\(\int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+(-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6) \log (x)+(9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9) \log ^2(x)}{(45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6) \log (x)+(9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9) \log ^2(x)} \, dx\) [1506]

Optimal result

Integrand size = 175, antiderivative size = 33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=-\frac {x^2}{3-x}+\log \left (x+\frac {5-x}{\left (x+x^3\right ) \log (x)}\right ) \]

[Out]

ln((5-x)/ln(x)/(x^3+x)+x)-x^2/(-x+3)

Rubi [F]

\[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx \]

[In]

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)*Lo
g[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]

[Out]

-9/(3 - x) + x + Log[x] - Log[Log[x]] + 3*Defer[Int][(5 - x + x^2*Log[x] + x^4*Log[x])^(-1), x] + (5 - I)*Defe
r[Int][1/((I - x)*(5 - x + x^2*Log[x] + x^4*Log[x])), x] - 10*Defer[Int][1/(x*(5 - x + x^2*Log[x] + x^4*Log[x]
)), x] + Defer[Int][x/(5 - x + x^2*Log[x] + x^4*Log[x]), x] + Defer[Int][x^3/(5 - x + x^2*Log[x] + x^4*Log[x])
, x] - (5 + I)*Defer[Int][1/((I + x)*(5 - x + x^2*Log[x] + x^4*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-3+x)^2 \left (-5+x-5 x^2+x^3\right )-\left (45-30 x+170 x^2-119 x^3+58 x^4-13 x^5+x^6\right ) \log (x)+\left (x+x^3\right )^2 \left (9-6 x-5 x^2+x^3\right ) \log ^2(x)}{(3-x)^2 x \left (1+x^2\right ) \log (x) \left (5-x+\left (x^2+x^4\right ) \log (x)\right )} \, dx \\ & = \int \left (\frac {9-6 x-5 x^2+x^3}{(-3+x)^2 x}-\frac {1}{x \log (x)}+\frac {x \left (1+x^2\right )}{5-x+x^2 \log (x)+x^4 \log (x)}+\frac {-10+x-20 x^2+3 x^3}{x \left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx \\ & = \int \frac {9-6 x-5 x^2+x^3}{(-3+x)^2 x} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {x \left (1+x^2\right )}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {-10+x-20 x^2+3 x^3}{x \left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx \\ & = \int \left (1-\frac {9}{(-3+x)^2}+\frac {1}{x}\right ) \, dx+\int \left (\frac {x}{5-x+x^2 \log (x)+x^4 \log (x)}+\frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)}\right ) \, dx+\int \left (\frac {3}{5-x+x^2 \log (x)+x^4 \log (x)}-\frac {10}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}-\frac {2 (1+5 x)}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \frac {1+5 x}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \left (\frac {1}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}+\frac {5 x}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \frac {1}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-10 \int \frac {x}{\left (1+x^2\right ) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-2 \int \left (\frac {i}{2 (i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}+\frac {i}{2 (i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-10 \int \left (-\frac {1}{2 (i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}+\frac {1}{2 (i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )}\right ) \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ & = -\frac {9}{3-x}+x+\log (x)-\log (\log (x))-i \int \frac {1}{(i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-i \int \frac {1}{(i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+3 \int \frac {1}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+5 \int \frac {1}{(i-x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-5 \int \frac {1}{(i+x) \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx-10 \int \frac {1}{x \left (5-x+x^2 \log (x)+x^4 \log (x)\right )} \, dx+\int \frac {x}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx+\int \frac {x^3}{5-x+x^2 \log (x)+x^4 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {9}{-3+x}+x-\log (x)-\log \left (1+x^2\right )-\log (\log (x))+\log \left (5-x+x^2 \log (x)+x^4 \log (x)\right ) \]

[In]

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]

[Out]

9/(-3 + x) + x - Log[x] - Log[1 + x^2] - Log[Log[x]] + Log[5 - x + x^2*Log[x] + x^4*Log[x]]

Maple [A] (verified)

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	\(\frac {-x \ln \left (\ln \left (x \right )\right )-\ln \left (x^{2}+1\right ) x +\ln \left (x^{4} \ln \left (x \right )+x^{2} \ln \left (x \right )-x +5\right ) x +x^{2}-x \ln \left (x \right )+3 \ln \left (\ln \left (x \right )\right )+3 \ln \left (x^{2}+1\right )-3 \ln \left (x^{4} \ln \left (x \right )+x^{2} \ln \left (x \right )-x +5\right )+3 \ln \left (x \right )}{-3+x}\)	\(88\)

[In]

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)

[Out]

-ln(x)+x+9/(-3+x)+ln(x^4+x^2-x/ln(x)+5/ln(x))-ln(x^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {x^{2} + {\left (x - 3\right )} \log \left (x\right ) + {\left (x - 3\right )} \log \left (\frac {{\left (x^{4} + x^{2}\right )} \log \left (x\right ) - x + 5}{x^{4} + x^{2}}\right ) - {\left (x - 3\right )} \log \left (\log \left (x\right )\right ) - 3 \, x + 9}{x - 3} \]

[In]

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-4
5)*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="fricas")

[Out]

(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)

Sympy [F(-2)]

Exception generated. \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]

[In]

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**3-17
0*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)

[Out]

Exception raised: PolynomialError >> 1/(x**7 + 2*x**5 + x**3) contains an element of the set of generators.

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\frac {x^{2} - 3 \, x + 9}{x - 3} + \log \left (x\right ) + \log \left (\frac {{\left (x^{4} + x^{2}\right )} \log \left (x\right ) - x + 5}{x^{4} + x^{2}}\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

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-4
5)*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="maxima")

[Out]

(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))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=x + \frac {9}{x - 3} + \log \left (x^{4} \log \left (x\right ) + x^{2} \log \left (x\right ) - x + 5\right ) - \log \left (x^{2} + 1\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right ) \]

[In]

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-4
5)*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="giac")

[Out]

x + 9/(x - 3) + log(x^4*log(x) + x^2*log(x) - x + 5) - log(x^2 + 1) - log(x) - log(log(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {-45+39 x-56 x^2+40 x^3-11 x^4+x^5+\left (-45+30 x-170 x^2+119 x^3-58 x^4+13 x^5-x^6\right ) \log (x)+\left (9 x^2-6 x^3+13 x^4-11 x^5-x^6-4 x^7-5 x^8+x^9\right ) \log ^2(x)}{\left (45 x-39 x^2+56 x^3-40 x^4+11 x^5-x^6\right ) \log (x)+\left (9 x^3-6 x^4+19 x^5-12 x^6+11 x^7-6 x^8+x^9\right ) \log ^2(x)} \, dx=\int -\frac {\ln \left (x\right )\,\left (x^6-13\,x^5+58\,x^4-119\,x^3+170\,x^2-30\,x+45\right )-39\,x+{\ln \left (x\right )}^2\,\left (-x^9+5\,x^8+4\,x^7+x^6+11\,x^5-13\,x^4+6\,x^3-9\,x^2\right )+56\,x^2-40\,x^3+11\,x^4-x^5+45}{\left (x^9-6\,x^8+11\,x^7-12\,x^6+19\,x^5-6\,x^4+9\,x^3\right )\,{\ln \left (x\right )}^2+\left (-x^6+11\,x^5-40\,x^4+56\,x^3-39\,x^2+45\,x\right )\,\ln \left (x\right )} \,d x \]

[In]

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)

[Out]

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)