Integrand size = 159, antiderivative size = 23 \begin {dmath*} \int \frac {-3+15 x-106 x^2+294 x^3-516 x^4+588 x^5-420 x^6+156 x^7+6 x^8-34 x^9+14 x^{10}-2 x^{11}+\left (-2+26 x-138 x^2+402 x^3-708 x^4+756 x^5-420 x^6-12 x^7+198 x^8-142 x^9+46 x^{10}-6 x^{11}\right ) \log (x)}{-1+9 x-36 x^2+84 x^3-126 x^4+126 x^5-84 x^6+36 x^7-9 x^8+x^9} \, dx=\frac {x}{(-1+x)^8}-2 \left (-x+x^2 (2+x)\right ) \log (x) \end {dmath*}
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \begin {dmath*} \int \frac {-3+15 x-106 x^2+294 x^3-516 x^4+588 x^5-420 x^6+156 x^7+6 x^8-34 x^9+14 x^{10}-2 x^{11}+\left (-2+26 x-138 x^2+402 x^3-708 x^4+756 x^5-420 x^6-12 x^7+198 x^8-142 x^9+46 x^{10}-6 x^{11}\right ) \log (x)}{-1+9 x-36 x^2+84 x^3-126 x^4+126 x^5-84 x^6+36 x^7-9 x^8+x^9} \, dx=x \left (\frac {1}{(-1+x)^8}-2 \left (-1+2 x+x^2\right ) \log (x)\right ) \end {dmath*}
Integrate[(-3 + 15*x - 106*x^2 + 294*x^3 - 516*x^4 + 588*x^5 - 420*x^6 + 1 56*x^7 + 6*x^8 - 34*x^9 + 14*x^10 - 2*x^11 + (-2 + 26*x - 138*x^2 + 402*x^ 3 - 708*x^4 + 756*x^5 - 420*x^6 - 12*x^7 + 198*x^8 - 142*x^9 + 46*x^10 - 6 *x^11)*Log[x])/(-1 + 9*x - 36*x^2 + 84*x^3 - 126*x^4 + 126*x^5 - 84*x^6 + 36*x^7 - 9*x^8 + x^9),x]
Leaf count is larger than twice the leaf count of optimal. \(176\) vs. \(2(23)=46\).
Time = 0.90 (sec) , antiderivative size = 176, normalized size of antiderivative = 7.65, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2007, 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.
\(\displaystyle \int \frac {-2 x^{11}+14 x^{10}-34 x^9+6 x^8+156 x^7-420 x^6+588 x^5-516 x^4+294 x^3-106 x^2+15 x+\left (-6 x^{11}+46 x^{10}-142 x^9+198 x^8-12 x^7-420 x^6+756 x^5-708 x^4+402 x^3-138 x^2+26 x-2\right ) \log (x)-3}{x^9-9 x^8+36 x^7-84 x^6+126 x^5-126 x^4+84 x^3-36 x^2+9 x-1} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-2 x^{11}+14 x^{10}-34 x^9+6 x^8+156 x^7-420 x^6+588 x^5-516 x^4+294 x^3-106 x^2+15 x+\left (-6 x^{11}+46 x^{10}-142 x^9+198 x^8-12 x^7-420 x^6+756 x^5-708 x^4+402 x^3-138 x^2+26 x-2\right ) \log (x)-3}{(x-1)^9}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 x^{11}}{(x-1)^9}+\frac {14 x^{10}}{(x-1)^9}-\frac {34 x^9}{(x-1)^9}+\frac {6 x^8}{(x-1)^9}+\frac {156 x^7}{(x-1)^9}-\frac {420 x^6}{(x-1)^9}+\frac {588 x^5}{(x-1)^9}-\frac {516 x^4}{(x-1)^9}+\frac {294 x^3}{(x-1)^9}-\frac {106 x^2}{(x-1)^9}-2 \left (3 x^2+4 x-1\right ) \log (x)+\frac {15 x}{(x-1)^9}-\frac {3}{(x-1)^9}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {39 x^8}{2 (1-x)^8}+\frac {15 x^7}{2 (1-x)^7}+\frac {105 x^7}{2 (1-x)^8}-\frac {7 x^6}{2 (1-x)^6}-\frac {21 x^6}{(1-x)^7}-\frac {147 x^6}{2 (1-x)^8}-2 x^3 \log (x)-4 x^2 \log (x)-\frac {156}{1-x}+\frac {336}{(1-x)^2}-\frac {448}{(1-x)^3}+\frac {525}{(1-x)^4}-\frac {588}{(1-x)^5}+\frac {476}{(1-x)^6}-\frac {217}{(1-x)^7}+\frac {83}{2 (1-x)^8}+2 x \log (x)\) |
