log(x)*log(2)*x - 110*lo g(x)*x**3 - 8*log(x)*x**2 + 20*log(2)**2 - 80*log(2) - 605*x**2)/(11*(2*lo g(x)*log(2)*x - 10*log(x)*x**2 + 10*log(2) - 55*x))