Integrand size = 154, antiderivative size = 34 \[ \int \frac {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx=\log \left (-5+\frac {\frac {3}{5+x \left (4-x^2\right )}+\log (x)}{4 (1+x) \log (x)}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(34)=68\).
Time = 21.47 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.06 \[ \int \frac {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx=-\log (1+x)+\log (19+20 x)-\log \left ((19+20 x) \left (5+4 x-x^3\right )\right )-\log (\log (x))+\log \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right ) \]
Integrate[(15 + 27*x + 12*x^2 - 3*x^3 - 3*x^4 + (27*x + 24*x^2 - 9*x^3 - 1 2*x^4)*Log[x] + (25*x + 40*x^2 + 16*x^3 - 10*x^4 - 8*x^5 + x^7)*Log[x]^2)/ ((-15*x - 27*x^2 - 12*x^3 + 3*x^4 + 3*x^5)*Log[x] + (475*x + 1735*x^2 + 23 64*x^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)*Log[x]^ 2),x]
-Log[1 + x] + Log[19 + 20*x] - Log[(19 + 20*x)*(5 + 4*x - x^3)] - Log[Log[ x]] + Log[3 - 95*Log[x] - 176*x*Log[x] - 80*x^2*Log[x] + 19*x^3*Log[x] + 2 0*x^4*Log[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.
| \(\displaystyle \int \frac {-3 x^4-3 x^3+12 x^2+\left (-12 x^4-9 x^3+24 x^2+27 x\right ) \log (x)+\left (x^7-8 x^5-10 x^4+16 x^3+40 x^2+25 x\right ) \log ^2(x)+27 x+15}{\left (3 x^5+3 x^4-12 x^3-27 x^2-15 x\right ) \log (x)+\left (20 x^9+39 x^8-141 x^7-512 x^6-222 x^5+1234 x^4+2364 x^3+1735 x^2+475 x\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 x^4+3 x^3-12 x^2-\left (-12 x^4-9 x^3+24 x^2+27 x\right ) \log (x)-\left (x^7-8 x^5-10 x^4+16 x^3+40 x^2+25 x\right ) \log ^2(x)-27 x-15}{x \left (-x^4-x^3+4 x^2+9 x+5\right ) \log (x) \left (20 x^4 \log (x)+19 x^3 \log (x)-80 x^2 \log (x)-176 x \log (x)-95 \log (x)+3\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {3 x^4+3 x^3-12 x^2-\left (-12 x^4-9 x^3+24 x^2+27 x\right ) \log (x)-\left (x^7-8 x^5-10 x^4+16 x^3+40 x^2+25 x\right ) \log ^2(x)-27 x-15}{2 x (x+1) \log (x) \left (20 x^4 \log (x)+19 x^3 \log (x)-80 x^2 \log (x)-176 x \log (x)-95 \log (x)+3\right )}+\frac {\left (-x^2+x+3\right ) \left (3 x^4+3 x^3-12 x^2-\left (-12 x^4-9 x^3+24 x^2+27 x\right ) \log (x)-\left (x^7-8 x^5-10 x^4+16 x^3+40 x^2+25 x\right ) \log ^2(x)-27 x-15\right )}{2 x \left (x^3-4 x-5\right ) \log (x) \left (20 x^4 \log (x)+19 x^3 \log (x)-80 x^2 \log (x)-176 x \log (x)-95 \log (x)+3\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-x \left (x^3-4 x-5\right )^2 \log ^2(x)+3 x \left (4 x^3+3 x^2-8 x-9\right ) \log (x)+3 \left (x^4+x^3-4 x^2-9 x-5\right )}{x (x+1) \left (-x^3+4 x+5\right ) \log (x) \left (\left (20 x^4+19 x^3-80 x^2-176 x-95\right ) \log (x)+3\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{20 x^2+39 x+19}-\frac {3 \left (80 x^3+57 x^2-160 x-176\right )}{(20 x+19) \left (x^3-4 x-5\right ) \left (20 x^4 \log (x)+19 x^3 \log (x)-80 x^2 \log (x)-176 x \log (x)-95 \log (x)+3\right )}+\frac {20 x^4+19 x^3-80 x^2-176 x-95}{x \left (20 x^4 \log (x)+19 x^3 \log (x)-80 x^2 \log (x)-176 x \log (x)-95 \log (x)+3\right )}-\frac {1}{x \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -176 \int \frac {1}{20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3}dx-95 \int \frac {1}{x \left (20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3\right )}dx-80 \int \frac {x}{20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3}dx+19 \int \frac {x^2}{20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3}dx+20 \int \frac {x^3}{20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3}dx-60 \int \frac {1}{(20 x+19) \left (20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3\right )}dx+12 \int \frac {1}{\left (x^3-4 x-5\right ) \left (20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3\right )}dx-9 \int \frac {x^2}{\left (x^3-4 x-5\right ) \left (20 \log (x) x^4+19 \log (x) x^3-80 \log (x) x^2-176 \log (x) x-95 \log (x)+3\right )}dx-\log (x+1)+\log (20 x+19)-\log (\log (x))\) |
