Integrand size = 175, antiderivative size = 33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=-\frac {x^2}{3-x}+\log \left (x+\frac {5-x}{\left (x+x^3\right ) \log (x)}\right ) \]
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {9}{-3+x}+x-\log (x)-\log \left (1+x^2\right )-\log (\log (x))+\log \left (5-x+x^2 \log (x)+x^4 \log (x)\right ) \]
Integrate[(-45 + 39*x - 56*x^2 + 40*x^3 - 11*x^4 + x^5 + (-45 + 30*x - 170 *x^2 + 119*x^3 - 58*x^4 + 13*x^5 - x^6)*Log[x] + (9*x^2 - 6*x^3 + 13*x^4 - 11*x^5 - x^6 - 4*x^7 - 5*x^8 + x^9)*Log[x]^2)/((45*x - 39*x^2 + 56*x^3 - 40*x^4 + 11*x^5 - x^6)*Log[x] + (9*x^3 - 6*x^4 + 19*x^5 - 12*x^6 + 11*x^7 - 6*x^8 + x^9)*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.
| \(\displaystyle \int \frac {x^5-11 x^4+40 x^3-56 x^2+\left (-x^6+13 x^5-58 x^4+119 x^3-170 x^2+30 x-45\right ) \log (x)+\left (x^9-5 x^8-4 x^7-x^6-11 x^5+13 x^4-6 x^3+9 x^2\right ) \log ^2(x)+39 x-45}{\left (-x^6+11 x^5-40 x^4+56 x^3-39 x^2+45 x\right ) \log (x)+\left (x^9-6 x^8+11 x^7-12 x^6+19 x^5-6 x^4+9 x^3\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x^5-11 x^4+40 x^3-56 x^2+\left (-x^6+13 x^5-58 x^4+119 x^3-170 x^2+30 x-45\right ) \log (x)+\left (x^9-5 x^8-4 x^7-x^6-11 x^5+13 x^4-6 x^3+9 x^2\right ) \log ^2(x)+39 x-45}{(3-x)^2 x \left (x^2+1\right ) \log (x) \left (x^4 \log (x)+x^2 \log (x)-x+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {x \left (x^2+1\right )}{x^4 \log (x)+x^2 \log (x)-x+5}+\frac {x^3-5 x^2-6 x+9}{(x-3)^2 x}+\frac {3 x^3-20 x^2+x-10}{x \left (x^2+1\right ) \left (x^4 \log (x)+x^2 \log (x)-x+5\right )}-\frac {1}{x \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {1}{\log (x) x^4+\log (x) x^2-x+5}dx+(5-i) \int \frac {1}{(i-x) \left (\log (x) x^4+\log (x) x^2-x+5\right )}dx-10 \int \frac {1}{x \left (\log (x) x^4+\log (x) x^2-x+5\right )}dx+\int \frac {x}{\log (x) x^4+\log (x) x^2-x+5}dx-(5+i) \int \frac {1}{(x+i) \left (\log (x) x^4+\log (x) x^2-x+5\right )}dx+\int \frac {x^3}{\log (x) x^4+\log (x) x^2-x+5}dx+x-\frac {9}{3-x}+\log (x)-\log (\log (x))\) |
Int[(-45 + 39*x - 56*x^2 + 40*x^3 - 11*x^4 + x^5 + (-45 + 30*x - 170*x^2 + 119*x^3 - 58*x^4 + 13*x^5 - x^6)*Log[x] + (9*x^2 - 6*x^3 + 13*x^4 - 11*x^ 5 - x^6 - 4*x^7 - 5*x^8 + x^9)*Log[x]^2)/((45*x - 39*x^2 + 56*x^3 - 40*x^4 + 11*x^5 - x^6)*Log[x] + (9*x^3 - 6*x^4 + 19*x^5 - 12*x^6 + 11*x^7 - 6*x^ 8 + x^9)*Log[x]^2),x]
3.16.6.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.54 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(-\ln \left (x \right )+x +\frac {9}{-3+x}+\ln \left (x^{4}+x^{2}-\frac {x}{\ln \left (x \right )}+\frac {5}{\ln \left (x \right )}\right )-\ln \left (x^{2}+1\right )\) | \(43\) |
| risch | \(\frac {x \ln \left (x \right )+x^{2}-3 \ln \left (x \right )-3 x +9}{-3+x}-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (x \right )-\frac {-5+x}{x^{2} \left (x^{2}+1\right )}\right )\) | \(48\) |
| parallelrisch | \(\frac {-x \ln \left (\ln \left (x \right )\right )-\ln \left (x^{2}+1\right ) x +\ln \left (x^{4} \ln \left (x \right )+x^{2} \ln \left (x \right )-x +5\right ) x +x^{2}-x \ln \left (x \right )+3 \ln \left (\ln \left (x \right )\right )+3 \ln \left (x^{2}+1\right )-3 \ln \left (x^{4} \ln \left (x \right )+x^{2} \ln \left (x \right )-x +5\right )+3 \ln \left (x \right )}{-3+x}\) | \(88\) |
