Integrand size = 175, antiderivative size = 25 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=x \left (5+\log \left (\frac {(x+\log (x))^2}{\left (-\frac {1}{x^2}+(3+x)^2\right )^2}\right )\right ) \]
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=5 x+x \log \left (\frac {x^4 (x+\log (x))^2}{\left (-1+9 x^2+6 x^3+x^4\right )^2}\right ) \]
Integrate[(-2 - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + (-9 + 45*x^2 + 1 8*x^3 + x^4)*Log[x] + (-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^ 4)*Log[x])*Log[(x^6 + 2*x^5*Log[x] + x^4*Log[x]^2)/(1 - 18*x^2 - 12*x^3 + 79*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8)])/(-x + 9*x^3 + 6*x^4 + x^5 + (- 1 + 9*x^2 + 6*x^3 + x^4)*Log[x]),x]
Leaf count is larger than twice the leaf count of optimal. \(318\) vs. \(2(25)=50\).
Time = 10.26 (sec) , antiderivative size = 318, normalized size of antiderivative = 12.72, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7292, 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.
\(\displaystyle \int \frac {3 x^5+32 x^4+75 x^3+18 x^2+\left (x^4+18 x^3+45 x^2-9\right ) \log (x)+\left (x^5+6 x^4+9 x^3+\left (x^4+6 x^3+9 x^2-1\right ) \log (x)-x\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{x^8+12 x^7+54 x^6+108 x^5+79 x^4-12 x^3-18 x^2+1}\right )-11 x-2}{x^5+6 x^4+9 x^3+\left (x^4+6 x^3+9 x^2-1\right ) \log (x)-x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x^5-32 x^4-75 x^3-18 x^2-\left (x^4+18 x^3+45 x^2-9\right ) \log (x)-\left (x^5+6 x^4+9 x^3+\left (x^4+6 x^3+9 x^2-1\right ) \log (x)-x\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{x^8+12 x^7+54 x^6+108 x^5+79 x^4-12 x^3-18 x^2+1}\right )+11 x+2}{\left (-x^4-6 x^3-9 x^2+1\right ) (x+\log (x))}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {-3 x^5-32 x^4-75 x^3-18 x^2-\left (x^4+18 x^3+45 x^2-9\right ) \log (x)-\left (x^5+6 x^4+9 x^3+\left (x^4+6 x^3+9 x^2-1\right ) \log (x)-x\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{x^8+12 x^7+54 x^6+108 x^5+79 x^4-12 x^3-18 x^2+1}\right )+11 x+2}{2 \left (x^2+3 x+1\right ) (x+\log (x))}-\frac {-3 x^5-32 x^4-75 x^3-18 x^2-\left (x^4+18 x^3+45 x^2-9\right ) \log (x)-\left (x^5+6 x^4+9 x^3+\left (x^4+6 x^3+9 x^2-1\right ) \log (x)-x\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{x^8+12 x^7+54 x^6+108 x^5+79 x^4-12 x^3-18 x^2+1}\right )+11 x+2}{2 \left (x^2+3 x-1\right ) (x+\log (x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x \log \left (\frac {x^4 (x+\log (x))^2}{\left (-x^4-6 x^3-9 x^2+1\right )^2}\right )+5 x+\left (3-\sqrt {5}\right ) \log \left (2 x-\sqrt {5}+3\right )+\frac {1}{12} \left (9-5 \sqrt {5}\right ) \log \left (2 x-\sqrt {5}+3\right )-\frac {1}{12} \left (45-17 \sqrt {5}\right ) \log \left (2 x-\sqrt {5}+3\right )-\frac {1}{12} \left (45+17 \sqrt {5}\right ) \log \left (2 x+\sqrt {5}+3\right )+\frac {1}{12} \left (9+5 \sqrt {5}\right ) \log \left (2 x+\sqrt {5}+3\right )+\left (3+\sqrt {5}\right ) \log \left (2 x+\sqrt {5}+3\right )+\frac {1}{12} \left (9-\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+3\right )+\left (3-\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+3\right )-\frac {1}{12} \left (45-13 \sqrt {13}\right ) \log \left (2 x-\sqrt {13}+3\right )-\frac {1}{12} \left (45+13 \sqrt {13}\right ) \log \left (2 x+\sqrt {13}+3\right )+\frac {1}{12} \left (9+\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+3\right )+\left (3+\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+3\right )\) |
Int[(-2 - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + (-9 + 45*x^2 + 18*x^3 + x^4)*Log[x] + (-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9*x^2 + 6*x^3 + x^4)*Log [x])*Log[(x^6 + 2*x^5*Log[x] + x^4*Log[x]^2)/(1 - 18*x^2 - 12*x^3 + 79*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8)])/(-x + 9*x^3 + 6*x^4 + x^5 + (-1 + 9* x^2 + 6*x^3 + x^4)*Log[x]),x]
5*x - ((45 - 17*Sqrt[5])*Log[3 - Sqrt[5] + 2*x])/12 + ((9 - 5*Sqrt[5])*Log [3 - Sqrt[5] + 2*x])/12 + (3 - Sqrt[5])*Log[3 - Sqrt[5] + 2*x] + (3 + Sqrt [5])*Log[3 + Sqrt[5] + 2*x] + ((9 + 5*Sqrt[5])*Log[3 + Sqrt[5] + 2*x])/12 - ((45 + 17*Sqrt[5])*Log[3 + Sqrt[5] + 2*x])/12 - ((45 - 13*Sqrt[13])*Log[ 3 - Sqrt[13] + 2*x])/12 + (3 - Sqrt[13])*Log[3 - Sqrt[13] + 2*x] + ((9 - S qrt[13])*Log[3 - Sqrt[13] + 2*x])/12 + (3 + Sqrt[13])*Log[3 + Sqrt[13] + 2 *x] + ((9 + Sqrt[13])*Log[3 + Sqrt[13] + 2*x])/12 - ((45 + 13*Sqrt[13])*Lo g[3 + Sqrt[13] + 2*x])/12 + x*Log[(x^4*(x + Log[x])^2)/(1 - 9*x^2 - 6*x^3 - x^4)^2]
3.20.11.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(25)=50\).
Time = 9.67 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64
method | result | size |
parallelrisch | \(-60+\ln \left (\frac {x^{4} \ln \left (x \right )^{2}+2 x^{5} \ln \left (x \right )+x^{6}}{x^{8}+12 x^{7}+54 x^{6}+108 x^{5}+79 x^{4}-12 x^{3}-18 x^{2}+1}\right ) x +5 x\) | \(66\) |
risch | \(\text {Expression too large to display}\) | \(844\) |
int((((x^4+6*x^3+9*x^2-1)*ln(x)+x^5+6*x^4+9*x^3-x)*ln((x^4*ln(x)^2+2*x^5*l n(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x^4+18*x^3+ 45*x^2-9)*ln(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^3+9*x^2-1)*ln (x)+x^5+6*x^4+9*x^3-x),x,method=_RETURNVERBOSE)
-60+ln((x^4*ln(x)^2+2*x^5*ln(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5+79*x^4-12* x^3-18*x^2+1))*x+5*x
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=x \log \left (\frac {x^{6} + 2 \, x^{5} \log \left (x\right ) + x^{4} \log \left (x\right )^{2}}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 79 \, x^{4} - 12 \, x^{3} - 18 \, x^{2} + 1}\right ) + 5 \, x \]
integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^ 2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x ^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^3+ 9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm=\
x*log((x^6 + 2*x^5*log(x) + x^4*log(x)^2)/(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 79*x^4 - 12*x^3 - 18*x^2 + 1)) + 5*x
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (22) = 44\).
