Integrand size = 105, antiderivative size = 33 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=(x+5 \log (4-x)) \left (-3+x^2+\frac {\left (-1+\frac {\log (x)}{3}\right )^2}{5 x}\right ) \] Output:
(x^2+1/5*(1/3*ln(x)-1)^2/x-3)*(5*ln(4-x)+x)
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=\frac {1}{45} \left (45 x \left (-3+x^2\right )-6 \log (x)+\log ^2(x)+\frac {5 \log (4-x) \left (9-135 x+45 x^3-6 \log (x)+\log ^2(x)\right )}{x}\right ) \] Input:
Integrate[(69*x - 141*x^2 - 135*x^3 - 315*x^4 + 135*x^5 + (300 - 75*x - 18 00*x^3 + 450*x^4)*Log[4 - x] + (-38*x + 2*x^2 + (-160 + 40*x)*Log[4 - x])* Log[x] + (5*x + (20 - 5*x)*Log[4 - x])*Log[x]^2)/(-180*x^2 + 45*x^3),x]
Output:
(45*x*(-3 + x^2) - 6*Log[x] + Log[x]^2 + (5*Log[4 - x]*(9 - 135*x + 45*x^3 - 6*Log[x] + Log[x]^2))/x)/45
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.34 (sec) , antiderivative size = 249, normalized size of antiderivative = 7.55, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2026, 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.
\(\displaystyle \int \frac {135 x^5-315 x^4-135 x^3-141 x^2+\left (2 x^2-38 x+(40 x-160) \log (4-x)\right ) \log (x)+\left (450 x^4-1800 x^3-75 x+300\right ) \log (4-x)+69 x+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{45 x^3-180 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {135 x^5-315 x^4-135 x^3-141 x^2+\left (2 x^2-38 x+(40 x-160) \log (4-x)\right ) \log (x)+\left (450 x^4-1800 x^3-75 x+300\right ) \log (4-x)+69 x+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{x^2 (45 x-180)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {(-x+x \log (4-x)-4 \log (4-x)) \log ^2(x)}{9 (x-4) x^2}+\frac {2 \left (x^2-19 x+20 x \log (4-x)-80 \log (4-x)\right ) \log (x)}{45 (x-4) x^2}+\frac {45 x^5-105 x^4+150 x^4 \log (4-x)-45 x^3-600 x^3 \log (4-x)-47 x^2+23 x-25 x \log (4-x)+100 \log (4-x)}{15 (x-4) x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right )+\frac {\operatorname {PolyLog}\left (2,\frac {4}{x}\right )}{18}+\frac {2 \operatorname {PolyLog}\left (2,\frac {x}{4}\right )}{9}+\frac {\operatorname {PolyLog}\left (3,\frac {4}{x}\right )}{18}-\frac {\operatorname {PolyLog}\left (3,\frac {x}{4}\right )}{18}+\frac {1}{18} \operatorname {PolyLog}\left (2,\frac {4}{x}\right ) \log (x)+\frac {1}{18} \operatorname {PolyLog}\left (2,\frac {x}{4}\right ) \log (x)+x^3+5 x^2 \log (4-x)-3 x-\frac {\log ^3(x)}{108}-\frac {1}{36} \log \left (1-\frac {4}{x}\right ) \log ^2(x)+\frac {1}{36} \log \left (1-\frac {x}{4}\right ) \log ^2(x)-\frac {\log ^2(x)}{180}+\frac {\log (4-x) \log ^2(x)}{9 x}-15 \log (4-x)-\frac {1}{6} \log (4) \log (x-4)-\frac {1}{18} \log \left (1-\frac {4}{x}\right ) \log (x)+\frac {2}{9} \log (4-x) \log (x)-\frac {2}{9} \log (4) \log (x)-\frac {2 \log (x)}{15}+\frac {\log (4-x)}{x}-\frac {2 \log (4-x) \log (x)}{3 x}\) |