Int[(-3 + 15*x - 106*x^2 + 294*x^3 - 516*x^4 + 588*x^5 - 420*x^6 + 156*x^7 + 6*x^8 - 34*x^9 + 14*x^10 - 2*x^11 + (-2 + 26*x - 138*x^2 + 402*x^3 - 70 8*x^4 + 756*x^5 - 420*x^6 - 12*x^7 + 198*x^8 - 142*x^9 + 46*x^10 - 6*x^11) *Log[x])/(-1 + 9*x - 36*x^2 + 84*x^3 - 126*x^4 + 126*x^5 - 84*x^6 + 36*x^7 - 9*x^8 + x^9),x]
83/(2*(1 - x)^8) - 217/(1 - x)^7 + 476/(1 - x)^6 - 588/(1 - x)^5 + 525/(1 - x)^4 - 448/(1 - x)^3 + 336/(1 - x)^2 - 156/(1 - x) - (147*x^6)/(2*(1 - x )^8) - (21*x^6)/(1 - x)^7 - (7*x^6)/(2*(1 - x)^6) + (105*x^7)/(2*(1 - x)^8 ) + (15*x^7)/(2*(1 - x)^7) - (39*x^8)/(2*(1 - x)^8) + 2*x*Log[x] - 4*x^2*L og[x] - 2*x^3*Log[x]
3.1.81.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35
method | result | size |
default | \(-2 x^{3} \ln \left (x \right )-4 x^{2} \ln \left (x \right )+2 x \ln \left (x \right )+\frac {1}{\left (-1+x \right )^{8}}+\frac {1}{\left (-1+x \right )^{7}}\) | \(31\) |
parts | \(-2 x^{3} \ln \left (x \right )-4 x^{2} \ln \left (x \right )+2 x \ln \left (x \right )+\frac {1}{\left (-1+x \right )^{8}}+\frac {1}{\left (-1+x \right )^{7}}\) | \(31\) |
risch | \(\left (-2 x^{3}-4 x^{2}+2 x \right ) \ln \left (x \right )+\frac {x}{x^{8}-8 x^{7}+28 x^{6}-56 x^{5}+70 x^{4}-56 x^{3}+28 x^{2}-8 x +1}\) | \(61\) |
norman | \(\frac {x +2 x \ln \left (x \right )-20 x^{2} \ln \left (x \right )+86 x^{3} \ln \left (x \right )-208 x^{4} \ln \left (x \right )+308 x^{5} \ln \left (x \right )-280 x^{6} \ln \left (x \right )+140 x^{7} \ln \left (x \right )-16 x^{8} \ln \left (x \right )-22 x^{9} \ln \left (x \right )+12 x^{10} \ln \left (x \right )-2 x^{11} \ln \left (x \right )}{\left (-1+x \right )^{8}}\) | \(84\) |
parallelrisch | \(\frac {280 x -6160 x^{9} \ln \left (x \right )-560 x^{11} \ln \left (x \right )-78400 x^{6} \ln \left (x \right )+39200 x^{7} \ln \left (x \right )+86240 x^{5} \ln \left (x \right )-4480 x^{8} \ln \left (x \right )-58240 x^{4} \ln \left (x \right )+560 x \ln \left (x \right )+24080 x^{3} \ln \left (x \right )-5600 x^{2} \ln \left (x \right )+3360 x^{10} \ln \left (x \right )}{280 x^{8}-2240 x^{7}+7840 x^{6}-15680 x^{5}+19600 x^{4}-15680 x^{3}+7840 x^{2}-2240 x +280}\) | \(122\) |
int(((-6*x^11+46*x^10-142*x^9+198*x^8-12*x^7-420*x^6+756*x^5-708*x^4+402*x ^3-138*x^2+26*x-2)*ln(x)-2*x^11+14*x^10-34*x^9+6*x^8+156*x^7-420*x^6+588*x ^5-516*x^4+294*x^3-106*x^2+15*x-3)/(x^9-9*x^8+36*x^7-84*x^6+126*x^5-126*x^ 4+84*x^3-36*x^2+9*x-1),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.43 \begin {dmath*} \int \frac {-3+15 x-106 x^2+294 x^3-516 x^4+588 x^5-420 x^6+156 x^7+6 x^8-34 x^9+14 x^{10}-2 x^{11}+\left (-2+26 x-138 x^2+402 x^3-708 x^4+756 x^5-420 x^6-12 x^7+198 x^8-142 x^9+46 x^{10}-6 x^{11}\right ) \log (x)}{-1+9 x-36 x^2+84 x^3-126 x^4+126 x^5-84 x^6+36 x^7-9 x^8+x^9} \, dx=-\frac {2 \, {\left (x^{11} - 6 \, x^{10} + 11 \, x^{9} + 8 \, x^{8} - 70 \, x^{7} + 140 \, x^{6} - 154 \, x^{5} + 104 \, x^{4} - 43 \, x^{3} + 10 \, x^{2} - x\right )} \log \left (x\right ) - x}{x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1} \end {dmath*}
