Integrand size = 82, antiderivative size = 32 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=4-x+\left (2-\frac {5}{4} \left (-\frac {1+x}{x^2}+4 \left (2+\frac {x}{\log (x)}\right )\right )\right )^2 \] Output:
(-8-5*x/ln(x)+5/4*(1+x)/x^2)^2+4-x
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(32)=64\).
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=\frac {25}{16 x^4}+\frac {25}{8 x^3}-\frac {295}{16 x^2}-\frac {20}{x}-x+\frac {25 x^2}{\log ^2(x)}-\frac {25}{2 \log (x)}-\frac {25}{2 x \log (x)}+\frac {80 x}{\log (x)} \] Input:
Integrate[(-400*x^6 + (100*x^3 + 100*x^4 - 640*x^5 + 400*x^6)*Log[x] + (10 0*x^3 + 640*x^5)*Log[x]^2 + (-50 - 75*x + 295*x^2 + 160*x^3 - 8*x^5)*Log[x ]^3)/(8*x^5*Log[x]^3),x]
Output:
25/(16*x^4) + 25/(8*x^3) - 295/(16*x^2) - 20/x - x + (25*x^2)/Log[x]^2 - 2 5/(2*Log[x]) - 25/(2*x*Log[x]) + (80*x)/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.
\(\displaystyle \int \frac {-400 x^6+\left (640 x^5+100 x^3\right ) \log ^2(x)+\left (-8 x^5+160 x^3+295 x^2-75 x-50\right ) \log ^3(x)+\left (400 x^6-640 x^5+100 x^4+100 x^3\right ) \log (x)}{8 x^5 \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{8} \int -\frac {400 x^6+\left (8 x^5-160 x^3-295 x^2+75 x+50\right ) \log ^3(x)-20 \left (32 x^5+5 x^3\right ) \log ^2(x)-20 \left (20 x^6-32 x^5+5 x^4+5 x^3\right ) \log (x)}{x^5 \log ^3(x)}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{8} \int \frac {400 x^6+\left (8 x^5-160 x^3-295 x^2+75 x+50\right ) \log ^3(x)-20 \left (32 x^5+5 x^3\right ) \log ^2(x)-20 \left (20 x^6-32 x^5+5 x^4+5 x^3\right ) \log (x)}{x^5 \log ^3(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{8} \int \left (\frac {400 x}{\log ^3(x)}-\frac {20 \left (32 x^2+5\right )}{\log (x) x^2}-\frac {20 \left (20 x^3-32 x^2+5 x+5\right )}{\log ^2(x) x^2}+\frac {8 x^5-160 x^3-295 x^2+75 x+50}{x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} \left (20 \int \frac {20 x^3-32 x^2+5 x+5}{x^2 \log ^2(x)}dx+100 \operatorname {ExpIntegralEi}(-\log (x))-800 \operatorname {ExpIntegralEi}(2 \log (x))+640 \operatorname {LogIntegral}(x)+\frac {25}{2 x^4}+\frac {25}{x^3}-\frac {295}{2 x^2}+\frac {200 x^2}{\log ^2(x)}+\frac {400 x^2}{\log (x)}-8 x-\frac {160}{x}\right )\) |
Input:
Int[(-400*x^6 + (100*x^3 + 100*x^4 - 640*x^5 + 400*x^6)*Log[x] + (100*x^3 + 640*x^5)*Log[x]^2 + (-50 - 75*x + 295*x^2 + 160*x^3 - 8*x^5)*Log[x]^3)/( 8*x^5*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).