Int[(15 + 27*x + 12*x^2 - 3*x^3 - 3*x^4 + (27*x + 24*x^2 - 9*x^3 - 12*x^4) *Log[x] + (25*x + 40*x^2 + 16*x^3 - 10*x^4 - 8*x^5 + x^7)*Log[x]^2)/((-15* x - 27*x^2 - 12*x^3 + 3*x^4 + 3*x^5)*Log[x] + (475*x + 1735*x^2 + 2364*x^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)*Log[x]^2),x]
3.15.11.3.1 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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.41 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38
| method | result | size |
| risch | \(-\ln \left (1+x \right )+\ln \left (20 x +19\right )+\ln \left (\ln \left (x \right )+\frac {3}{20 x^{4}+19 x^{3}-80 x^{2}-176 x -95}\right )-\ln \left (\ln \left (x \right )\right )\) | \(47\) |
| parallelrisch | \(-\ln \left (\ln \left (x \right )\right )-\ln \left (1+x \right )+\ln \left (x^{4} \ln \left (x \right )+\frac {19 x^{3} \ln \left (x \right )}{20}-4 x^{2} \ln \left (x \right )-\frac {44 x \ln \left (x \right )}{5}-\frac {19 \ln \left (x \right )}{4}+\frac {3}{20}\right )-\ln \left (x^{3}-4 x -5\right )\) | \(56\) |
| default | \(-\ln \left (\ln \left (x \right )\right )-\ln \left (1+x \right )-\ln \left (x^{3}-4 x -5\right )+\ln \left (20 x^{4} \ln \left (x \right )+19 x^{3} \ln \left (x \right )-80 x^{2} \ln \left (x \right )-176 x \ln \left (x \right )-95 \ln \left (x \right )+3\right )\) | \(57\) |
int(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*ln(x)^2+(-12*x^4-9*x^3+24*x^2+2 7*x)*ln(x)-3*x^4-3*x^3+12*x^2+27*x+15)/((20*x^9+39*x^8-141*x^7-512*x^6-222 *x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*ln(x)^2+(3*x^5+3*x^4-12*x^3-27*x^2- 15*x)*ln(x)),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97 \[ \int \frac {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx=\log \left (20 \, x + 19\right ) - \log \left (x + 1\right ) + \log \left (\frac {{\left (20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95\right )} \log \left (x\right ) + 3}{20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*log(x)^2+(-12*x^4-9*x^3+2 4*x^2+27*x)*log(x)-3*x^4-3*x^3+12*x^2+27*x+15)/((20*x^9+39*x^8-141*x^7-512 *x^6-222*x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*log(x)^2+(3*x^5+3*x^4-12*x^ 3-27*x^2-15*x)*log(x)),x, algorithm=\
log(20*x + 19) - log(x + 1) + log(((20*x^4 + 19*x^3 - 80*x^2 - 176*x - 95) *log(x) + 3)/(20*x^4 + 19*x^3 - 80*x^2 - 176*x - 95)) - log(log(x))
Exception generated. \[ \int \frac {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((x**7-8*x**5-10*x**4+16*x**3+40*x**2+25*x)*ln(x)**2+(-12*x**4-9 *x**3+24*x**2+27*x)*ln(x)-3*x**4-3*x**3+12*x**2+27*x+15)/((20*x**9+39*x**8 -141*x**7-512*x**6-222*x**5+1234*x**4+2364*x**3+1735*x**2+475*x)*ln(x)**2+ (3*x**5+3*x**4-12*x**3-27*x**2-15*x)*ln(x)),x)
Exception raised: PolynomialError >> 1/(400*x**9 + 760*x**8 - 2839*x**7 - 10080*x**6 - 4088*x**5 + 24550*x**4 + 46176*x**3 + 33440*x**2 + 9025*x) co ntains an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (32) = 64\).
