\(\int \frac {-16+8 x-13 x^2+2 x^3+(4080-2024 x+247 x^2+5 x^3-x^4) \log ^2(\frac {-3060+753 x+3 x^2-3 x^3}{-4+x})}{-4080+2024 x-247 x^2-5 x^3+x^4+(-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5) \log (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x})+(-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6) \log ^2(\frac {-3060+753 x+3 x^2-3 x^3}{-4+x})} \, dx\) [8771]

Optimal result

Integrand size = 177, antiderivative size = 29 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {x}{x^2+\frac {x}{\log \left (3 \left (255+x+\frac {x^3}{4-x}\right )\right )}} \]

[Out]

x/(x^2+x/ln(765+3*x^3/(-x+4)+3*x))

Rubi [F]

\[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx \]

[In]

Int[(-16 + 8*x - 13*x^2 + 2*x^3 + (4080 - 2024*x + 247*x^2 + 5*x^3 - x^4)*Log[(-3060 + 753*x + 3*x^2 - 3*x^3)/
(-4 + x)]^2)/(-4080 + 2024*x - 247*x^2 - 5*x^3 + x^4 + (-8160*x + 4048*x^2 - 494*x^3 - 10*x^4 + 2*x^5)*Log[(-3
060 + 753*x + 3*x^2 - 3*x^3)/(-4 + x)] + (-4080*x^2 + 2024*x^3 - 247*x^4 - 5*x^5 + x^6)*Log[(-3060 + 753*x + 3
*x^2 - 3*x^3)/(-4 + x)]^2),x]

[Out]

x^(-1) - Defer[Int][1/((-4 + x)*(1 + x*Log[(-3*(1020 - 251*x - x^2 + x^3))/(-4 + x)])^2), x] - Defer[Int][1/(x
^2*(1 + x*Log[(-3*(1020 - 251*x - x^2 + x^3))/(-4 + x)])^2), x] - 251*Defer[Int][1/((1020 - 251*x - x^2 + x^3)
*(1 + x*Log[(-3*(1020 - 251*x - x^2 + x^3))/(-4 + x)])^2), x] - 2*Defer[Int][x/((1020 - 251*x - x^2 + x^3)*(1
+ x*Log[(-3*(1020 - 251*x - x^2 + x^3))/(-4 + x)])^2), x] + 3*Defer[Int][x^2/((1020 - 251*x - x^2 + x^3)*(1 +
x*Log[(-3*(1020 - 251*x - x^2 + x^3))/(-4 + x)])^2), x] + 2*Defer[Int][1/(x^2*(1 + x*Log[(-3*(1020 - 251*x - x
^2 + x^3))/(-4 + x)])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {16-8 x+13 x^2-2 x^3-\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )}{\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx \\ & = \int \left (-\frac {1}{x^2}+\frac {4080-2024 x+231 x^2+13 x^3-14 x^4+2 x^5}{(-4+x) x^2 \left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2}+\frac {2}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )}\right ) \, dx \\ & = \frac {1}{x}+2 \int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )} \, dx+\int \frac {4080-2024 x+231 x^2+13 x^3-14 x^4+2 x^5}{(-4+x) x^2 \left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx \\ & = \frac {1}{x}+2 \int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )} \, dx+\int \left (-\frac {1}{(-4+x) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2}-\frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2}+\frac {-251-2 x+3 x^2}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2}\right ) \, dx \\ & = \frac {1}{x}+2 \int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )} \, dx-\int \frac {1}{(-4+x) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx-\int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx+\int \frac {-251-2 x+3 x^2}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx \\ & = \frac {1}{x}+2 \int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )} \, dx-\int \frac {1}{(-4+x) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx-\int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx+\int \left (-\frac {251}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2}-\frac {2 x}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2}+\frac {3 x^2}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2}\right ) \, dx \\ & = \frac {1}{x}-2 \int \frac {x}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx+2 \int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )} \, dx+3 \int \frac {x^2}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx-251 \int \frac {1}{\left (1020-251 x-x^2+x^3\right ) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx-\int \frac {1}{(-4+x) \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx-\int \frac {1}{x^2 \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {1}{x}-\frac {1}{x \left (1+x \log \left (-\frac {3 \left (1020-251 x-x^2+x^3\right )}{-4+x}\right )\right )} \]