Time = 0.66 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.08 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=5 x + \left (x + \frac {1}{5}\right ) \log {\left (\frac {x^{6} + 2 x^{5} \log {\left (x \right )} + x^{4} \log {\left (x \right )}^{2}}{x^{8} + 12 x^{7} + 54 x^{6} + 108 x^{5} + 79 x^{4} - 12 x^{3} - 18 x^{2} + 1} \right )} - \frac {4 \log {\left (x \right )}}{5} - \frac {2 \log {\left (x + \log {\left (x \right )} \right )}}{5} + \frac {2 \log {\left (x^{4} + 6 x^{3} + 9 x^{2} - 1 \right )}}{5} \]
integrate((((x**4+6*x**3+9*x**2-1)*ln(x)+x**5+6*x**4+9*x**3-x)*ln((x**4*ln (x)**2+2*x**5*ln(x)+x**6)/(x**8+12*x**7+54*x**6+108*x**5+79*x**4-12*x**3-1 8*x**2+1))+(x**4+18*x**3+45*x**2-9)*ln(x)+3*x**5+32*x**4+75*x**3+18*x**2-1 1*x-2)/((x**4+6*x**3+9*x**2-1)*ln(x)+x**5+6*x**4+9*x**3-x),x)
5*x + (x + 1/5)*log((x**6 + 2*x**5*log(x) + x**4*log(x)**2)/(x**8 + 12*x** 7 + 54*x**6 + 108*x**5 + 79*x**4 - 12*x**3 - 18*x**2 + 1)) - 4*log(x)/5 - 2*log(x + log(x))/5 + 2*log(x**4 + 6*x**3 + 9*x**2 - 1)/5
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=-2 \, x \log \left (x^{2} + 3 \, x + 1\right ) - 2 \, x \log \left (x^{2} + 3 \, x - 1\right ) + 2 \, x \log \left (x + \log \left (x\right )\right ) + 4 \, x \log \left (x\right ) + 5 \, x \]
integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^ 2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x ^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^3+ 9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.93 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=-x \log \left (x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 79 \, x^{4} - 12 \, x^{3} - 18 \, x^{2} + 1\right ) + x \log \left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 4 \, x \log \left (x\right ) + 5 \, x \]
integrate((((x^4+6*x^3+9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x)*log((x^4*log(x)^ 2+2*x^5*log(x)+x^6)/(x^8+12*x^7+54*x^6+108*x^5+79*x^4-12*x^3-18*x^2+1))+(x ^4+18*x^3+45*x^2-9)*log(x)+3*x^5+32*x^4+75*x^3+18*x^2-11*x-2)/((x^4+6*x^3+ 9*x^2-1)*log(x)+x^5+6*x^4+9*x^3-x),x, algorithm=\
-x*log(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 79*x^4 - 12*x^3 - 18*x^2 + 1) + x *log(x^2 + 2*x*log(x) + log(x)^2) + 4*x*log(x) + 5*x
Time = 12.95 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.48 \[ \int \frac {-2-11 x+18 x^2+75 x^3+32 x^4+3 x^5+\left (-9+45 x^2+18 x^3+x^4\right ) \log (x)+\left (-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {x^6+2 x^5 \log (x)+x^4 \log ^2(x)}{1-18 x^2-12 x^3+79 x^4+108 x^5+54 x^6+12 x^7+x^8}\right )}{-x+9 x^3+6 x^4+x^5+\left (-1+9 x^2+6 x^3+x^4\right ) \log (x)} \, dx=x\,\left (\ln \left (\frac {x^6+2\,x^5\,\ln \left (x\right )+x^4\,{\ln \left (x\right )}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+79\,x^4-12\,x^3-18\,x^2+1}\right )+5\right ) \]
int((log((2*x^5*log(x) + x^4*log(x)^2 + x^6)/(79*x^4 - 12*x^3 - 18*x^2 + 1 08*x^5 + 54*x^6 + 12*x^7 + x^8 + 1))*(9*x^3 - x + 6*x^4 + x^5 + log(x)*(9* x^2 + 6*x^3 + x^4 - 1)) - 11*x + 18*x^2 + 75*x^3 + 32*x^4 + 3*x^5 + log(x) *(45*x^2 + 18*x^3 + x^4 - 9) - 2)/(9*x^3 - x + 6*x^4 + x^5 + log(x)*(9*x^2 + 6*x^3 + x^4 - 1)),x)