\(\int \frac {18-12 x+2 x^2+x^2 (i \pi +\log (\frac {25}{4}))+(-9+6 x-x^2) \log (x^2)}{(3 x^2-x^3) (i \pi +\log (\frac {25}{4}))+(9 x-6 x^2+x^3) \log (x^2)} \, dx\) [2165]

Optimal result

Integrand size = 82, antiderivative size = 32 \[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=\log \left (\frac {1}{3} \left (\frac {i \pi +\log \left (\frac {25}{4}\right )}{3-x}+\frac {\log \left (x^2\right )}{x}\right )\right ) \]

[Out]

ln(1/3*(ln(25/4)+I*Pi)/(-x+3)+1/3*ln(x^2)/x)

Rubi [F]

\[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=\int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx \]

[In]

Int[(18 - 12*x + 2*x^2 + x^2*(I*Pi + Log[25/4]) + (-9 + 6*x - x^2)*Log[x^2])/((3*x^2 - x^3)*(I*Pi + Log[25/4])
+ (9*x - 6*x^2 + x^3)*Log[x^2]),x]

[Out]

-Log[x] + (6*I)*Defer[Int][1/(x*(-(x*(Pi - I*Log[25/4])) - I*(-3 + x)*Log[x^2])), x] + (2*I)*Defer[Int][(x*(Pi
- I*Log[25/4]) + I*(-3 + x)*Log[x^2])^(-1), x] + 3*(Pi - I*Log[25/4])*Defer[Int][1/((3 - x)*(x*(Pi - I*Log[25
/4]) + I*(-3 + x)*Log[x^2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {18-12 x+x^2 \left (2+i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx \\ & = \int \frac {18-12 x+x^2 \left (2+i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{(3-x) x \left (i \pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )+3 \log \left (x^2\right )-x \log \left (x^2\right )\right )} \, dx \\ & = \int \left (-\frac {1}{x}+\frac {i \left (-18-2 x^2+3 x \left (4-i \pi -\log \left (\frac {25}{4}\right )\right )\right )}{(3-x) x \left (\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )-3 i \log \left (x^2\right )+i x \log \left (x^2\right )\right )}\right ) \, dx \\ & = -\log (x)+i \int \frac {-18-2 x^2+3 x \left (4-i \pi -\log \left (\frac {25}{4}\right )\right )}{(3-x) x \left (\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )-3 i \log \left (x^2\right )+i x \log \left (x^2\right )\right )} \, dx \\ & = -\log (x)+i \int \frac {-18-2 x^2+3 x \left (4-i \pi -\log \left (\frac {25}{4}\right )\right )}{(3-x) x \left (x \left (\pi -i \log \left (\frac {25}{4}\right )\right )+i (-3+x) \log \left (x^2\right )\right )} \, dx \\ & = -\log (x)+i \int \left (\frac {6}{x \left (-\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )+3 i \log \left (x^2\right )-i x \log \left (x^2\right )\right )}+\frac {2}{\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )-3 i \log \left (x^2\right )+i x \log \left (x^2\right )}+\frac {3 \left (-i \pi -\log \left (\frac {25}{4}\right )\right )}{(3-x) \left (\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )-3 i \log \left (x^2\right )+i x \log \left (x^2\right )\right )}\right ) \, dx \\ & = -\log (x)+2 i \int \frac {1}{\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )-3 i \log \left (x^2\right )+i x \log \left (x^2\right )} \, dx+6 i \int \frac {1}{x \left (-\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )+3 i \log \left (x^2\right )-i x \log \left (x^2\right )\right )} \, dx+\left (3 \left (\pi -i \log \left (\frac {25}{4}\right )\right )\right ) \int \frac {1}{(3-x) \left (\pi x \left (1-\frac {i \log \left (\frac {25}{4}\right )}{\pi }\right )-3 i \log \left (x^2\right )+i x \log \left (x^2\right )\right )} \, dx \\ & = -\log (x)+2 i \int \frac {1}{x \left (\pi -i \log \left (\frac {25}{4}\right )\right )+i (-3+x) \log \left (x^2\right )} \, dx+6 i \int \frac {1}{x \left (-x \left (\pi -i \log \left (\frac {25}{4}\right )\right )-i (-3+x) \log \left (x^2\right )\right )} \, dx+\left (3 \left (\pi -i \log \left (\frac {25}{4}\right )\right )\right ) \int \frac {1}{(3-x) \left (x \left (\pi -i \log \left (\frac {25}{4}\right )\right )+i (-3+x) \log \left (x^2\right )\right )} \, dx \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(32)=64\).

Time = 5.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.78 \[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=-i \arctan \left (\frac {\pi x}{-x \log \left (\frac {25}{4}\right )-3 \log \left (x^2\right )+x \log \left (x^2\right )}\right )-\frac {1}{2} \log \left ((3-x)^2\right )-\log (x)+\frac {1}{2} \log \left (\pi ^2 x^2+x^2 \log ^2\left (\frac {25}{4}\right )+6 x \log \left (\frac {25}{4}\right ) \log \left (x^2\right )-2 x^2 \log \left (\frac {25}{4}\right ) \log \left (x^2\right )+9 \log ^2\left (x^2\right )-6 x \log ^2\left (x^2\right )+x^2 \log ^2\left (x^2\right )\right ) \]

[In]

Integrate[(18 - 12*x + 2*x^2 + x^2*(I*Pi + Log[25/4]) + (-9 + 6*x - x^2)*Log[x^2])/((3*x^2 - x^3)*(I*Pi + Log[
25/4]) + (9*x - 6*x^2 + x^3)*Log[x^2]),x]

[Out]