[In]

Integrate[(-16 + 8*x - 13*x^2 + 2*x^3 + (4080 - 2024*x + 247*x^2 + 5*x^3 - x^4)*Log[(-3060 + 753*x + 3*x^2 - 3
*x^3)/(-4 + x)]^2)/(-4080 + 2024*x - 247*x^2 - 5*x^3 + x^4 + (-8160*x + 4048*x^2 - 494*x^3 - 10*x^4 + 2*x^5)*L
og[(-3060 + 753*x + 3*x^2 - 3*x^3)/(-4 + x)] + (-4080*x^2 + 2024*x^3 - 247*x^4 - 5*x^5 + x^6)*Log[(-3060 + 753
*x + 3*x^2 - 3*x^3)/(-4 + x)]^2),x]

[Out]

x^(-1) - 1/(x*(1 + x*Log[(-3*(1020 - 251*x - x^2 + x^3))/(-4 + x)]))

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31

method	result	size
risch	\(\frac {1}{x}-\frac {1}{x \left (x \ln \left (\frac {-3 x^{3}+3 x^{2}+753 x -3060}{x -4}\right )+1\right )}\)	\(38\)
parallelrisch	\(\frac {\ln \left (-\frac {3 \left (x^{3}-x^{2}-251 x +1020\right )}{x -4}\right )}{x \ln \left (-\frac {3 \left (x^{3}-x^{2}-251 x +1020\right )}{x -4}\right )+1}\)	\(50\)
norman	\(\frac {\ln \left (\frac {-3 x^{3}+3 x^{2}+753 x -3060}{x -4}\right )}{x \ln \left (\frac {-3 x^{3}+3 x^{2}+753 x -3060}{x -4}\right )+1}\)	\(52\)

[In]

int(((-x^4+5*x^3+247*x^2-2024*x+4080)*ln((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+2*x^3-13*x^2+8*x-16)/((x^6-5*x^5-2
47*x^4+2024*x^3-4080*x^2)*ln((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)*ln((-3*
x^3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x,method=_RETURNVERBOSE)

[Out]

1/x-1/x/(x*ln((-3*x^3+3*x^2+753*x-3060)/(x-4))+1)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {\log \left (-\frac {3 \, {\left (x^{3} - x^{2} - 251 \, x + 1020\right )}}{x - 4}\right )}{x \log \left (-\frac {3 \, {\left (x^{3} - x^{2} - 251 \, x + 1020\right )}}{x - 4}\right ) + 1} \]

[In]

integrate(((-x^4+5*x^3+247*x^2-2024*x+4080)*log((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+2*x^3-13*x^2+8*x-16)/((x^6-
5*x^5-247*x^4+2024*x^3-4080*x^2)*log((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)
*log((-3*x^3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x, algorithm="fricas")

[Out]

log(-3*(x^3 - x^2 - 251*x + 1020)/(x - 4))/(x*log(-3*(x^3 - x^2 - 251*x + 1020)/(x - 4)) + 1)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=- \frac {1}{x^{2} \log {\left (\frac {- 3 x^{3} + 3 x^{2} + 753 x - 3060}{x - 4} \right )} + x} + \frac {1}{x} \]

[In]

integrate(((-x**4+5*x**3+247*x**2-2024*x+4080)*ln((-3*x**3+3*x**2+753*x-3060)/(x-4))**2+2*x**3-13*x**2+8*x-16)
/((x**6-5*x**5-247*x**4+2024*x**3-4080*x**2)*ln((-3*x**3+3*x**2+753*x-3060)/(x-4))**2+(2*x**5-10*x**4-494*x**3
+4048*x**2-8160*x)*ln((-3*x**3+3*x**2+753*x-3060)/(x-4))+x**4-5*x**3-247*x**2+2024*x-4080),x)