int(((x^9-5*x^8-4*x^7-x^6-11*x^5+13*x^4-6*x^3+9*x^2)*ln(x)^2+(-x^6+13*x^5- 58*x^4+119*x^3-170*x^2+30*x-45)*ln(x)+x^5-11*x^4+40*x^3-56*x^2+39*x-45)/(( x^9-6*x^8+11*x^7-12*x^6+19*x^5-6*x^4+9*x^3)*ln(x)^2+(-x^6+11*x^5-40*x^4+56 *x^3-39*x^2+45*x)*ln(x)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {x^{2} + {\left (x - 3\right )} \log \left (x\right ) + {\left (x - 3\right )} \log \left (\frac {{\left (x^{4} + x^{2}\right )} \log \left (x\right ) - x + 5}{x^{4} + x^{2}}\right ) - {\left (x - 3\right )} \log \left (\log \left (x\right )\right ) - 3 \, x + 9}{x - 3} \]
integrate(((x^9-5*x^8-4*x^7-x^6-11*x^5+13*x^4-6*x^3+9*x^2)*log(x)^2+(-x^6+ 13*x^5-58*x^4+119*x^3-170*x^2+30*x-45)*log(x)+x^5-11*x^4+40*x^3-56*x^2+39* x-45)/((x^9-6*x^8+11*x^7-12*x^6+19*x^5-6*x^4+9*x^3)*log(x)^2+(-x^6+11*x^5- 40*x^4+56*x^3-39*x^2+45*x)*log(x)),x, algorithm=\
(x^2 + (x - 3)*log(x) + (x - 3)*log(((x^4 + x^2)*log(x) - x + 5)/(x^4 + x^ 2)) - (x - 3)*log(log(x)) - 3*x + 9)/(x - 3)
Exception generated. \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((x**9-5*x**8-4*x**7-x**6-11*x**5+13*x**4-6*x**3+9*x**2)*ln(x)** 2+(-x**6+13*x**5-58*x**4+119*x**3-170*x**2+30*x-45)*ln(x)+x**5-11*x**4+40* x**3-56*x**2+39*x-45)/((x**9-6*x**8+11*x**7-12*x**6+19*x**5-6*x**4+9*x**3) *ln(x)**2+(-x**6+11*x**5-40*x**4+56*x**3-39*x**2+45*x)*ln(x)),x)
Exception raised: PolynomialError >> 1/(x**7 + 2*x**5 + x**3) contains an element of the set of generators.
Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {x^{2} - 3 \, x + 9}{x - 3} + \log \left (x\right ) + \log \left (\frac {{\left (x^{4} + x^{2}\right )} \log \left (x\right ) - x + 5}{x^{4} + x^{2}}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((x^9-5*x^8-4*x^7-x^6-11*x^5+13*x^4-6*x^3+9*x^2)*log(x)^2+(-x^6+ 13*x^5-58*x^4+119*x^3-170*x^2+30*x-45)*log(x)+x^5-11*x^4+40*x^3-56*x^2+39* x-45)/((x^9-6*x^8+11*x^7-12*x^6+19*x^5-6*x^4+9*x^3)*log(x)^2+(-x^6+11*x^5- 40*x^4+56*x^3-39*x^2+45*x)*log(x)),x, algorithm=\
(x^2 - 3*x + 9)/(x - 3) + log(x) + log(((x^4 + x^2)*log(x) - x + 5)/(x^4 + x^2)) - log(log(x))
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=x + \frac {9}{x - 3} + \log \left (x^{4} \log \left (x\right ) + x^{2} \log \left (x\right ) - x + 5\right ) - \log \left (x^{2} + 1\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right ) \]
integrate(((x^9-5*x^8-4*x^7-x^6-11*x^5+13*x^4-6*x^3+9*x^2)*log(x)^2+(-x^6+ 13*x^5-58*x^4+119*x^3-170*x^2+30*x-45)*log(x)+x^5-11*x^4+40*x^3-56*x^2+39* x-45)/((x^9-6*x^8+11*x^7-12*x^6+19*x^5-6*x^4+9*x^3)*log(x)^2+(-x^6+11*x^5- 40*x^4+56*x^3-39*x^2+45*x)*log(x)),x, algorithm=\
Timed out. \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\int -\frac {\ln \left (x\right )\,\left (x^6-13\,x^5+58\,x^4-119\,x^3+170\,x^2-30\,x+45\right )-39\,x+{\ln \left (x\right )}^2\,\left (-x^9+5\,x^8+4\,x^7+x^6+11\,x^5-13\,x^4+6\,x^3-9\,x^2\right )+56\,x^2-40\,x^3+11\,x^4-x^5+45}{\left (x^9-6\,x^8+11\,x^7-12\,x^6+19\,x^5-6\,x^4+9\,x^3\right )\,{\ln \left (x\right )}^2+\left (-x^6+11\,x^5-40\,x^4+56\,x^3-39\,x^2+45\,x\right )\,\ln \left (x\right )} \,d x \]
int(-(log(x)*(170*x^2 - 30*x - 119*x^3 + 58*x^4 - 13*x^5 + x^6 + 45) - 39* x + log(x)^2*(6*x^3 - 9*x^2 - 13*x^4 + 11*x^5 + x^6 + 4*x^7 + 5*x^8 - x^9) + 56*x^2 - 40*x^3 + 11*x^4 - x^5 + 45)/(log(x)^2*(9*x^3 - 6*x^4 + 19*x^5 - 12*x^6 + 11*x^7 - 6*x^8 + x^9) + log(x)*(45*x - 39*x^2 + 56*x^3 - 40*x^4 + 11*x^5 - x^6)),x)
int(-(log(x)*(170*x^2 - 30*x - 119*x^3 + 58*x^4 - 13*x^5 + x^6 + 45) - 39* x + log(x)^2*(6*x^3 - 9*x^2 - 13*x^4 + 11*x^5 + x^6 + 4*x^7 + 5*x^8 - x^9) + 56*x^2 - 40*x^3 + 11*x^4 - x^5 + 45)/(log(x)^2*(9*x^3 - 6*x^4 + 19*x^5 - 12*x^6 + 11*x^7 - 6*x^8 + x^9) + log(x)*(45*x - 39*x^2 + 56*x^3 - 40*x^4 + 11*x^5 - x^6)), x)