Integrand size = 81, antiderivative size = 28 \[ \int \frac {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx=\frac {4 (5-x)^2 x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \]
Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx=\frac {4 (-5+x)^2 x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )} \]
Integrate[((200*x - 80*x^2 + 8*x^3)*Log[25/x^2] + (400*x - 160*x^2 + 16*x^ 3 + (200*x - 120*x^2 + 16*x^3)*Log[25/x^2])*Log[12/x])/(Log[25/x^2]^3*Log[ 12/x]^3),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.
| \(\displaystyle \int \frac {\left (8 x^3-80 x^2+200 x\right ) \log \left (\frac {25}{x^2}\right )+\left (16 x^3-160 x^2+\left (16 x^3-120 x^2+200 x\right ) \log \left (\frac {25}{x^2}\right )+400 x\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 x (x-5)^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}+\frac {8 x (x-5) \left (2 x \log \left (\frac {25}{x^2}\right )-5 \log \left (\frac {25}{x^2}\right )+2 x-10\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 200 \int \frac {x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}dx-120 \int \frac {x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}dx+200 \int \frac {x}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}dx-80 \int \frac {x^2}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}dx+400 \int \frac {x}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}dx-160 \int \frac {x^2}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}dx+16 \int \frac {x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}dx+8 \int \frac {x^3}{\log ^2\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )}dx+16 \int \frac {x^3}{\log ^3\left (\frac {25}{x^2}\right ) \log ^2\left (\frac {12}{x}\right )}dx\) |
Int[((200*x - 80*x^2 + 8*x^3)*Log[25/x^2] + (400*x - 160*x^2 + 16*x^3 + (2 00*x - 120*x^2 + 16*x^3)*Log[25/x^2])*Log[12/x])/(Log[25/x^2]^3*Log[12/x]^ 3),x]
3.12.54.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 10.16 (sec) , antiderivative size = 44256, normalized size of antiderivative = 1580.57
\[\text {output too large to display}\]
int((((16*x^3-120*x^2+200*x)*ln(25/x^2)+16*x^3-160*x^2+400*x)*ln(12/x)+(8* x^3-80*x^2+200*x)*ln(25/x^2))/ln(25/x^2)^3/ln(12/x)^3,x)
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx=\frac {16 \, {\left (x^{4} - 10 \, x^{3} + 25 \, x^{2}\right )}}{\log \left (\frac {144}{25}\right )^{2} \log \left (\frac {25}{x^{2}}\right )^{2} + 2 \, \log \left (\frac {144}{25}\right ) \log \left (\frac {25}{x^{2}}\right )^{3} + \log \left (\frac {25}{x^{2}}\right )^{4}} \]
integrate((((16*x^3-120*x^2+200*x)*log(25/x^2)+16*x^3-160*x^2+400*x)*log(1 2/x)+(8*x^3-80*x^2+200*x)*log(25/x^2))/log(25/x^2)^3/log(12/x)^3,x, algori thm=\
16*(x^4 - 10*x^3 + 25*x^2)/(log(144/25)^2*log(25/x^2)^2 + 2*log(144/25)*lo g(25/x^2)^3 + log(25/x^2)^4)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (24) = 48\).
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx=\frac {16 x^{4} - 160 x^{3} + 400 x^{2}}{\log {\left (\frac {25}{x^{2}} \right )}^{4} + \left (- 4 \log {\left (5 \right )} + 4 \log {\left (12 \right )}\right ) \log {\left (\frac {25}{x^{2}} \right )}^{3} + \left (- 8 \log {\left (5 \right )} \log {\left (12 \right )} + 4 \log {\left (5 \right )}^{2} + 4 \log {\left (12 \right )}^{2}\right ) \log {\left (\frac {25}{x^{2}} \right )}^{2}} \]
integrate((((16*x**3-120*x**2+200*x)*ln(25/x**2)+16*x**3-160*x**2+400*x)*l n(12/x)+(8*x**3-80*x**2+200*x)*ln(25/x**2))/ln(25/x**2)**3/ln(12/x)**3,x)
(16*x**4 - 160*x**3 + 400*x**2)/(log(25/x**2)**4 + (-4*log(5) + 4*log(12)) *log(25/x**2)**3 + (-8*log(5)*log(12) + 4*log(5)**2 + 4*log(12)**2)*log(25 /x**2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (26) = 52\).
Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.18 \[ \int \frac {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx=\frac {x^{4} - 10 \, x^{3} + 25 \, x^{2}}{\log \left (5\right )^{2} \log \left (3\right )^{2} + 4 \, \log \left (5\right )^{2} \log \left (3\right ) \log \left (2\right ) + 4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} - 2 \, {\left (\log \left (5\right ) + \log \left (3\right ) + 2 \, \log \left (2\right )\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} + {\left (\log \left (5\right )^{2} + 4 \, \log \left (5\right ) \log \left (3\right ) + \log \left (3\right )^{2} + 4 \, {\left (2 \, \log \left (5\right ) + \log \left (3\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} - 2 \, {\left (\log \left (5\right )^{2} \log \left (3\right ) + \log \left (5\right ) \log \left (3\right )^{2} + 4 \, \log \left (5\right ) \log \left (2\right )^{2} + 2 \, {\left (\log \left (5\right )^{2} + 2 \, \log \left (5\right ) \log \left (3\right )\right )} \log \left (2\right )\right )} \log \left (x\right )} \]
integrate((((16*x^3-120*x^2+200*x)*log(25/x^2)+16*x^3-160*x^2+400*x)*log(1 2/x)+(8*x^3-80*x^2+200*x)*log(25/x^2))/log(25/x^2)^3/log(12/x)^3,x, algori thm=\
(x^4 - 10*x^3 + 25*x^2)/(log(5)^2*log(3)^2 + 4*log(5)^2*log(3)*log(2) + 4* log(5)^2*log(2)^2 - 2*(log(5) + log(3) + 2*log(2))*log(x)^3 + log(x)^4 + ( log(5)^2 + 4*log(5)*log(3) + log(3)^2 + 4*(2*log(5) + log(3))*log(2) + 4*l og(2)^2)*log(x)^2 - 2*(log(5)^2*log(3) + log(5)*log(3)^2 + 4*log(5)*log(2) ^2 + 2*(log(5)^2 + 2*log(5)*log(3))*log(2))*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 284, normalized size of antiderivative = 10.14 \[ \int \frac {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx=\frac {x^{4} \log \left (12\right )^{2} - 2 \, x^{4} \log \left (12\right ) \log \left (5\right ) + x^{4} \log \left (5\right )^{2} - 10 \, x^{3} \log \left (12\right )^{2} + 20 \, x^{3} \log \left (12\right ) \log \left (5\right ) - 10 \, x^{3} \log \left (5\right )^{2} + 25 \, x^{2} \log \left (12\right )^{2} - 50 \, x^{2} \log \left (12\right ) \log \left (5\right ) + 25 \, x^{2} \log \left (5\right )^{2}}{\log \left (12\right )^{4} \log \left (5\right )^{2} - 2 \, \log \left (12\right )^{3} \log \left (5\right )^{3} + \log \left (12\right )^{2} \log \left (5\right )^{4} - 2 \, \log \left (12\right )^{4} \log \left (5\right ) \log \left (x\right ) + 2 \, \log \left (12\right )^{3} \log \left (5\right )^{2} \log \left (x\right ) + 2 \, \log \left (12\right )^{2} \log \left (5\right )^{3} \log \left (x\right ) - 2 \, \log \left (12\right ) \log \left (5\right )^{4} \log \left (x\right ) + \log \left (12\right )^{4} \log \left (x\right )^{2} + 2 \, \log \left (12\right )^{3} \log \left (5\right ) \log \left (x\right )^{2} - 6 \, \log \left (12\right )^{2} \log \left (5\right )^{2} \log \left (x\right )^{2} + 2 \, \log \left (12\right ) \log \left (5\right )^{3} \log \left (x\right )^{2} + \log \left (5\right )^{4} \log \left (x\right )^{2} - 2 \, \log \left (12\right )^{3} \log \left (x\right )^{3} + 2 \, \log \left (12\right )^{2} \log \left (5\right ) \log \left (x\right )^{3} + 2 \, \log \left (12\right ) \log \left (5\right )^{2} \log \left (x\right )^{3} - 2 \, \log \left (5\right )^{3} \log \left (x\right )^{3} + \log \left (12\right )^{2} \log \left (x\right )^{4} - 2 \, \log \left (12\right ) \log \left (5\right ) \log \left (x\right )^{4} + \log \left (5\right )^{2} \log \left (x\right )^{4}} \]
integrate((((16*x^3-120*x^2+200*x)*log(25/x^2)+16*x^3-160*x^2+400*x)*log(1 2/x)+(8*x^3-80*x^2+200*x)*log(25/x^2))/log(25/x^2)^3/log(12/x)^3,x, algori thm=\
(x^4*log(12)^2 - 2*x^4*log(12)*log(5) + x^4*log(5)^2 - 10*x^3*log(12)^2 + 20*x^3*log(12)*log(5) - 10*x^3*log(5)^2 + 25*x^2*log(12)^2 - 50*x^2*log(12 )*log(5) + 25*x^2*log(5)^2)/(log(12)^4*log(5)^2 - 2*log(12)^3*log(5)^3 + l og(12)^2*log(5)^4 - 2*log(12)^4*log(5)*log(x) + 2*log(12)^3*log(5)^2*log(x ) + 2*log(12)^2*log(5)^3*log(x) - 2*log(12)*log(5)^4*log(x) + log(12)^4*lo g(x)^2 + 2*log(12)^3*log(5)*log(x)^2 - 6*log(12)^2*log(5)^2*log(x)^2 + 2*l og(12)*log(5)^3*log(x)^2 + log(5)^4*log(x)^2 - 2*log(12)^3*log(x)^3 + 2*lo g(12)^2*log(5)*log(x)^3 + 2*log(12)*log(5)^2*log(x)^3 - 2*log(5)^3*log(x)^ 3 + log(12)^2*log(x)^4 - 2*log(12)*log(5)*log(x)^4 + log(5)^2*log(x)^4)
Time = 10.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {\left (200 x-80 x^2+8 x^3\right ) \log \left (\frac {25}{x^2}\right )+\left (400 x-160 x^2+16 x^3+\left (200 x-120 x^2+16 x^3\right ) \log \left (\frac {25}{x^2}\right )\right ) \log \left (\frac {12}{x}\right )}{\log ^3\left (\frac {25}{x^2}\right ) \log ^3\left (\frac {12}{x}\right )} \, dx=\frac {4\,x^2\,{\left (x-5\right )}^2}{4\,{\ln \left (\frac {12}{x}\right )}^4-4\,{\ln \left (\frac {12}{x}\right )}^3\,\left (2\,\ln \left (\frac {1}{x}\right )+\ln \left (\frac {144\,x^2}{25}\right )\right )+{\ln \left (\frac {12}{x}\right )}^2\,{\left (2\,\ln \left (\frac {1}{x}\right )+\ln \left (\frac {144\,x^2}{25}\right )\right )}^2} \]
int((log(12/x)*(400*x + log(25/x^2)*(200*x - 120*x^2 + 16*x^3) - 160*x^2 + 16*x^3) + log(25/x^2)*(200*x - 80*x^2 + 8*x^3))/(log(12/x)^3*log(25/x^2)^ 3),x)