(-I)*ArcTan[(Pi*x)/(-(x*Log[25/4]) - 3*Log[x^2] + x*Log[x^2])] - Log[(3 - x)^2]/2 - Log[x] + Log[Pi^2*x^2 + x^
2*Log[25/4]^2 + 6*x*Log[25/4]*Log[x^2] - 2*x^2*Log[25/4]*Log[x^2] + 9*Log[x^2]^2 - 6*x*Log[x^2]^2 + x^2*Log[x^
2]^2]/2

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\ln \left (x \right )+\ln \left (\ln \left (x^{2}\right )-\frac {x \left (2 \ln \left (5\right )-2 \ln \left (2\right )+i \pi \right )}{-3+x}\right )\)	\(33\)
parallelrisch	\(-\frac {\ln \left (x^{2}\right )}{2}+\ln \left (i \left (i \ln \left (\frac {25}{4}\right ) x -i x \ln \left (x^{2}\right )+3 i \ln \left (x^{2}\right )-\pi x \right )\right )-\ln \left (-3+x \right )\)	\(44\)

[In]

int(((-x^2+6*x-9)*ln(x^2)+x^2*(ln(25/4)+I*Pi)+2*x^2-12*x+18)/((x^3-6*x^2+9*x)*ln(x^2)+(-x^3+3*x^2)*(ln(25/4)+I
*Pi)),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(ln(x^2)-x*(2*ln(5)-2*ln(2)+I*Pi)/(-3+x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=-\frac {1}{2} \, \log \left (x^{2}\right ) + \log \left (\frac {-i \, \pi x - x \log \left (\frac {25}{4}\right ) + {\left (x - 3\right )} \log \left (x^{2}\right )}{x - 3}\right ) \]

[In]

integrate(((-x^2+6*x-9)*log(x^2)+x^2*(log(25/4)+I*pi)+2*x^2-12*x+18)/((x^3-6*x^2+9*x)*log(x^2)+(-x^3+3*x^2)*(l
og(25/4)+I*pi)),x, algorithm="fricas")

[Out]

-1/2*log(x^2) + log((-I*pi*x - x*log(25/4) + (x - 3)*log(x^2))/(x - 3))

Sympy [A] (verification not implemented)

Time = 72.48 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=- \log {\left (x \right )} + \log {\left (\log {\left (x^{2} \right )} + \frac {- 2 x \log {\left (5 \right )} + 2 x \log {\left (2 \right )} - i \pi x}{x - 3} \right )} \]

[In]

integrate(((-x**2+6*x-9)*ln(x**2)+x**2*(ln(25/4)+I*pi)+2*x**2-12*x+18)/((x**3-6*x**2+9*x)*ln(x**2)+(-x**3+3*x*
*2)*(ln(25/4)+I*pi)),x)

[Out]

-log(x) + log(log(x**2) + (-2*x*log(5) + 2*x*log(2) - I*pi*x)/(x - 3))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=-\log \left (x\right ) + \log \left (\frac {{\left (-i \, \pi - 2 \, \log \left (5\right ) + 2 \, \log \left (2\right )\right )} x + 2 \, {\left (x - 3\right )} \log \left (x\right )}{2 \, {\left (x - 3\right )}}\right ) \]

[In]

integrate(((-x^2+6*x-9)*log(x^2)+x^2*(log(25/4)+I*pi)+2*x^2-12*x+18)/((x^3-6*x^2+9*x)*log(x^2)+(-x^3+3*x^2)*(l
og(25/4)+I*pi)),x, algorithm="maxima")

[Out]

-log(x) + log(1/2*((-I*pi - 2*log(5) + 2*log(2))*x + 2*(x - 3)*log(x))/(x - 3))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=\log \left (\pi x - 2 i \, x \log \left (5\right ) + 2 i \, x \log \left (2\right ) + i \, x \log \left (x^{2}\right ) - 3 i \, \log \left (x^{2}\right )\right ) - \log \left (x - 3\right ) - \log \left (x\right ) \]

[In]

integrate(((-x^2+6*x-9)*log(x^2)+x^2*(log(25/4)+I*pi)+2*x^2-12*x+18)/((x^3-6*x^2+9*x)*log(x^2)+(-x^3+3*x^2)*(l
og(25/4)+I*pi)),x, algorithm="giac")

[Out]

log(pi*x - 2*I*x*log(5) + 2*I*x*log(2) + I*x*log(x^2) - 3*I*log(x^2)) - log(x - 3) - log(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {18-12 x+2 x^2+x^2 \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (-9+6 x-x^2\right ) \log \left (x^2\right )}{\left (3 x^2-x^3\right ) \left (i \pi +\log \left (\frac {25}{4}\right )\right )+\left (9 x-6 x^2+x^3\right ) \log \left (x^2\right )} \, dx=\int \frac {2\,x^2-\ln \left (x^2\right )\,\left (x^2-6\,x+9\right )+x^2\,\left (\ln \left (\frac {25}{4}\right )+\Pi \,1{}\mathrm {i}\right )-12\,x+18}{\left (\ln \left (\frac {25}{4}\right )+\Pi \,1{}\mathrm {i}\right )\,\left (3\,x^2-x^3\right )+\ln \left (x^2\right )\,\left (x^3-6\,x^2+9\,x\right )} \,d x \]

[In]

int((x^2*(Pi*1i + log(25/4)) - log(x^2)*(x^2 - 6*x + 9) - 12*x + 2*x^2 + 18)/((Pi*1i + log(25/4))*(3*x^2 - x^3
) + log(x^2)*(9*x - 6*x^2 + x^3)),x)

[Out]

int((x^2*(Pi*1i + log(25/4)) - log(x^2)*(x^2 - 6*x + 9) - 12*x + 2*x^2 + 18)/((Pi*1i + log(25/4))*(3*x^2 - x^3
) + log(x^2)*(9*x - 6*x^2 + x^3)), x)