Input:
Int[(69*x - 141*x^2 - 135*x^3 - 315*x^4 + 135*x^5 + (300 - 75*x - 1800*x^3 + 450*x^4)*Log[4 - x] + (-38*x + 2*x^2 + (-160 + 40*x)*Log[4 - x])*Log[x] + (5*x + (20 - 5*x)*Log[4 - x])*Log[x]^2)/(-180*x^2 + 45*x^3),x]
Output:
-3*x + x^3 - 15*Log[4 - x] + Log[4 - x]/x + 5*x^2*Log[4 - x] - (Log[4]*Log [-4 + x])/6 - (2*Log[x])/15 - (2*Log[4]*Log[x])/9 - (Log[1 - 4/x]*Log[x])/ 18 + (2*Log[4 - x]*Log[x])/9 - (2*Log[4 - x]*Log[x])/(3*x) - Log[x]^2/180 - (Log[1 - 4/x]*Log[x]^2)/36 + (Log[4 - x]*Log[x]^2)/(9*x) + (Log[1 - x/4] *Log[x]^2)/36 - Log[x]^3/108 + PolyLog[2, 1 - x/4]/6 + PolyLog[2, 4/x]/18 + (Log[x]*PolyLog[2, 4/x])/18 + (2*PolyLog[2, x/4])/9 + (Log[x]*PolyLog[2, x/4])/18 + PolyLog[3, 4/x]/18 - PolyLog[3, x/4]/18
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 133.95 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52
method | result | size |
risch | \(\frac {\left (45 x^{3}+\ln \left (x \right )^{2}-6 \ln \left (x \right )+9\right ) \ln \left (-x +4\right )}{9 x}+\frac {\ln \left (x \right )^{2}}{45}+x^{3}-3 x -\frac {2 \ln \left (x \right )}{15}-15 \ln \left (x -4\right )\) | \(50\) |
parallelrisch | \(\frac {360 x^{4}+1800 \ln \left (-x +4\right ) x^{3}+8 x \ln \left (x \right )^{2}+40 \ln \left (-x +4\right ) \ln \left (x \right )^{2}-48 x \ln \left (x \right )+23400 x \ln \left (x -4\right )-1080 x^{2}-28800 \ln \left (-x +4\right ) x -240 \ln \left (-x +4\right ) \ln \left (x \right )-2160 x +360 \ln \left (-x +4\right )}{360 x}\) | \(89\) |
default | \(-\frac {\left (-\ln \left (x \right )^{2}-2 \ln \left (x \right )-2\right ) \ln \left (-x +4\right )}{9 x}+\frac {785 \ln \left (x -4\right )}{12}-\frac {11 \ln \left (x \right )}{20}+\frac {\ln \left (x \right )^{2}}{45}-\frac {2 \ln \left (x \right ) \ln \left (1-\frac {x}{4}\right )}{45}-\frac {2 \operatorname {polylog}\left (2, \frac {x}{4}\right )}{45}+\frac {2 \left (-20-20 \ln \left (x \right )\right ) \ln \left (-x +4\right )}{45 x}+\frac {2 \left (\ln \left (x \right )-\ln \left (\frac {x}{4}\right )\right ) \ln \left (1-\frac {x}{4}\right )}{45}-\frac {2 \operatorname {dilog}\left (\frac {x}{4}\right )}{45}+5 \left (-x +4\right )^{2} \ln \left (-x +4\right )-3 x +120-40 \left (-x +4\right ) \ln \left (-x +4\right )+\frac {5 \ln \left (-x \right )}{12}+\frac {5 \left (-x +4\right ) \ln \left (-x +4\right )}{12 x}+x^{3}\) | \(155\) |