Time = 1.50 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75
method | result | size |
default | \(-x -\frac {20}{x}+\frac {80 x}{\ln \left (x \right )}+\frac {25 x^{2}}{\ln \left (x \right )^{2}}-\frac {295}{16 x^{2}}-\frac {25}{2 \ln \left (x \right )}+\frac {25}{8 x^{3}}-\frac {25}{2 x \ln \left (x \right )}+\frac {25}{16 x^{4}}\) | \(56\) |
parts | \(-x -\frac {20}{x}+\frac {80 x}{\ln \left (x \right )}+\frac {25 x^{2}}{\ln \left (x \right )^{2}}-\frac {295}{16 x^{2}}-\frac {25}{2 \ln \left (x \right )}+\frac {25}{8 x^{3}}-\frac {25}{2 x \ln \left (x \right )}+\frac {25}{16 x^{4}}\) | \(56\) |
risch | \(-\frac {16 x^{5}+320 x^{3}+295 x^{2}-50 x -25}{16 x^{4}}+\frac {25 x^{3}+80 x^{2} \ln \left (x \right )-\frac {25 x \ln \left (x \right )}{2}-\frac {25 \ln \left (x \right )}{2}}{x \ln \left (x \right )^{2}}\) | \(58\) |
parallelrisch | \(-\frac {16 x^{5} \ln \left (x \right )^{2}-400 x^{6}-1280 x^{5} \ln \left (x \right )+200 x^{4} \ln \left (x \right )+320 x^{3} \ln \left (x \right )^{2}+200 x^{3} \ln \left (x \right )+295 x^{2} \ln \left (x \right )^{2}-50 x \ln \left (x \right )^{2}-25 \ln \left (x \right )^{2}}{16 x^{4} \ln \left (x \right )^{2}}\) | \(77\) |
Input:
int(1/8*((-8*x^5+160*x^3+295*x^2-75*x-50)*ln(x)^3+(640*x^5+100*x^3)*ln(x)^ 2+(400*x^6-640*x^5+100*x^4+100*x^3)*ln(x)-400*x^6)/x^5/ln(x)^3,x,method=_R ETURNVERBOSE)
Output:
-x-20/x+80*x/ln(x)+25*x^2/ln(x)^2-295/16/x^2-25/2/ln(x)+25/8/x^3-25/2/x/ln (x)+25/16/x^4
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=\frac {400 \, x^{6} - {\left (16 \, x^{5} + 320 \, x^{3} + 295 \, x^{2} - 50 \, x - 25\right )} \log \left (x\right )^{2} + 40 \, {\left (32 \, x^{5} - 5 \, x^{4} - 5 \, x^{3}\right )} \log \left (x\right )}{16 \, x^{4} \log \left (x\right )^{2}} \] Input:
integrate(1/8*((-8*x^5+160*x^3+295*x^2-75*x-50)*log(x)^3+(640*x^5+100*x^3) *log(x)^2+(400*x^6-640*x^5+100*x^4+100*x^3)*log(x)-400*x^6)/x^5/log(x)^3,x , algorithm="fricas")
Output:
1/16*(400*x^6 - (16*x^5 + 320*x^3 + 295*x^2 - 50*x - 25)*log(x)^2 + 40*(32 *x^5 - 5*x^4 - 5*x^3)*log(x))/(x^4*log(x)^2)
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=- x + \frac {50 x^{3} + \left (160 x^{2} - 25 x - 25\right ) \log {\left (x \right )}}{2 x \log {\left (x \right )}^{2}} - \frac {320 x^{3} + 295 x^{2} - 50 x - 25}{16 x^{4}} \] Input:
integrate(1/8*((-8*x**5+160*x**3+295*x**2-75*x-50)*ln(x)**3+(640*x**5+100* x**3)*ln(x)**2+(400*x**6-640*x**5+100*x**4+100*x**3)*ln(x)-400*x**6)/x**5/ ln(x)**3,x)
Output:
-x + (50*x**3 + (160*x**2 - 25*x - 25)*log(x))/(2*x*log(x)**2) - (320*x**3 + 295*x**2 - 50*x - 25)/(16*x**4)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.25 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=-x - \frac {20}{x} - \frac {25}{2 \, \log \left (x\right )} - \frac {295}{16 \, x^{2}} + \frac {25}{8 \, x^{3}} + \frac {25}{16 \, x^{4}} + \frac {25}{2} \, {\rm Ei}\left (-\log \left (x\right )\right ) + 80 \, {\rm Ei}\left (\log \left (x\right )\right ) - 80 \, \Gamma \left (-1, -\log \left (x\right )\right ) + 100 \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - \frac {25}{2} \, \Gamma \left (-1, \log \left (x\right )\right ) + 200 \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) \] Input:
integrate(1/8*((-8*x^5+160*x^3+295*x^2-75*x-50)*log(x)^3+(640*x^5+100*x^3) *log(x)^2+(400*x^6-640*x^5+100*x^4+100*x^3)*log(x)-400*x^6)/x^5/log(x)^3,x , algorithm="maxima")
Output:
-x - 20/x - 25/2/log(x) - 295/16/x^2 + 25/8/x^3 + 25/16/x^4 + 25/2*Ei(-log (x)) + 80*Ei(log(x)) - 80*gamma(-1, -log(x)) + 100*gamma(-1, -2*log(x)) - 25/2*gamma(-1, log(x)) + 200*gamma(-2, -2*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (26) = 52\).
