\(\int \frac {-1+2 x+(12 x-4 x^2) \log (x)+(17 x-12 x^2+2 x^3) \log ^2(x)}{x+(6 x-2 x^2) \log (x)+(9 x-6 x^2+x^3) \log ^2(x)} \, dx\) [8739]

Optimal result

Integrand size = 70, antiderivative size = 17 \[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=2 x-\frac {1}{3-x+\frac {1}{\log (x)}} \]

[Out]

2*x-1/(1/ln(x)-x+3)

Rubi [F]

\[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx \]

[In]

Int[(-1 + 2*x + (12*x - 4*x^2)*Log[x] + (17*x - 12*x^2 + 2*x^3)*Log[x]^2)/(x + (6*x - 2*x^2)*Log[x] + (9*x - 6
*x^2 + x^3)*Log[x]^2),x]

[Out]

(-3 + x)^(-1) + 2*x - Defer[Int][1/((-3 + x)^2*(-1 - 3*Log[x] + x*Log[x])^2), x] - Defer[Int][1/(x*(-1 - 3*Log
[x] + x*Log[x])^2), x] - 2*Defer[Int][1/((-3 + x)^2*(-1 - 3*Log[x] + x*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+2 x-4 (-3+x) x \log (x)+x \left (17-12 x+2 x^2\right ) \log ^2(x)}{x (1-(-3+x) \log (x))^2} \, dx \\ & = \int \left (\frac {17-12 x+2 x^2}{(-3+x)^2}+\frac {-9+5 x-x^2}{(-3+x)^2 x (-1-3 \log (x)+x \log (x))^2}-\frac {2}{(-3+x)^2 (-1-3 \log (x)+x \log (x))}\right ) \, dx \\ & = -\left (2 \int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))} \, dx\right )+\int \frac {17-12 x+2 x^2}{(-3+x)^2} \, dx+\int \frac {-9+5 x-x^2}{(-3+x)^2 x (-1-3 \log (x)+x \log (x))^2} \, dx \\ & = -\left (2 \int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))} \, dx\right )+\int \left (2-\frac {1}{(-3+x)^2}\right ) \, dx+\int \left (-\frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))^2}-\frac {1}{x (-1-3 \log (x)+x \log (x))^2}\right ) \, dx \\ & = \frac {1}{-3+x}+2 x-2 \int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))} \, dx-\int \frac {1}{(-3+x)^2 (-1-3 \log (x)+x \log (x))^2} \, dx-\int \frac {1}{x (-1-3 \log (x)+x \log (x))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=\frac {1}{-3+x}+2 x+\frac {1}{(-3+x) (-1-3 \log (x)+x \log (x))} \]

[In]

Integrate[(-1 + 2*x + (12*x - 4*x^2)*Log[x] + (17*x - 12*x^2 + 2*x^3)*Log[x]^2)/(x + (6*x - 2*x^2)*Log[x] + (9
*x - 6*x^2 + x^3)*Log[x]^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76

method	result	size
default	\(\frac {-17 \ln \left (x \right )+2 x^{2} \ln \left (x \right )-2 x -6}{x \ln \left (x \right )-3 \ln \left (x \right )-1}\)	\(30\)
norman	\(\frac {-17 \ln \left (x \right )+2 x^{2} \ln \left (x \right )-2 x -6}{x \ln \left (x \right )-3 \ln \left (x \right )-1}\)	\(30\)
parallelrisch	\(\frac {-17 \ln \left (x \right )+2 x^{2} \ln \left (x \right )-2 x -6}{x \ln \left (x \right )-3 \ln \left (x \right )-1}\)	\(30\)
risch	\(\frac {2 x^{2}-6 x +1}{-3+x}+\frac {1}{\left (-3+x \right ) \left (x \ln \left (x \right )-3 \ln \left (x \right )-1\right )}\)	\(36\)

[In]

int(((2*x^3-12*x^2+17*x)*ln(x)^2+(-4*x^2+12*x)*ln(x)+2*x-1)/((x^3-6*x^2+9*x)*ln(x)^2+(-2*x^2+6*x)*ln(x)+x),x,m
ethod=_RETURNVERBOSE)

[Out]

(-17*ln(x)+2*x^2*ln(x)-2*x-6)/(x*ln(x)-3*ln(x)-1)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=\frac {{\left (2 \, x^{2} - 6 \, x + 1\right )} \log \left (x\right ) - 2 \, x}{{\left (x - 3\right )} \log \left (x\right ) - 1} \]

[In]

integrate(((2*x^3-12*x^2+17*x)*log(x)^2+(-4*x^2+12*x)*log(x)+2*x-1)/((x^3-6*x^2+9*x)*log(x)^2+(-2*x^2+6*x)*log
(x)+x),x, algorithm="fricas")

[Out]

((2*x^2 - 6*x + 1)*log(x) - 2*x)/((x - 3)*log(x) - 1)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=2 x + \frac {1}{- x + \left (x^{2} - 6 x + 9\right ) \log {\left (x \right )} + 3} + \frac {1}{x - 3} \]

[In]

integrate(((2*x**3-12*x**2+17*x)*ln(x)**2+(-4*x**2+12*x)*ln(x)+2*x-1)/((x**3-6*x**2+9*x)*ln(x)**2+(-2*x**2+6*x
)*ln(x)+x),x)

[Out]

2*x + 1/(-x + (x**2 - 6*x + 9)*log(x) + 3) + 1/(x - 3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=\frac {{\left (2 \, x^{2} - 6 \, x + 1\right )} \log \left (x\right ) - 2 \, x}{{\left (x - 3\right )} \log \left (x\right ) - 1} \]

[In]

integrate(((2*x^3-12*x^2+17*x)*log(x)^2+(-4*x^2+12*x)*log(x)+2*x-1)/((x^3-6*x^2+9*x)*log(x)^2+(-2*x^2+6*x)*log
(x)+x),x, algorithm="maxima")

[Out]

((2*x^2 - 6*x + 1)*log(x) - 2*x)/((x - 3)*log(x) - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=2 \, x + \frac {1}{x^{2} \log \left (x\right ) - 6 \, x \log \left (x\right ) - x + 9 \, \log \left (x\right ) + 3} + \frac {1}{x - 3} \]

[In]

integrate(((2*x^3-12*x^2+17*x)*log(x)^2+(-4*x^2+12*x)*log(x)+2*x-1)/((x^3-6*x^2+9*x)*log(x)^2+(-2*x^2+6*x)*log
(x)+x),x, algorithm="giac")

[Out]

2*x + 1/(x^2*log(x) - 6*x*log(x) - x + 9*log(x) + 3) + 1/(x - 3)

Mupad [B] (verification not implemented)

Time = 13.72 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {-1+2 x+\left (12 x-4 x^2\right ) \log (x)+\left (17 x-12 x^2+2 x^3\right ) \log ^2(x)}{x+\left (6 x-2 x^2\right ) \log (x)+\left (9 x-6 x^2+x^3\right ) \log ^2(x)} \, dx=2\,x-\frac {\ln \left (x\right )}{3\,\ln \left (x\right )-x\,\ln \left (x\right )+1} \]

[In]

int((2*x + log(x)^2*(17*x - 12*x^2 + 2*x^3) + log(x)*(12*x - 4*x^2) - 1)/(x + log(x)*(6*x - 2*x^2) + log(x)^2*
(9*x - 6*x^2 + x^3)),x)

[Out]

2*x - log(x)/(3*log(x) - x*log(x) + 1)