Integrand size = 148, antiderivative size = 33 \[ \int \frac {-1125-2325 x-335 x^2+849 x^3-216 x^4+16 x^5+\left (-150 x-140 x^2+74 x^3-8 x^4\right ) \log (5-x)+\left (-500 x+300 x^2-60 x^3+4 x^4\right ) \log ^2(5-x)+\left (-90-222 x-112 x^2+32 x^3\right ) \log (x)+\left (-18 x-48 x^2-32 x^3\right ) \log ^2(x)}{-1125 x-2325 x^2-335 x^3+849 x^4-216 x^5+16 x^6} \, dx=-3-\frac {\log ^2(5-x)}{3+4 x}+\log (x)+\frac {\log ^2(x)}{(5-x)^2} \]
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {-1125-2325 x-335 x^2+849 x^3-216 x^4+16 x^5+\left (-150 x-140 x^2+74 x^3-8 x^4\right ) \log (5-x)+\left (-500 x+300 x^2-60 x^3+4 x^4\right ) \log ^2(5-x)+\left (-90-222 x-112 x^2+32 x^3\right ) \log (x)+\left (-18 x-48 x^2-32 x^3\right ) \log ^2(x)}{-1125 x-2325 x^2-335 x^3+849 x^4-216 x^5+16 x^6} \, dx=-\frac {\log ^2(5-x)}{3+4 x}+\log (x)+\frac {\log ^2(x)}{(-5+x)^2}-\frac {2}{23} \log \left (\frac {23}{4}\right ) \log (3+4 x)+\frac {2}{23} \log \left (\frac {23}{4}\right ) \log (6+8 x) \]
Integrate[(-1125 - 2325*x - 335*x^2 + 849*x^3 - 216*x^4 + 16*x^5 + (-150*x - 140*x^2 + 74*x^3 - 8*x^4)*Log[5 - x] + (-500*x + 300*x^2 - 60*x^3 + 4*x ^4)*Log[5 - x]^2 + (-90 - 222*x - 112*x^2 + 32*x^3)*Log[x] + (-18*x - 48*x ^2 - 32*x^3)*Log[x]^2)/(-1125*x - 2325*x^2 - 335*x^3 + 849*x^4 - 216*x^5 + 16*x^6),x]
-(Log[5 - x]^2/(3 + 4*x)) + Log[x] + Log[x]^2/(-5 + x)^2 - (2*Log[23/4]*Lo g[3 + 4*x])/23 + (2*Log[23/4]*Log[6 + 8*x])/23
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.87 (sec) , antiderivative size = 276, normalized size of antiderivative = 8.36, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2026, 2463, 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.
\(\displaystyle \int \frac {16 x^5-216 x^4+849 x^3-335 x^2+\left (-32 x^3-48 x^2-18 x\right ) \log ^2(x)+\left (32 x^3-112 x^2-222 x-90\right ) \log (x)+\left (4 x^4-60 x^3+300 x^2-500 x\right ) \log ^2(5-x)+\left (-8 x^4+74 x^3-140 x^2-150 x\right ) \log (5-x)-2325 x-1125}{16 x^6-216 x^5+849 x^4-335 x^3-2325 x^2-1125 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {16 x^5-216 x^4+849 x^3-335 x^2+\left (-32 x^3-48 x^2-18 x\right ) \log ^2(x)+\left (32 x^3-112 x^2-222 x-90\right ) \log (x)+\left (4 x^4-60 x^3+300 x^2-500 x\right ) \log ^2(5-x)+\left (-8 x^4+74 x^3-140 x^2-150 x\right ) \log (5-x)-2325 x-1125}{x \left (16 x^5-216 x^4+849 x^3-335 x^2-2325 x-1125\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {48 \left (16 x^5-216 x^4+849 x^3-335 x^2+\left (-32 x^3-48 x^2-18 x\right ) \log ^2(x)+\left (32 x^3-112 x^2-222 x-90\right ) \log (x)+\left (4 x^4-60 x^3+300 x^2-500 x\right ) \log ^2(5-x)+\left (-8 x^4+74 x^3-140 x^2-150 x\right ) \log (5-x)-2325 x-1125\right )}{279841 (x-5) x}-\frac {8 \left (16 x^5-216 x^4+849 x^3-335 x^2+\left (-32 x^3-48 x^2-18 x\right ) \log ^2(x)+\left (32 x^3-112 x^2-222 x-90\right ) \log (x)+\left (4 x^4-60 x^3+300 x^2-500 x\right ) \log ^2(5-x)+\left (-8 x^4+74 x^3-140 x^2-150 x\right ) \log (5-x)-2325 x-1125\right )}{12167 (x-5)^2 x}+\frac {16 x^5-216 x^4+849 x^3-335 x^2+\left (-32 x^3-48 x^2-18 x\right ) \log ^2(x)+\left (32 x^3-112 x^2-222 x-90\right ) \log (x)+\left (4 x^4-60 x^3+300 x^2-500 x\right ) \log ^2(5-x)+\left (-8 x^4+74 x^3-140 x^2-150 x\right ) \log (5-x)-2325 x-1125}{529 (x-5)^3 x}-\frac {192 \left (16 x^5-216 x^4+849 x^3-335 x^2+\left (-32 x^3-48 x^2-18 x\right ) \log ^2(x)+\left (32 x^3-112 x^2-222 x-90\right ) \log (x)+\left (4 x^4-60 x^3+300 x^2-500 x\right ) \log ^2(5-x)+\left (-8 x^4+74 x^3-140 x^2-150 x\right ) \log (5-x)-2325 x-1125\right )}{279841 x (4 x+3)}-\frac {64 \left (16 x^5-216 x^4+849 x^3-335 x^2+\left (-32 x^3-48 x^2-18 x\right ) \log ^2(x)+\left (32 x^3-112 x^2-222 x-90\right ) \log (x)+\left (4 x^4-60 x^3+300 x^2-500 x\right ) \log ^2(5-x)+\left (-8 x^4+74 x^3-140 x^2-150 x\right ) \log (5-x)-2325 x-1125\right )}{12167 x (4 x+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{25} \operatorname {PolyLog}\left (2,1-\frac {x}{5}\right )-\frac {2 \operatorname {PolyLog}\left (2,\frac {5}{x}\right )}{25}+\frac {192 x^4}{279841}-\frac {8576 x^3}{839523}+\frac {31024 x^2}{279841}-\frac {16 \left (8 x^3-51 x^2-90 x+725\right ) \log (5-x)}{279841}-\frac {192 (5-x)^4}{279841}+\frac {16 (5-x)^2}{12167}-\frac {2 (4 x+3)^3}{36501}-\frac {384 (x+3)^2}{279841}-\frac {88360 x}{279841}-\frac {4 (5-x) \log ^2(5-x)}{23 (4 x+3)}-\frac {1}{23} \log ^2(5-x)+\frac {\log ^2(x)}{25}+\frac {\log ^2(x)}{(5-x)^2}-\frac {128 (5-x)^3 \log (5-x)}{279841}+\frac {16 (5-x)^2 \log (5-x)}{12167}+\frac {16}{529} (5-x) \log (5-x)+\frac {2 (4 x+3)^2 \log (5-x)}{12167}-\frac {2}{23} \log (5-x)-\frac {2}{25} \log (5) \log (x-5)+\frac {2}{25} \log \left (1-\frac {5}{x}\right ) \log (x)+\log (x)\) |
Int[(-1125 - 2325*x - 335*x^2 + 849*x^3 - 216*x^4 + 16*x^5 + (-150*x - 140 *x^2 + 74*x^3 - 8*x^4)*Log[5 - x] + (-500*x + 300*x^2 - 60*x^3 + 4*x^4)*Lo g[5 - x]^2 + (-90 - 222*x - 112*x^2 + 32*x^3)*Log[x] + (-18*x - 48*x^2 - 3 2*x^3)*Log[x]^2)/(-1125*x - 2325*x^2 - 335*x^3 + 849*x^4 - 216*x^5 + 16*x^ 6),x]
(16*(5 - x)^2)/12167 - (192*(5 - x)^4)/279841 - (88360*x)/279841 + (31024* x^2)/279841 - (8576*x^3)/839523 + (192*x^4)/279841 - (384*(3 + x)^2)/27984 1 - (2*(3 + 4*x)^3)/36501 - (2*Log[5 - x])/23 + (16*(5 - x)*Log[5 - x])/52 9 + (16*(5 - x)^2*Log[5 - x])/12167 - (128*(5 - x)^3*Log[5 - x])/279841 + (2*(3 + 4*x)^2*Log[5 - x])/12167 - (16*(725 - 90*x - 51*x^2 + 8*x^3)*Log[5 - x])/279841 - Log[5 - x]^2/23 - (4*(5 - x)*Log[5 - x]^2)/(23*(3 + 4*x)) - (2*Log[5]*Log[-5 + x])/25 + Log[x] + (2*Log[1 - 5/x]*Log[x])/25 + Log[x] ^2/25 + Log[x]^2/(5 - x)^2 + (2*PolyLog[2, 1 - x/5])/25 - (2*PolyLog[2, 5/ x])/25
3.28.84.3.1 Defintions of rubi rules used
