Integrand size = 180, antiderivative size = 28 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=\frac {-4+3 x+\frac {5 \log (x)}{x+4 x^4}}{-3-x+\log (x)} \]
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {x \left (-4+3 x-16 x^3+12 x^4\right )+5 \log (x)}{x \left (1+4 x^3\right ) (3+x-\log (x))} \]
Integrate[(-15 - x - 16*x^2 - 60*x^3 + 12*x^4 - 128*x^5 + 64*x^7 - 256*x^8 + (15 + 10*x + 3*x^2 + 240*x^3 + 100*x^4 + 24*x^5 + 48*x^8)*Log[x] + (-5 - 80*x^3)*Log[x]^2)/(9*x^2 + 6*x^3 + x^4 + 72*x^5 + 48*x^6 + 8*x^7 + 144*x ^8 + 96*x^9 + 16*x^10 + (-6*x^2 - 2*x^3 - 48*x^5 - 16*x^6 - 96*x^8 - 32*x^ 9)*Log[x] + (x^2 + 8*x^5 + 16*x^8)*Log[x]^2),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.
\(\displaystyle \int \frac {-256 x^8+64 x^7-128 x^5+12 x^4-60 x^3+\left (-80 x^3-5\right ) \log ^2(x)-16 x^2+\left (48 x^8+24 x^5+100 x^4+240 x^3+3 x^2+10 x+15\right ) \log (x)-x-15}{16 x^{10}+96 x^9+144 x^8+8 x^7+48 x^6+72 x^5+x^4+6 x^3+9 x^2+\left (16 x^8+8 x^5+x^2\right ) \log ^2(x)+\left (-32 x^9-96 x^8-16 x^6-48 x^5-2 x^3-6 x^2\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-256 x^8+64 x^7-128 x^5+12 x^4-60 x^3-\left (80 x^3+5\right ) \log ^2(x)-16 x^2+\left (48 x^8+24 x^5+100 x^4+240 x^3+3 x^2+10 x+15\right ) \log (x)-x-15}{x^2 \left (4 x^3+1\right )^2 (x-\log (x)+3)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {5 \left (16 x^3+1\right )}{x^2 \left (4 x^3+1\right )^2}-\frac {3 \left (16 x^8+8 x^5-20 x^4-80 x^3+x^2-5\right )}{x^2 \left (4 x^3+1\right )^2 (x-\log (x)+3)}+\frac {12 x^6-28 x^5+16 x^4+3 x^3-2 x^2+14 x-15}{x^2 \left (4 x^3+1\right ) (x-\log (x)+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 180 \int \frac {x}{\left (4 x^3+1\right )^2 (x-\log (x)+3)}dx-15 \int \frac {1}{x^2 (x-\log (x)+3)^2}dx+15 \int \frac {1}{x^2 (x-\log (x)+3)}dx+60 \int \frac {x^2}{\left (4 x^3+1\right )^2 (x-\log (x)+3)}dx-7 \int \frac {1}{(x-\log (x)+3)^2}dx+14 \int \frac {1}{x (x-\log (x)+3)^2}dx+3 \int \frac {x}{(x-\log (x)+3)^2}dx-\frac {5}{3} \int \frac {1}{\left (-(-2)^{2/3} x-1\right ) (x-\log (x)+3)^2}dx+10 \sqrt [3]{-2} \int \frac {1}{\left ((-2)^{2/3} x+1\right ) (x-\log (x)+3)^2}dx-\frac {5}{3} \int \frac {1}{\left (-2^{2/3} x-1\right ) (x-\log (x)+3)^2}dx-\frac {10}{3} 2^{2/3} \int \frac {1}{\left (2^{2/3} x+1\right ) (x-\log (x)+3)^2}dx-10 \sqrt [3]{2} \int \frac {1}{\left (2^{2/3} x+1\right ) (x-\log (x)+3)^2}dx-\frac {10}{3} 2^{2/3} \int \frac {1}{\left (2^{2/3} x-\sqrt [3]{-1}\right ) (x-\log (x)+3)^2}dx-\frac {10}{3} 2^{2/3} \int \frac {1}{\left (2^{2/3} x+(-1)^{2/3}\right ) (x-\log (x)+3)^2}dx-10 (-1)^{2/3} \sqrt [3]{2} \int \frac {1}{\left (1-\sqrt [3]{-1} 2^{2/3} x\right ) (x-\log (x)+3)^2}dx-\frac {5}{3} \int \frac {1}{\left (\sqrt [3]{-1} 2^{2/3} x-1\right ) (x-\log (x)+3)^2}dx-3 \int \frac {1}{x-\log (x)+3}dx-10 \sqrt [3]{-2} \int \frac {1}{\left ((-2)^{2/3} x+1\right ) (x-\log (x)+3)}dx+10 \sqrt [3]{2} \int \frac {1}{\left (2^{2/3} x+1\right ) (x-\log (x)+3)}dx+10 (-1)^{2/3} \sqrt [3]{2} \int \frac {1}{\left (1-\sqrt [3]{-1} 2^{2/3} x\right ) (x-\log (x)+3)}dx+\frac {5}{x \left (4 x^3+1\right )}\) |
