Integrand size = 88, antiderivative size = 32 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=x \left (1-2 x+\frac {\left (-(7-x)^2+x\right )^2}{(5+\log (3))^2}-\log (\log (x))\right ) \]
Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x \left (2426+323 x^2-30 x^3+x^4+10 \log (3)+\log ^2(3)-2 x \left (760+10 \log (3)+\log ^2(3)\right )-(5+\log (3))^2 \log (\log (x))\right )}{(5+\log (3))^2} \]
Integrate[(-25 - 10*Log[3] - Log[3]^2 + (2426 - 3040*x + 969*x^2 - 120*x^3 + 5*x^4 + (10 - 40*x)*Log[3] + (1 - 4*x)*Log[3]^2)*Log[x] + (-25 - 10*Log [3] - Log[3]^2)*Log[x]*Log[Log[x]])/((25 + 10*Log[3] + Log[3]^2)*Log[x]),x ]
(x*(2426 + 323*x^2 - 30*x^3 + x^4 + 10*Log[3] + Log[3]^2 - 2*x*(760 + 10*L og[3] + Log[3]^2) - (5 + Log[3])^2*Log[Log[x]]))/(5 + Log[3])^2
Time = 0.47 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {27, 25, 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.
\(\displaystyle \int \frac {\left (5 x^4-120 x^3+969 x^2-3040 x+(1-4 x) \log ^2(3)+(10-40 x) \log (3)+2426\right ) \log (x)+\left (-25-\log ^2(3)-10 \log (3)\right ) \log (\log (x)) \log (x)-25-\log ^2(3)-10 \log (3)}{\left (25+\log ^2(3)+10 \log (3)\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {-\left (\left (5 x^4-120 x^3+969 x^2-3040 x+(1-4 x) \log ^2(3)+10 (1-4 x) \log (3)+2426\right ) \log (x)\right )+(5+\log (3))^2 \log (\log (x)) \log (x)+(5+\log (3))^2}{\log (x)}dx}{(5+\log (3))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {-\left (\left (5 x^4-120 x^3+969 x^2-3040 x+(1-4 x) \log ^2(3)+10 (1-4 x) \log (3)+2426\right ) \log (x)\right )+(5+\log (3))^2 \log (\log (x)) \log (x)+(5+\log (3))^2}{\log (x)}dx}{(5+\log (3))^2}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {-5 \log (x) x^4+120 \log (x) x^3-969 \log (x) x^2+3040 \left (1+\frac {1}{760} \log (3) (10+\log (3))\right ) \log (x) x-2426 \left (1+\frac {\log (3) (10+\log (3))}{2426}\right ) \log (x)+25 \left (1+\frac {1}{25} \log (3) (10+\log (3))\right )}{\log (x)}+(5+\log (3))^2 \log (\log (x))\right )dx}{(5+\log (3))^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-x^5+30 x^4-323 x^3+2 x^2 (760+\log (3) (10+\log (3)))-x \left (2426+\log ^2(3)+10 \log (3)\right )+x (5+\log (3))^2 \log (\log (x))}{(5+\log (3))^2}\) |
Int[(-25 - 10*Log[3] - Log[3]^2 + (2426 - 3040*x + 969*x^2 - 120*x^3 + 5*x ^4 + (10 - 40*x)*Log[3] + (1 - 4*x)*Log[3]^2)*Log[x] + (-25 - 10*Log[3] - Log[3]^2)*Log[x]*Log[Log[x]])/((25 + 10*Log[3] + Log[3]^2)*Log[x]),x]
-((-323*x^3 + 30*x^4 - x^5 - x*(2426 + 10*Log[3] + Log[3]^2) + 2*x^2*(760 + Log[3]*(10 + Log[3])) + x*(5 + Log[3])^2*Log[Log[x]])/(5 + Log[3])^2)
3.7.96.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(32)=64\).