parts | \(-\frac {\left (-\ln \left (x \right )^{2}-2 \ln \left (x \right )-2\right ) \ln \left (-x +4\right )}{9 x}+\frac {785 \ln \left (x -4\right )}{12}-\frac {11 \ln \left (x \right )}{20}+\frac {\ln \left (x \right )^{2}}{45}-\frac {2 \ln \left (x \right ) \ln \left (1-\frac {x}{4}\right )}{45}-\frac {2 \operatorname {polylog}\left (2, \frac {x}{4}\right )}{45}+\frac {2 \left (-20-20 \ln \left (x \right )\right ) \ln \left (-x +4\right )}{45 x}+\frac {2 \left (\ln \left (x \right )-\ln \left (\frac {x}{4}\right )\right ) \ln \left (1-\frac {x}{4}\right )}{45}-\frac {2 \operatorname {dilog}\left (\frac {x}{4}\right )}{45}+5 \left (-x +4\right )^{2} \ln \left (-x +4\right )-3 x +120-40 \left (-x +4\right ) \ln \left (-x +4\right )+\frac {5 \ln \left (-x \right )}{12}+\frac {5 \left (-x +4\right ) \ln \left (-x +4\right )}{12 x}+x^{3}\) | \(155\) |
orering | \(\text {Expression too large to display}\) | \(10054\) |
Input:
int((((-5*x+20)*ln(-x+4)+5*x)*ln(x)^2+((40*x-160)*ln(-x+4)+2*x^2-38*x)*ln( x)+(450*x^4-1800*x^3-75*x+300)*ln(-x+4)+135*x^5-315*x^4-135*x^3-141*x^2+69 *x)/(45*x^3-180*x^2),x,method=_RETURNVERBOSE)
Output:
1/9*(45*x^3+ln(x)^2-6*ln(x)+9)/x*ln(-x+4)+1/45*ln(x)^2+x^3-3*x-2/15*ln(x)- 15*ln(x-4)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=\frac {45 \, x^{4} + {\left (x + 5 \, \log \left (-x + 4\right )\right )} \log \left (x\right )^{2} - 135 \, x^{2} - 6 \, {\left (x + 5 \, \log \left (-x + 4\right )\right )} \log \left (x\right ) + 45 \, {\left (5 \, x^{3} - 15 \, x + 1\right )} \log \left (-x + 4\right )}{45 \, x} \] Input:
integrate((((-5*x+20)*log(-x+4)+5*x)*log(x)^2+((40*x-160)*log(-x+4)+2*x^2- 38*x)*log(x)+(450*x^4-1800*x^3-75*x+300)*log(-x+4)+135*x^5-315*x^4-135*x^3 -141*x^2+69*x)/(45*x^3-180*x^2),x, algorithm="fricas")
Output:
1/45*(45*x^4 + (x + 5*log(-x + 4))*log(x)^2 - 135*x^2 - 6*(x + 5*log(-x + 4))*log(x) + 45*(5*x^3 - 15*x + 1)*log(-x + 4))/x
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 1.58 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=x^{3} - 3 x + \frac {\log {\left (x \right )}^{2}}{45} - \frac {2 \log {\left (x \right )}}{15} - 15 \log {\left (x - 4 \right )} + \frac {\left (45 x^{3} + \log {\left (x \right )}^{2} - 6 \log {\left (x \right )} + 9\right ) \log {\left (4 - x \right )}}{9 x} \] Input:
integrate((((-5*x+20)*ln(-x+4)+5*x)*ln(x)**2+((40*x-160)*ln(-x+4)+2*x**2-3 8*x)*ln(x)+(450*x**4-1800*x**3-75*x+300)*ln(-x+4)+135*x**5-315*x**4-135*x* *3-141*x**2+69*x)/(45*x**3-180*x**2),x)
Output:
x**3 - 3*x + log(x)**2/45 - 2*log(x)/15 - 15*log(x - 4) + (45*x**3 + log(x )**2 - 6*log(x) + 9)*log(4 - x)/(9*x)
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=\frac {45 \, x^{4} + x \log \left (x\right )^{2} - 135 \, x^{2} - 6 \, x \log \left (x\right ) + 5 \, {\left (45 \, x^{3} + \log \left (x\right )^{2} - 135 \, x - 6 \, \log \left (x\right ) + 9\right )} \log \left (-x + 4\right )}{45 \, x} \] Input:
integrate((((-5*x+20)*log(-x+4)+5*x)*log(x)^2+((40*x-160)*log(-x+4)+2*x^2- 38*x)*log(x)+(450*x^4-1800*x^3-75*x+300)*log(-x+4)+135*x^5-315*x^4-135*x^3 -141*x^2+69*x)/(45*x^3-180*x^2),x, algorithm="maxima")