integrate(((-6*x^11+46*x^10-142*x^9+198*x^8-12*x^7-420*x^6+756*x^5-708*x^4 +402*x^3-138*x^2+26*x-2)*log(x)-2*x^11+14*x^10-34*x^9+6*x^8+156*x^7-420*x^ 6+588*x^5-516*x^4+294*x^3-106*x^2+15*x-3)/(x^9-9*x^8+36*x^7-84*x^6+126*x^5 -126*x^4+84*x^3-36*x^2+9*x-1),x, algorithm=\
-(2*(x^11 - 6*x^10 + 11*x^9 + 8*x^8 - 70*x^7 + 140*x^6 - 154*x^5 + 104*x^4 - 43*x^3 + 10*x^2 - x)*log(x) - x)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^ 4 - 56*x^3 + 28*x^2 - 8*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \begin {dmath*} \int \frac {-3+15 x-106 x^2+294 x^3-516 x^4+588 x^5-420 x^6+156 x^7+6 x^8-34 x^9+14 x^{10}-2 x^{11}+\left (-2+26 x-138 x^2+402 x^3-708 x^4+756 x^5-420 x^6-12 x^7+198 x^8-142 x^9+46 x^{10}-6 x^{11}\right ) \log (x)}{-1+9 x-36 x^2+84 x^3-126 x^4+126 x^5-84 x^6+36 x^7-9 x^8+x^9} \, dx=\frac {x}{x^{8} - 8 x^{7} + 28 x^{6} - 56 x^{5} + 70 x^{4} - 56 x^{3} + 28 x^{2} - 8 x + 1} + \left (- 2 x^{3} - 4 x^{2} + 2 x\right ) \log {\left (x \right )} \end {dmath*}
integrate(((-6*x**11+46*x**10-142*x**9+198*x**8-12*x**7-420*x**6+756*x**5- 708*x**4+402*x**3-138*x**2+26*x-2)*ln(x)-2*x**11+14*x**10-34*x**9+6*x**8+1 56*x**7-420*x**6+588*x**5-516*x**4+294*x**3-106*x**2+15*x-3)/(x**9-9*x**8+ 36*x**7-84*x**6+126*x**5-126*x**4+84*x**3-36*x**2+9*x-1),x)
x/(x**8 - 8*x**7 + 28*x**6 - 56*x**5 + 70*x**4 - 56*x**3 + 28*x**2 - 8*x + 1) + (-2*x**3 - 4*x**2 + 2*x)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 1945 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 1945, normalized size of antiderivative = 84.57 \begin {dmath*} \int \frac {-3+15 x-106 x^2+294 x^3-516 x^4+588 x^5-420 x^6+156 x^7+6 x^8-34 x^9+14 x^{10}-2 x^{11}+\left (-2+26 x-138 x^2+402 x^3-708 x^4+756 x^5-420 x^6-12 x^7+198 x^8-142 x^9+46 x^{10}-6 x^{11}\right ) \log (x)}{-1+9 x-36 x^2+84 x^3-126 x^4+126 x^5-84 x^6+36 x^7-9 x^8+x^9} \, dx=\text {Too large to display} \end {dmath*}
integrate(((-6*x^11+46*x^10-142*x^9+198*x^8-12*x^7-420*x^6+756*x^5-708*x^4 +402*x^3-138*x^2+26*x-2)*log(x)-2*x^11+14*x^10-34*x^9+6*x^8+156*x^7-420*x^ 6+588*x^5-516*x^4+294*x^3-106*x^2+15*x-3)/(x^9-9*x^8+36*x^7-84*x^6+126*x^5 -126*x^4+84*x^3-36*x^2+9*x-1),x, algorithm=\