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31
method | result | size |
parts | \(-\frac {-x^{5}+2 x^{2} \ln \left (3\right )^{2}+30 x^{4}-x \ln \left (3\right )^{2}+20 x^{2} \ln \left (3\right )-323 x^{3}-10 x \ln \left (3\right )+1520 x^{2}-2426 x}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-x \ln \left (\ln \left (x \right )\right )\) | \(74\) |
default | \(\frac {2426 x +10 \ln \left (3\right ) \left (-2 x^{2}+x -x \ln \left (\ln \left (x \right )\right )\right )+\ln \left (3\right )^{2} \left (-2 x^{2}+x -x \ln \left (\ln \left (x \right )\right )\right )-1520 x^{2}+323 x^{3}-30 x^{4}+x^{5}-25 x \ln \left (\ln \left (x \right )\right )}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}\) | \(77\) |
parallelrisch | \(\frac {x^{5}-2 x^{2} \ln \left (3\right )^{2}-x \ln \left (3\right )^{2} \ln \left (\ln \left (x \right )\right )-30 x^{4}+x \ln \left (3\right )^{2}-20 x^{2} \ln \left (3\right )-10 \ln \left (3\right ) x \ln \left (\ln \left (x \right )\right )+323 x^{3}+10 x \ln \left (3\right )-1520 x^{2}-25 x \ln \left (\ln \left (x \right )\right )+2426 x}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}\) | \(87\) |
risch | \(-x \ln \left (\ln \left (x \right )\right )+\frac {x^{5}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {2 x^{2} \ln \left (3\right )^{2}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {30 x^{4}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {x \ln \left (3\right )^{2}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {20 x^{2} \ln \left (3\right )}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {323 x^{3}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {10 x \ln \left (3\right )}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}-\frac {1520 x^{2}}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}+\frac {2426 x}{\ln \left (3\right )^{2}+10 \ln \left (3\right )+25}\) | \(165\) |
int(((-ln(3)^2-10*ln(3)-25)*ln(x)*ln(ln(x))+((-4*x+1)*ln(3)^2+(-40*x+10)*l n(3)+5*x^4-120*x^3+969*x^2-3040*x+2426)*ln(x)-ln(3)^2-10*ln(3)-25)/(ln(3)^ 2+10*ln(3)+25)/ln(x),x,method=_RETURNVERBOSE)
-1/(ln(3)^2+10*ln(3)+25)*(-x^5+2*x^2*ln(3)^2+30*x^4-x*ln(3)^2+20*x^2*ln(3) -323*x^3-10*x*ln(3)+1520*x^2-2426*x)-x*ln(ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (30) = 60\).
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.59 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5} - 30 \, x^{4} + 323 \, x^{3} - {\left (2 \, x^{2} - x\right )} \log \left (3\right )^{2} - 1520 \, x^{2} - 10 \, {\left (2 \, x^{2} - x\right )} \log \left (3\right ) - {\left (x \log \left (3\right )^{2} + 10 \, x \log \left (3\right ) + 25 \, x\right )} \log \left (\log \left (x\right )\right ) + 2426 \, x}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} \]
integrate(((-log(3)^2-10*log(3)-25)*log(x)*log(log(x))+((-4*x+1)*log(3)^2+ (-40*x+10)*log(3)+5*x^4-120*x^3+969*x^2-3040*x+2426)*log(x)-log(3)^2-10*lo g(3)-25)/(log(3)^2+10*log(3)+25)/log(x),x, algorithm=\
(x^5 - 30*x^4 + 323*x^3 - (2*x^2 - x)*log(3)^2 - 1520*x^2 - 10*(2*x^2 - x) *log(3) - (x*log(3)^2 + 10*x*log(3) + 25*x)*log(log(x)) + 2426*x)/(log(3)^ 2 + 10*log(3) + 25)
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5}}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} - \frac {30 x^{4}}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} + \frac {323 x^{3}}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} + \frac {x^{2} \left (-1520 - 20 \log {\left (3 \right )} - 2 \log {\left (3 \right )}^{2}\right )}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} - x \log {\left (\log {\left (x \right )} \right )} + \frac {x \left (\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 2426\right )}{\log {\left (3 \right )}^{2} + 10 \log {\left (3 \right )} + 25} \]