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])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52
method | result | size |
risch | \(-\frac {\ln \left (5-x \right )^{2}}{3+4 x}+\frac {x^{2} \ln \left (x \right )-10 x \ln \left (x \right )+\ln \left (x \right )^{2}+25 \ln \left (x \right )}{x^{2}-10 x +25}\) | \(50\) |
parallelrisch | \(-\frac {27000 \ln \left (5-x \right )^{2} x^{2}-270000 \ln \left (5-x \right )^{2} x -108000 x \ln \left (x \right )^{2}-1890000 x \ln \left (x \right )-2025000 \ln \left (x \right )-81000 \ln \left (x \right )^{2}-108000 x^{3} \ln \left (x \right )+999000 x^{2} \ln \left (x \right )+675000 \ln \left (5-x \right )^{2}}{27000 \left (3+4 x \right ) \left (-5+x \right )^{2}}\) | \(86\) |
int(((-32*x^3-48*x^2-18*x)*ln(x)^2+(32*x^3-112*x^2-222*x-90)*ln(x)+(4*x^4- 60*x^3+300*x^2-500*x)*ln(5-x)^2+(-8*x^4+74*x^3-140*x^2-150*x)*ln(5-x)+16*x ^5-216*x^4+849*x^3-335*x^2-2325*x-1125)/(16*x^6-216*x^5+849*x^4-335*x^3-23 25*x^2-1125*x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.97 \[ \int \frac {-1125-2325 x-335 x^2+849 x^3-216 x^4+16 x^5+\left (-150 x-140 x^2+74 x^3-8 x^4\right ) \log (5-x)+\left (-500 x+300 x^2-60 x^3+4 x^4\right ) \log ^2(5-x)+\left (-90-222 x-112 x^2+32 x^3\right ) \log (x)+\left (-18 x-48 x^2-32 x^3\right ) \log ^2(x)}{-1125 x-2325 x^2-335 x^3+849 x^4-216 x^5+16 x^6} \, dx=\frac {{\left (4 \, x + 3\right )} \log \left (x\right )^{2} - {\left (x^{2} - 10 \, x + 25\right )} \log \left (-x + 5\right )^{2} + {\left (4 \, x^{3} - 37 \, x^{2} + 70 \, x + 75\right )} \log \left (x\right )}{4 \, x^{3} - 37 \, x^{2} + 70 \, x + 75} \]
integrate(((-32*x^3-48*x^2-18*x)*log(x)^2+(32*x^3-112*x^2-222*x-90)*log(x) +(4*x^4-60*x^3+300*x^2-500*x)*log(5-x)^2+(-8*x^4+74*x^3-140*x^2-150*x)*log (5-x)+16*x^5-216*x^4+849*x^3-335*x^2-2325*x-1125)/(16*x^6-216*x^5+849*x^4- 335*x^3-2325*x^2-1125*x),x, algorithm=\
((4*x + 3)*log(x)^2 - (x^2 - 10*x + 25)*log(-x + 5)^2 + (4*x^3 - 37*x^2 + 70*x + 75)*log(x))/(4*x^3 - 37*x^2 + 70*x + 75)
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {-1125-2325 x-335 x^2+849 x^3-216 x^4+16 x^5+\left (-150 x-140 x^2+74 x^3-8 x^4\right ) \log (5-x)+\left (-500 x+300 x^2-60 x^3+4 x^4\right ) \log ^2(5-x)+\left (-90-222 x-112 x^2+32 x^3\right ) \log (x)+\left (-18 x-48 x^2-32 x^3\right ) \log ^2(x)}{-1125 x-2325 x^2-335 x^3+849 x^4-216 x^5+16 x^6} \, dx=\log {\left (x \right )} + \frac {\log {\left (x \right )}^{2}}{x^{2} - 10 x + 25} - \frac {\log {\left (5 - x \right )}^{2}}{4 x + 3} \]
integrate(((-32*x**3-48*x**2-18*x)*ln(x)**2+(32*x**3-112*x**2-222*x-90)*ln (x)+(4*x**4-60*x**3+300*x**2-500*x)*ln(5-x)**2+(-8*x**4+74*x**3-140*x**2-1 50*x)*ln(5-x)+16*x**5-216*x**4+849*x**3-335*x**2-2325*x-1125)/(16*x**6-216 *x**5+849*x**4-335*x**3-2325*x**2-1125*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (31) = 62\).
Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 4.48 \[ \int \frac {-1125-2325 x-335 x^2+849 x^3-216 x^4+16 x^5+\left (-150 x-140 x^2+74 x^3-8 x^4\right ) \log (5-x)+\left (-500 x+300 x^2-60 x^3+4 x^4\right ) \log ^2(5-x)+\left (-90-222 x-112 x^2+32 x^3\right ) \log (x)+\left (-18 x-48 x^2-32 x^3\right ) \log ^2(x)}{-1125 x-2325 x^2-335 x^3+849 x^4-216 x^5+16 x^6} \, dx=\frac {279841 \, {\left (4 \, x + 3\right )} \log \left (x\right )^{2} - 279841 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (-x + 5\right )^{2} + 2275620 \, x^{2} + 135441 \, {\left (4 \, x^{3} - 37 \, x^{2} + 70 \, x + 75\right )} \log \left (-x + 5\right ) - 14114985 \, x - 1828500}{279841 \, {\left (4 \, x^{3} - 37 \, x^{2} + 70 \, x + 75\right )}} + \frac {15 \, {\left (1688 \, x^{2} - 24194 \, x + 86705\right )}}{24334 \, {\left (4 \, x^{3} - 37 \, x^{2} + 70 \, x + 75\right )}} - \frac {2325 \, {\left (96 \, x^{2} - 684 \, x + 491\right )}}{24334 \, {\left (4 \, x^{3} - 37 \, x^{2} + 70 \, x + 75\right )}} - \frac {135441}{279841} \, \log \left (x - 5\right ) + \log \left (x\right ) \]
integrate(((-32*x^3-48*x^2-18*x)*log(x)^2+(32*x^3-112*x^2-222*x-90)*log(x) +(4*x^4-60*x^3+300*x^2-500*x)*log(5-x)^2+(-8*x^4+74*x^3-140*x^2-150*x)*log (5-x)+16*x^5-216*x^4+849*x^3-335*x^2-2325*x-1125)/(16*x^6-216*x^5+849*x^4- 335*x^3-2325*x^2-1125*x),x, algorithm=\
1/279841*(279841*(4*x + 3)*log(x)^2 - 279841*(x^2 - 10*x + 25)*log(-x + 5) ^2 + 2275620*x^2 + 135441*(4*x^3 - 37*x^2 + 70*x + 75)*log(-x + 5) - 14114 985*x - 1828500)/(4*x^3 - 37*x^2 + 70*x + 75) + 15/24334*(1688*x^2 - 24194 *x + 86705)/(4*x^3 - 37*x^2 + 70*x + 75) - 2325/24334*(96*x^2 - 684*x + 49 1)/(4*x^3 - 37*x^2 + 70*x + 75) - 135441/279841*log(x - 5) + log(x)
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {-1125-2325 x-335 x^2+849 x^3-216 x^4+16 x^5+\left (-150 x-140 x^2+74 x^3-8 x^4\right ) \log (5-x)+\left (-500 x+300 x^2-60 x^3+4 x^4\right ) \log ^2(5-x)+\left (-90-222 x-112 x^2+32 x^3\right ) \log (x)+\left (-18 x-48 x^2-32 x^3\right ) \log ^2(x)}{-1125 x-2325 x^2-335 x^3+849 x^4-216 x^5+16 x^6} \, dx=\frac {\log \left (x\right )^{2}}{x^{2} - 10 \, x + 25} - \frac {\log \left (-x + 5\right )^{2}}{4 \, x + 3} + \log \left (x\right ) \]
integrate(((-32*x^3-48*x^2-18*x)*log(x)^2+(32*x^3-112*x^2-222*x-90)*log(x) +(4*x^4-60*x^3+300*x^2-500*x)*log(5-x)^2+(-8*x^4+74*x^3-140*x^2-150*x)*log (5-x)+16*x^5-216*x^4+849*x^3-335*x^2-2325*x-1125)/(16*x^6-216*x^5+849*x^4- 335*x^3-2325*x^2-1125*x),x, algorithm=\
Time = 12.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {-1125-2325 x-335 x^2+849 x^3-216 x^4+16 x^5+\left (-150 x-140 x^2+74 x^3-8 x^4\right ) \log (5-x)+\left (-500 x+300 x^2-60 x^3+4 x^4\right ) \log ^2(5-x)+\left (-90-222 x-112 x^2+32 x^3\right ) \log (x)+\left (-18 x-48 x^2-32 x^3\right ) \log ^2(x)}{-1125 x-2325 x^2-335 x^3+849 x^4-216 x^5+16 x^6} \, dx=\ln \left (x\right )+\frac {{\ln \left (x\right )}^2}{x^2-10\,x+25}-\frac {{\ln \left (5-x\right )}^2}{4\,\left (x+\frac {3}{4}\right )} \]
int((2325*x + log(5 - x)^2*(500*x - 300*x^2 + 60*x^3 - 4*x^4) + log(x)^2*( 18*x + 48*x^2 + 32*x^3) + 335*x^2 - 849*x^3 + 216*x^4 - 16*x^5 + log(5 - x )*(150*x + 140*x^2 - 74*x^3 + 8*x^4) + log(x)*(222*x + 112*x^2 - 32*x^3 + 90) + 1125)/(1125*x + 2325*x^2 + 335*x^3 - 849*x^4 + 216*x^5 - 16*x^6),x)