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=-x + \frac {5 \, {\left (10 \, x^{3} + 32 \, x^{2} \log \left (x\right ) - 5 \, x \log \left (x\right ) - 5 \, \log \left (x\right )\right )}}{2 \, x \log \left (x\right )^{2}} - \frac {5 \, {\left (64 \, x^{3} + 59 \, x^{2} - 10 \, x - 5\right )}}{16 \, x^{4}} \] Input:
integrate(1/8*((-8*x^5+160*x^3+295*x^2-75*x-50)*log(x)^3+(640*x^5+100*x^3) *log(x)^2+(400*x^6-640*x^5+100*x^4+100*x^3)*log(x)-400*x^6)/x^5/log(x)^3,x , algorithm="giac")
Output:
-x + 5/2*(10*x^3 + 32*x^2*log(x) - 5*x*log(x) - 5*log(x))/(x*log(x)^2) - 5 /16*(64*x^3 + 59*x^2 - 10*x - 5)/x^4
Time = 3.66 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=-\frac {x^5+20\,x^3+\frac {295\,x^2}{16}-\frac {25\,x}{8}-\frac {25}{16}}{x^4}-\frac {\ln \left (x\right )\,\left (-80\,x^5+\frac {25\,x^4}{2}+\frac {25\,x^3}{2}\right )-25\,x^6}{x^4\,{\ln \left (x\right )}^2} \] Input:
int(((log(x)^2*(100*x^3 + 640*x^5))/8 - (log(x)^3*(75*x - 295*x^2 - 160*x^ 3 + 8*x^5 + 50))/8 + (log(x)*(100*x^3 + 100*x^4 - 640*x^5 + 400*x^6))/8 - 50*x^6)/(x^5*log(x)^3),x)
Output:
- ((295*x^2)/16 - (25*x)/8 + 20*x^3 + x^5 - 25/16)/x^4 - (log(x)*((25*x^3) /2 + (25*x^4)/2 - 80*x^5) - 25*x^6)/(x^4*log(x)^2)
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.38 \[ \int \frac {-400 x^6+\left (100 x^3+100 x^4-640 x^5+400 x^6\right ) \log (x)+\left (100 x^3+640 x^5\right ) \log ^2(x)+\left (-50-75 x+295 x^2+160 x^3-8 x^5\right ) \log ^3(x)}{8 x^5 \log ^3(x)} \, dx=\frac {-16 \mathrm {log}\left (x \right )^{2} x^{5}-320 \mathrm {log}\left (x \right )^{2} x^{3}-295 \mathrm {log}\left (x \right )^{2} x^{2}+50 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+1280 \,\mathrm {log}\left (x \right ) x^{5}-200 \,\mathrm {log}\left (x \right ) x^{4}-200 \,\mathrm {log}\left (x \right ) x^{3}+400 x^{6}}{16 \mathrm {log}\left (x \right )^{2} x^{4}} \] Input:
int(1/8*((-8*x^5+160*x^3+295*x^2-75*x-50)*log(x)^3+(640*x^5+100*x^3)*log(x )^2+(400*x^6-640*x^5+100*x^4+100*x^3)*log(x)-400*x^6)/x^5/log(x)^3,x)
Output:
( - 16*log(x)**2*x**5 - 320*log(x)**2*x**3 - 295*log(x)**2*x**2 + 50*log(x )**2*x + 25*log(x)**2 + 1280*log(x)*x**5 - 200*log(x)*x**4 - 200*log(x)*x* *3 + 400*x**6)/(16*log(x)**2*x**4)