Int[(-15 - x - 16*x^2 - 60*x^3 + 12*x^4 - 128*x^5 + 64*x^7 - 256*x^8 + (15 + 10*x + 3*x^2 + 240*x^3 + 100*x^4 + 24*x^5 + 48*x^8)*Log[x] + (-5 - 80*x ^3)*Log[x]^2)/(9*x^2 + 6*x^3 + x^4 + 72*x^5 + 48*x^6 + 8*x^7 + 144*x^8 + 9 6*x^9 + 16*x^10 + (-6*x^2 - 2*x^3 - 48*x^5 - 16*x^6 - 96*x^8 - 32*x^9)*Log [x] + (x^2 + 8*x^5 + 16*x^8)*Log[x]^2),x]
3.15.3.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.45 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96
method | result | size |
parallelrisch | \(\frac {16 x -48 x^{5}-20 \ln \left (x \right )+64 x^{4}-12 x^{2}}{4 x \left (-4 x^{3} \ln \left (x \right )+x +12 x^{3}+4 x^{4}-\ln \left (x \right )+3\right )}\) | \(55\) |
risch | \(\frac {5}{\left (4 x^{3}+1\right ) x}-\frac {12 x^{5}-16 x^{4}+3 x^{2}+x +15}{\left (4 x^{3}+1\right ) x \left (-\ln \left (x \right )+3+x \right )}\) | \(57\) |
default | \(\frac {22 \ln \left (x \right )+88 x^{3} \ln \left (x \right )-\frac {5 \ln \left (x \right )^{2}}{x}-12 x^{3} \ln \left (x \right )^{2}+\frac {15 \ln \left (x \right )}{x}-3 \ln \left (x \right )^{2}-156 x^{3}-39}{\left (\ln \left (x \right )-3\right ) \left (-4 x^{3} \ln \left (x \right )+x +12 x^{3}+4 x^{4}-\ln \left (x \right )+3\right )}\) | \(83\) |
int(((-80*x^3-5)*ln(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+15)*ln( x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^5+x^2)*l n(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*ln(x)+16*x^10+96*x^9+144 *x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {12 \, x^{5} - 16 \, x^{4} + 3 \, x^{2} - 4 \, x + 5 \, \log \left (x\right )}{4 \, x^{5} + 12 \, x^{4} + x^{2} - {\left (4 \, x^{4} + x\right )} \log \left (x\right ) + 3 \, x} \]
integrate(((-80*x^3-5)*log(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+ 15)*log(x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^ 5+x^2)*log(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*log(x)+16*x^10+ 96*x^9+144*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x, algorithm=\
-(12*x^5 - 16*x^4 + 3*x^2 - 4*x + 5*log(x))/(4*x^5 + 12*x^4 + x^2 - (4*x^4 + x)*log(x) + 3*x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=\frac {12 x^{5} - 16 x^{4} + 3 x^{2} + x + 15}{- 4 x^{5} - 12 x^{4} - x^{2} - 3 x + \left (4 x^{4} + x\right ) \log {\left (x \right )}} + \frac {5}{4 x^{4} + x} \]