Output:
1/45*(45*x^4 + x*log(x)^2 - 135*x^2 - 6*x*log(x) + 5*(45*x^3 + log(x)^2 - 135*x - 6*log(x) + 9)*log(-x + 4))/x
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=x^{3} + \frac {1}{45} \, \log \left (x\right )^{2} + \frac {1}{9} \, {\left (45 \, x^{2} + \frac {\log \left (x\right )^{2}}{x} - \frac {6 \, \log \left (x\right )}{x} + \frac {9}{x}\right )} \log \left (-x + 4\right ) - 3 \, x - 15 \, \log \left (x - 4\right ) - \frac {2}{15} \, \log \left (x\right ) \] Input:
integrate((((-5*x+20)*log(-x+4)+5*x)*log(x)^2+((40*x-160)*log(-x+4)+2*x^2- 38*x)*log(x)+(450*x^4-1800*x^3-75*x+300)*log(-x+4)+135*x^5-315*x^4-135*x^3 -141*x^2+69*x)/(45*x^3-180*x^2),x, algorithm="giac")
Output:
x^3 + 1/45*log(x)^2 + 1/9*(45*x^2 + log(x)^2/x - 6*log(x)/x + 9/x)*log(-x + 4) - 3*x - 15*log(x - 4) - 2/15*log(x)
Time = 0.61 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=\frac {{\ln \left (x\right )}^2}{45}-15\,\ln \left (x-4\right )-\frac {2\,\ln \left (x\right )}{15}-3\,x+x^3+\frac {\ln \left (4-x\right )}{x}+5\,x^2\,\ln \left (4-x\right )+\frac {\ln \left (4-x\right )\,{\ln \left (x\right )}^2}{9\,x}-\frac {2\,\ln \left (4-x\right )\,\ln \left (x\right )}{3\,x} \] Input:
int(-(69*x + log(x)^2*(5*x - log(4 - x)*(5*x - 20)) - log(4 - x)*(75*x + 1 800*x^3 - 450*x^4 - 300) - 141*x^2 - 135*x^3 - 315*x^4 + 135*x^5 + log(x)* (log(4 - x)*(40*x - 160) - 38*x + 2*x^2))/(180*x^2 - 45*x^3),x)
Output:
log(x)^2/45 - 15*log(x - 4) - (2*log(x))/15 - 3*x + x^3 + log(4 - x)/x + 5 *x^2*log(4 - x) + (log(4 - x)*log(x)^2)/(9*x) - (2*log(4 - x)*log(x))/(3*x )
Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.58 \[ \int \frac {69 x-141 x^2-135 x^3-315 x^4+135 x^5+\left (300-75 x-1800 x^3+450 x^4\right ) \log (4-x)+\left (-38 x+2 x^2+(-160+40 x) \log (4-x)\right ) \log (x)+(5 x+(20-5 x) \log (4-x)) \log ^2(x)}{-180 x^2+45 x^3} \, dx=\frac {20 \,\mathrm {log}\left (-x +4\right ) \mathrm {log}\left (x \right )^{2}-120 \,\mathrm {log}\left (-x +4\right ) \mathrm {log}\left (x \right )+900 \,\mathrm {log}\left (-x +4\right ) x^{3}-14445 \,\mathrm {log}\left (-x +4\right ) x +180 \,\mathrm {log}\left (-x +4\right )+11745 \,\mathrm {log}\left (x -4\right ) x +4 \mathrm {log}\left (x \right )^{2} x -24 \,\mathrm {log}\left (x \right ) x +180 x^{4}-540 x^{2}}{180 x} \] Input:
int((((-5*x+20)*log(-x+4)+5*x)*log(x)^2+((40*x-160)*log(-x+4)+2*x^2-38*x)* log(x)+(450*x^4-1800*x^3-75*x+300)*log(-x+4)+135*x^5-315*x^4-135*x^3-141*x ^2+69*x)/(45*x^3-180*x^2),x)
Output:
(20*log( - x + 4)*log(x)**2 - 120*log( - x + 4)*log(x) + 900*log( - x + 4) *x**3 - 14445*log( - x + 4)*x + 180*log( - x + 4) + 11745*log(x - 4)*x + 4 *log(x)**2*x - 24*log(x)*x + 180*x**4 - 540*x**2)/(180*x)