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.97 \[ \int \frac {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx=\log \left (20 \, x + 19\right ) - \log \left (x + 1\right ) + \log \left (\frac {{\left (20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95\right )} \log \left (x\right ) + 3}{20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*log(x)^2+(-12*x^4-9*x^3+2 4*x^2+27*x)*log(x)-3*x^4-3*x^3+12*x^2+27*x+15)/((20*x^9+39*x^8-141*x^7-512 *x^6-222*x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*log(x)^2+(3*x^5+3*x^4-12*x^ 3-27*x^2-15*x)*log(x)),x, algorithm=\
log(20*x + 19) - log(x + 1) + log(((20*x^4 + 19*x^3 - 80*x^2 - 176*x - 95) *log(x) + 3)/(20*x^4 + 19*x^3 - 80*x^2 - 176*x - 95)) - log(log(x))
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71 \[ \int \frac {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx=\log \left (20 \, x^{4} \log \left (x\right ) + 19 \, x^{3} \log \left (x\right ) - 80 \, x^{2} \log \left (x\right ) - 176 \, x \log \left (x\right ) - 95 \, \log \left (x\right ) + 3\right ) - \log \left (x^{4} + x^{3} - 4 \, x^{2} - 9 \, x - 5\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((x^7-8*x^5-10*x^4+16*x^3+40*x^2+25*x)*log(x)^2+(-12*x^4-9*x^3+2 4*x^2+27*x)*log(x)-3*x^4-3*x^3+12*x^2+27*x+15)/((20*x^9+39*x^8-141*x^7-512 *x^6-222*x^5+1234*x^4+2364*x^3+1735*x^2+475*x)*log(x)^2+(3*x^5+3*x^4-12*x^ 3-27*x^2-15*x)*log(x)),x, algorithm=\
log(20*x^4*log(x) + 19*x^3*log(x) - 80*x^2*log(x) - 176*x*log(x) - 95*log( x) + 3) - log(x^4 + x^3 - 4*x^2 - 9*x - 5) - log(log(x))
Timed out. \[ \int \frac {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(x)} \, dx=\int -\frac {27\,x+\ln \left (x\right )\,\left (-12\,x^4-9\,x^3+24\,x^2+27\,x\right )+{\ln \left (x\right )}^2\,\left (x^7-8\,x^5-10\,x^4+16\,x^3+40\,x^2+25\,x\right )+12\,x^2-3\,x^3-3\,x^4+15}{\ln \left (x\right )\,\left (-3\,x^5-3\,x^4+12\,x^3+27\,x^2+15\,x\right )-{\ln \left (x\right )}^2\,\left (20\,x^9+39\,x^8-141\,x^7-512\,x^6-222\,x^5+1234\,x^4+2364\,x^3+1735\,x^2+475\,x\right )} \,d x \]
int(-(27*x + log(x)*(27*x + 24*x^2 - 9*x^3 - 12*x^4) + log(x)^2*(25*x + 40 *x^2 + 16*x^3 - 10*x^4 - 8*x^5 + x^7) + 12*x^2 - 3*x^3 - 3*x^4 + 15)/(log( x)*(15*x + 27*x^2 + 12*x^3 - 3*x^4 - 3*x^5) - log(x)^2*(475*x + 1735*x^2 + 2364*x^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)),x)
int(-(27*x + log(x)*(27*x + 24*x^2 - 9*x^3 - 12*x^4) + log(x)^2*(25*x + 40 *x^2 + 16*x^3 - 10*x^4 - 8*x^5 + x^7) + 12*x^2 - 3*x^3 - 3*x^4 + 15)/(log( x)*(15*x + 27*x^2 + 12*x^3 - 3*x^4 - 3*x^5) - log(x)^2*(475*x + 1735*x^2 + 2364*x^3 + 1234*x^4 - 222*x^5 - 512*x^6 - 141*x^7 + 39*x^8 + 20*x^9)), x)