integrate(((-80*x**3-5)*ln(x)**2+(48*x**8+24*x**5+100*x**4+240*x**3+3*x**2 +10*x+15)*ln(x)-256*x**8+64*x**7-128*x**5+12*x**4-60*x**3-16*x**2-x-15)/(( 16*x**8+8*x**5+x**2)*ln(x)**2+(-32*x**9-96*x**8-16*x**6-48*x**5-2*x**3-6*x **2)*ln(x)+16*x**10+96*x**9+144*x**8+8*x**7+48*x**6+72*x**5+x**4+6*x**3+9* x**2),x)
(12*x**5 - 16*x**4 + 3*x**2 + x + 15)/(-4*x**5 - 12*x**4 - x**2 - 3*x + (4 *x**4 + x)*log(x)) + 5/(4*x**4 + x)
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {12 \, x^{5} - 16 \, x^{4} + 3 \, x^{2} - 4 \, x + 5 \, \log \left (x\right )}{4 \, x^{5} + 12 \, x^{4} + x^{2} - {\left (4 \, x^{4} + x\right )} \log \left (x\right ) + 3 \, x} \]
integrate(((-80*x^3-5)*log(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+ 15)*log(x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^ 5+x^2)*log(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*log(x)+16*x^10+ 96*x^9+144*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x, algorithm=\
-(12*x^5 - 16*x^4 + 3*x^2 - 4*x + 5*log(x))/(4*x^5 + 12*x^4 + x^2 - (4*x^4 + x)*log(x) + 3*x)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {20 \, x^{2}}{4 \, x^{3} + 1} - \frac {12 \, x^{5} - 16 \, x^{4} + 3 \, x^{2} + x + 15}{4 \, x^{5} - 4 \, x^{4} \log \left (x\right ) + 12 \, x^{4} + x^{2} - x \log \left (x\right ) + 3 \, x} + \frac {5}{x} \]
integrate(((-80*x^3-5)*log(x)^2+(48*x^8+24*x^5+100*x^4+240*x^3+3*x^2+10*x+ 15)*log(x)-256*x^8+64*x^7-128*x^5+12*x^4-60*x^3-16*x^2-x-15)/((16*x^8+8*x^ 5+x^2)*log(x)^2+(-32*x^9-96*x^8-16*x^6-48*x^5-2*x^3-6*x^2)*log(x)+16*x^10+ 96*x^9+144*x^8+8*x^7+48*x^6+72*x^5+x^4+6*x^3+9*x^2),x, algorithm=\
-20*x^2/(4*x^3 + 1) - (12*x^5 - 16*x^4 + 3*x^2 + x + 15)/(4*x^5 - 4*x^4*lo g(x) + 12*x^4 + x^2 - x*log(x) + 3*x) + 5/x
Time = 12.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-15-x-16 x^2-60 x^3+12 x^4-128 x^5+64 x^7-256 x^8+\left (15+10 x+3 x^2+240 x^3+100 x^4+24 x^5+48 x^8\right ) \log (x)+\left (-5-80 x^3\right ) \log ^2(x)}{9 x^2+6 x^3+x^4+72 x^5+48 x^6+8 x^7+144 x^8+96 x^9+16 x^{10}+\left (-6 x^2-2 x^3-48 x^5-16 x^6-96 x^8-32 x^9\right ) \log (x)+\left (x^2+8 x^5+16 x^8\right ) \log ^2(x)} \, dx=-\frac {5\,\ln \left (x\right )+x\,\left (3\,\ln \left (x\right )-13\right )+x^4\,\left (12\,\ln \left (x\right )-52\right )}{x\,\left (4\,x^3+1\right )\,\left (x-\ln \left (x\right )+3\right )} \]
int(-(x + log(x)^2*(80*x^3 + 5) - log(x)*(10*x + 3*x^2 + 240*x^3 + 100*x^4 + 24*x^5 + 48*x^8 + 15) + 16*x^2 + 60*x^3 - 12*x^4 + 128*x^5 - 64*x^7 + 2 56*x^8 + 15)/(log(x)^2*(x^2 + 8*x^5 + 16*x^8) + 9*x^2 + 6*x^3 + x^4 + 72*x ^5 + 48*x^6 + 8*x^7 + 144*x^8 + 96*x^9 + 16*x^10 - log(x)*(6*x^2 + 2*x^3 + 48*x^5 + 16*x^6 + 96*x^8 + 32*x^9)),x)