-2/3*x^3 - 2*x^2 + 2*x + 3/2*(8*x^7 - 28*x^6 + 56*x^5 - 70*x^4 + 56*x^3 - 28*x^2 + 8*x - 1)*log(x)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 15/2*(28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8 *x + 1)*log(x)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) - 9/2*(56*x^5 - 70*x^4 + 56*x^3 - 28*x^2 + 8*x - 1)*log(x)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 177/70*( 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1)*log(x)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) - 201/140*(56*x^3 - 28*x^2 + 8*x - 1 )*log(x)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 23/28*(28*x^2 - 8*x + 1)*log(x)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70* x^4 - 56*x^3 + 28*x^2 - 8*x + 1) - 13/28*(8*x - 1)*log(x)/(x^8 - 8*x^7 + 2 8*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 2/105*(35*x^11 - 17 5*x^10 + 35*x^9 + 1820*x^8 - 7840*x^7 + 15680*x^6 - 15680*x^5 + 4900*x^4 + 5488*x^3 - 6664*x^2 - 105*(x^11 - 6*x^10 + 11*x^9)*log(x) + 2864*x - 463) /(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 1/ 84*(55440*x^7 - 349272*x^6 + 957264*x^5 - 1473780*x^4 + 1373064*x^3 - 7727 72*x^2 + 242968*x - 32891)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^ 3 + 28*x^2 - 8*x + 1) + 17/140*(10080*x^7 - 58800*x^6 + 152880*x^5 - 22638 0*x^4 + 204624*x^3 - 112392*x^2 + 34632*x - 4609)/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) - 1/140*(6720*x^7 - 35280*...
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \begin {dmath*} \int \frac {-3+15 x-106 x^2+294 x^3-516 x^4+588 x^5-420 x^6+156 x^7+6 x^8-34 x^9+14 x^{10}-2 x^{11}+\left (-2+26 x-138 x^2+402 x^3-708 x^4+756 x^5-420 x^6-12 x^7+198 x^8-142 x^9+46 x^{10}-6 x^{11}\right ) \log (x)}{-1+9 x-36 x^2+84 x^3-126 x^4+126 x^5-84 x^6+36 x^7-9 x^8+x^9} \, dx=-2 \, {\left (x^{3} + 2 \, x^{2} - x\right )} \log \left (x\right ) + \frac {x}{x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1} \end {dmath*}
integrate(((-6*x^11+46*x^10-142*x^9+198*x^8-12*x^7-420*x^6+756*x^5-708*x^4 +402*x^3-138*x^2+26*x-2)*log(x)-2*x^11+14*x^10-34*x^9+6*x^8+156*x^7-420*x^ 6+588*x^5-516*x^4+294*x^3-106*x^2+15*x-3)/(x^9-9*x^8+36*x^7-84*x^6+126*x^5 -126*x^4+84*x^3-36*x^2+9*x-1),x, algorithm=\
-2*(x^3 + 2*x^2 - x)*log(x) + x/(x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1)
Time = 14.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \begin {dmath*} \int \frac {-3+15 x-106 x^2+294 x^3-516 x^4+588 x^5-420 x^6+156 x^7+6 x^8-34 x^9+14 x^{10}-2 x^{11}+\left (-2+26 x-138 x^2+402 x^3-708 x^4+756 x^5-420 x^6-12 x^7+198 x^8-142 x^9+46 x^{10}-6 x^{11}\right ) \log (x)}{-1+9 x-36 x^2+84 x^3-126 x^4+126 x^5-84 x^6+36 x^7-9 x^8+x^9} \, dx=\frac {x}{x^8-8\,x^7+28\,x^6-56\,x^5+70\,x^4-56\,x^3+28\,x^2-8\,x+1}-\ln \left (x\right )\,\left (2\,x^3+4\,x^2-2\,x\right ) \end {dmath*}
int(-(log(x)*(138*x^2 - 26*x - 402*x^3 + 708*x^4 - 756*x^5 + 420*x^6 + 12* x^7 - 198*x^8 + 142*x^9 - 46*x^10 + 6*x^11 + 2) - 15*x + 106*x^2 - 294*x^3 + 516*x^4 - 588*x^5 + 420*x^6 - 156*x^7 - 6*x^8 + 34*x^9 - 14*x^10 + 2*x^ 11 + 3)/(9*x - 36*x^2 + 84*x^3 - 126*x^4 + 126*x^5 - 84*x^6 + 36*x^7 - 9*x ^8 + x^9 - 1),x)