[Out]

-1/(x**2*log((-3*x**3 + 3*x**2 + 753*x - 3060)/(x - 4)) + x) + 1/x

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {i \, \pi + \log \left (3\right ) + \log \left (x^{3} - x^{2} - 251 \, x + 1020\right ) - \log \left (x - 4\right )}{{\left (i \, \pi + \log \left (3\right )\right )} x + x \log \left (x^{3} - x^{2} - 251 \, x + 1020\right ) - x \log \left (x - 4\right ) + 1} \]

[In]

integrate(((-x^4+5*x^3+247*x^2-2024*x+4080)*log((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+2*x^3-13*x^2+8*x-16)/((x^6-
5*x^5-247*x^4+2024*x^3-4080*x^2)*log((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)
*log((-3*x^3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x, algorithm="maxima")

[Out]

(I*pi + log(3) + log(x^3 - x^2 - 251*x + 1020) - log(x - 4))/((I*pi + log(3))*x + x*log(x^3 - x^2 - 251*x + 10
20) - x*log(x - 4) + 1)

Giac [A] (verification not implemented)

none

Time = 0.73 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=-\frac {1}{x^{2} \log \left (-\frac {3 \, {\left (x^{3} - x^{2} - 251 \, x + 1020\right )}}{x - 4}\right ) + x} + \frac {1}{x} \]

[In]

integrate(((-x^4+5*x^3+247*x^2-2024*x+4080)*log((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+2*x^3-13*x^2+8*x-16)/((x^6-
5*x^5-247*x^4+2024*x^3-4080*x^2)*log((-3*x^3+3*x^2+753*x-3060)/(x-4))^2+(2*x^5-10*x^4-494*x^3+4048*x^2-8160*x)
*log((-3*x^3+3*x^2+753*x-3060)/(x-4))+x^4-5*x^3-247*x^2+2024*x-4080),x, algorithm="giac")

[Out]

-1/(x^2*log(-3*(x^3 - x^2 - 251*x + 1020)/(x - 4)) + x) + 1/x

Mupad [B] (verification not implemented)

Time = 15.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-16+8 x-13 x^2+2 x^3+\left (4080-2024 x+247 x^2+5 x^3-x^4\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )}{-4080+2024 x-247 x^2-5 x^3+x^4+\left (-8160 x+4048 x^2-494 x^3-10 x^4+2 x^5\right ) \log \left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )+\left (-4080 x^2+2024 x^3-247 x^4-5 x^5+x^6\right ) \log ^2\left (\frac {-3060+753 x+3 x^2-3 x^3}{-4+x}\right )} \, dx=\frac {1}{x}-\frac {1}{x+x^2\,\ln \left (\frac {-3\,x^3+3\,x^2+753\,x-3060}{x-4}\right )} \]

[In]

int(-(8*x + log((753*x + 3*x^2 - 3*x^3 - 3060)/(x - 4))^2*(247*x^2 - 2024*x + 5*x^3 - x^4 + 4080) - 13*x^2 + 2
*x^3 - 16)/(log((753*x + 3*x^2 - 3*x^3 - 3060)/(x - 4))*(8160*x - 4048*x^2 + 494*x^3 + 10*x^4 - 2*x^5) - 2024*
x + 247*x^2 + 5*x^3 - x^4 + log((753*x + 3*x^2 - 3*x^3 - 3060)/(x - 4))^2*(4080*x^2 - 2024*x^3 + 247*x^4 + 5*x
^5 - x^6) + 4080),x)

[Out]

1/x - 1/(x + x^2*log((753*x + 3*x^2 - 3*x^3 - 3060)/(x - 4)))