integrate(((-ln(3)**2-10*ln(3)-25)*ln(x)*ln(ln(x))+((-4*x+1)*ln(3)**2+(-40 *x+10)*ln(3)+5*x**4-120*x**3+969*x**2-3040*x+2426)*ln(x)-ln(3)**2-10*ln(3) -25)/(ln(3)**2+10*ln(3)+25)/ln(x),x)
x**5/(log(3)**2 + 10*log(3) + 25) - 30*x**4/(log(3)**2 + 10*log(3) + 25) + 323*x**3/(log(3)**2 + 10*log(3) + 25) + x**2*(-1520 - 20*log(3) - 2*log(3 )**2)/(log(3)**2 + 10*log(3) + 25) - x*log(log(x)) + x*(log(3)**2 + 10*log (3) + 2426)/(log(3)**2 + 10*log(3) + 25)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.62 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5} - 30 \, x^{4} - 2 \, x^{2} \log \left (3\right )^{2} + 323 \, x^{3} - 20 \, x^{2} \log \left (3\right ) - {\left (x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right )\right )} \log \left (3\right )^{2} + x \log \left (3\right )^{2} - {\rm Ei}\left (\log \left (x\right )\right ) \log \left (3\right )^{2} - 1520 \, x^{2} - 10 \, {\left (x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right )\right )} \log \left (3\right ) + 10 \, x \log \left (3\right ) - 10 \, {\rm Ei}\left (\log \left (x\right )\right ) \log \left (3\right ) - 25 \, x \log \left (\log \left (x\right )\right ) + 2426 \, x}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} \]
integrate(((-log(3)^2-10*log(3)-25)*log(x)*log(log(x))+((-4*x+1)*log(3)^2+ (-40*x+10)*log(3)+5*x^4-120*x^3+969*x^2-3040*x+2426)*log(x)-log(3)^2-10*lo g(3)-25)/(log(3)^2+10*log(3)+25)/log(x),x, algorithm=\
(x^5 - 30*x^4 - 2*x^2*log(3)^2 + 323*x^3 - 20*x^2*log(3) - (x*log(log(x)) - Ei(log(x)))*log(3)^2 + x*log(3)^2 - Ei(log(x))*log(3)^2 - 1520*x^2 - 10* (x*log(log(x)) - Ei(log(x)))*log(3) + 10*x*log(3) - 10*Ei(log(x))*log(3) - 25*x*log(log(x)) + 2426*x)/(log(3)^2 + 10*log(3) + 25)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.69 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {x^{5} - 30 \, x^{4} - 2 \, x^{2} \log \left (3\right )^{2} - x \log \left (3\right )^{2} \log \left (\log \left (x\right )\right ) + 323 \, x^{3} - 20 \, x^{2} \log \left (3\right ) + x \log \left (3\right )^{2} - 10 \, x \log \left (3\right ) \log \left (\log \left (x\right )\right ) - 1520 \, x^{2} + 10 \, x \log \left (3\right ) - 25 \, x \log \left (\log \left (x\right )\right ) + 2426 \, x}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} \]
integrate(((-log(3)^2-10*log(3)-25)*log(x)*log(log(x))+((-4*x+1)*log(3)^2+ (-40*x+10)*log(3)+5*x^4-120*x^3+969*x^2-3040*x+2426)*log(x)-log(3)^2-10*lo g(3)-25)/(log(3)^2+10*log(3)+25)/log(x),x, algorithm=\
(x^5 - 30*x^4 - 2*x^2*log(3)^2 - x*log(3)^2*log(log(x)) + 323*x^3 - 20*x^2 *log(3) + x*log(3)^2 - 10*x*log(3)*log(log(x)) - 1520*x^2 + 10*x*log(3) - 25*x*log(log(x)) + 2426*x)/(log(3)^2 + 10*log(3) + 25)
Time = 7.94 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.50 \[ \int \frac {-25-10 \log (3)-\log ^2(3)+\left (2426-3040 x+969 x^2-120 x^3+5 x^4+(10-40 x) \log (3)+(1-4 x) \log ^2(3)\right ) \log (x)+\left (-25-10 \log (3)-\log ^2(3)\right ) \log (x) \log (\log (x))}{\left (25+10 \log (3)+\log ^2(3)\right ) \log (x)} \, dx=\frac {323\,x^3}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}-x\,\ln \left (\ln \left (x\right )\right )-\frac {30\,x^4}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}+\frac {x^5}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}+\frac {x\,\left (10\,\ln \left (3\right )+{\ln \left (3\right )}^2+2426\right )}{10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25}-\frac {x^2\,\left (40\,\ln \left (3\right )+4\,{\ln \left (3\right )}^2+3040\right )}{2\,\left (10\,\ln \left (3\right )+{\ln \left (3\right )}^2+25\right )} \]
int(-(10*log(3) + log(x)*(3040*x + log(3)*(40*x - 10) + log(3)^2*(4*x - 1) - 969*x^2 + 120*x^3 - 5*x^4 - 2426) + log(3)^2 + log(log(x))*log(x)*(10*l og(3) + log(3)^2 + 25) + 25)/(log(x)*(10*log(3) + log(3)^2 + 25)),x)