Integrand size = 105, antiderivative size = 20 \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=x+\frac {5}{(-x+\log (x)) \log \left (-x^2\right )} \] Output:
x+5/ln(-x^2)/(ln(x)-x)
Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=x-\frac {5}{(x-\log (x)) \log \left (-x^2\right )} \] Input:
Integrate[(10*x + (-5 + 5*x)*Log[-x^2] + x^3*Log[-x^2]^2 + x*Log[x]^2*Log[ -x^2]^2 + Log[x]*(-10 - 2*x^2*Log[-x^2]^2))/(x^3*Log[-x^2]^2 - 2*x^2*Log[x ]*Log[-x^2]^2 + x*Log[x]^2*Log[-x^2]^2),x]
Output:
x - 5/((x - Log[x])*Log[-x^2])
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.
\(\displaystyle \int \frac {x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-2 x^2 \log ^2\left (-x^2\right )-10\right )+(5 x-5) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+10 x}{-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+x^3 \log ^2\left (-x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-2 x^2 \log ^2\left (-x^2\right )-10\right )+(5 x-5) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+10 x}{x (x-\log (x))^2 \log ^2\left (-x^2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {10}{x (x-\log (x)) \log ^2\left (-x^2\right )}+\frac {5 (x-1)}{x (x-\log (x))^2 \log \left (-x^2\right )}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 10 \int \frac {1}{x (x-\log (x)) \log ^2\left (-x^2\right )}dx+5 \int \frac {1}{(x-\log (x))^2 \log \left (-x^2\right )}dx-5 \int \frac {1}{x (x-\log (x))^2 \log \left (-x^2\right )}dx+x\) |
Input:
Int[(10*x + (-5 + 5*x)*Log[-x^2] + x^3*Log[-x^2]^2 + x*Log[x]^2*Log[-x^2]^ 2 + Log[x]*(-10 - 2*x^2*Log[-x^2]^2))/(x^3*Log[-x^2]^2 - 2*x^2*Log[x]*Log[ -x^2]^2 + x*Log[x]^2*Log[-x^2]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(42\) vs. \(2(20)=40\).
Time = 8.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.15
method | result | size |
parallelrisch | \(\frac {-60+12 x^{2} \ln \left (-x^{2}\right )-12 x \ln \left (-x^{2}\right ) \ln \left (x \right )}{12 \ln \left (-x^{2}\right ) \left (x -\ln \left (x \right )\right )}\) | \(43\) |
risch | \(x +\frac {10 i}{-2 x \pi \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi x \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{2}-\ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+4 i \ln \left (x \right )^{2}-4 i x \ln \left (x \right )+2 \pi x -2 \pi \ln \left (x \right )}\) | \(157\) |
Input:
int((x*ln(-x^2)^2*ln(x)^2+(-2*x^2*ln(-x^2)^2-10)*ln(x)+x^3*ln(-x^2)^2+(5*x -5)*ln(-x^2)+10*x)/(x*ln(-x^2)^2*ln(x)^2-2*x^2*ln(-x^2)^2*ln(x)+x^3*ln(-x^ 2)^2),x,method=_RETURNVERBOSE)
Output:
1/12*(-60+12*x^2*ln(-x^2)-12*x*ln(-x^2)*ln(x))/ln(-x^2)/(x-ln(x))
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=\frac {x \log \left (-x^{2}\right )^{2} + {\left (i \, \pi x - 2 \, x^{2}\right )} \log \left (-x^{2}\right ) + 10}{{\left (i \, \pi - 2 \, x\right )} \log \left (-x^{2}\right ) + \log \left (-x^{2}\right )^{2}} \] Input:
integrate((x*log(-x^2)^2*log(x)^2+(-2*x^2*log(-x^2)^2-10)*log(x)+x^3*log(- x^2)^2+(5*x-5)*log(-x^2)+10*x)/(x*log(-x^2)^2*log(x)^2-2*x^2*log(-x^2)^2*l og(x)+x^3*log(-x^2)^2),x, algorithm="fricas")
Output:
(x*log(-x^2)^2 + (I*pi*x - 2*x^2)*log(-x^2) + 10)/((I*pi - 2*x)*log(-x^2) + log(-x^2)^2)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=x + \frac {5}{- i \pi x + \left (- 2 x + i \pi \right ) \log {\left (x \right )} + 2 \log {\left (x \right )}^{2}} \] Input:
integrate((x*ln(-x**2)**2*ln(x)**2+(-2*x**2*ln(-x**2)**2-10)*ln(x)+x**3*ln (-x**2)**2+(5*x-5)*ln(-x**2)+10*x)/(x*ln(-x**2)**2*ln(x)**2-2*x**2*ln(-x** 2)**2*ln(x)+x**3*ln(-x**2)**2),x)
Output:
x + 5/(-I*pi*x + (-2*x + I*pi)*log(x) + 2*log(x)**2)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.60 \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=\frac {-i \, \pi x^{2} + 2 \, x \log \left (x\right )^{2} + {\left (i \, \pi x - 2 \, x^{2}\right )} \log \left (x\right ) + 5}{-i \, \pi x + {\left (i \, \pi - 2 \, x\right )} \log \left (x\right ) + 2 \, \log \left (x\right )^{2}} \] Input:
integrate((x*log(-x^2)^2*log(x)^2+(-2*x^2*log(-x^2)^2-10)*log(x)+x^3*log(- x^2)^2+(5*x-5)*log(-x^2)+10*x)/(x*log(-x^2)^2*log(x)^2-2*x^2*log(-x^2)^2*l og(x)+x^3*log(-x^2)^2),x, algorithm="maxima")
Output:
(-I*pi*x^2 + 2*x*log(x)^2 + (I*pi*x - 2*x^2)*log(x) + 5)/(-I*pi*x + (I*pi - 2*x)*log(x) + 2*log(x)^2)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=x + \frac {5}{-i \, \pi x + i \, \pi \log \left (x\right ) - 2 \, x \log \left (x\right ) + 2 \, \log \left (x\right )^{2}} \] Input:
integrate((x*log(-x^2)^2*log(x)^2+(-2*x^2*log(-x^2)^2-10)*log(x)+x^3*log(- x^2)^2+(5*x-5)*log(-x^2)+10*x)/(x*log(-x^2)^2*log(x)^2-2*x^2*log(-x^2)^2*l og(x)+x^3*log(-x^2)^2),x, algorithm="giac")
Output:
x + 5/(-I*pi*x + I*pi*log(x) - 2*x*log(x) + 2*log(x)^2)
Timed out. \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=\int \frac {10\,x+\ln \left (-x^2\right )\,\left (5\,x-5\right )+x^3\,{\ln \left (-x^2\right )}^2-\ln \left (x\right )\,\left (2\,x^2\,{\ln \left (-x^2\right )}^2+10\right )+x\,{\ln \left (-x^2\right )}^2\,{\ln \left (x\right )}^2}{x^3\,{\ln \left (-x^2\right )}^2-2\,x^2\,{\ln \left (-x^2\right )}^2\,\ln \left (x\right )+x\,{\ln \left (-x^2\right )}^2\,{\ln \left (x\right )}^2} \,d x \] Input:
int((10*x + log(-x^2)*(5*x - 5) + x^3*log(-x^2)^2 - log(x)*(2*x^2*log(-x^2 )^2 + 10) + x*log(-x^2)^2*log(x)^2)/(x^3*log(-x^2)^2 + x*log(-x^2)^2*log(x )^2 - 2*x^2*log(-x^2)^2*log(x)),x)
Output:
int((10*x + log(-x^2)*(5*x - 5) + x^3*log(-x^2)^2 - log(x)*(2*x^2*log(-x^2 )^2 + 10) + x*log(-x^2)^2*log(x)^2)/(x^3*log(-x^2)^2 + x*log(-x^2)^2*log(x )^2 - 2*x^2*log(-x^2)^2*log(x)), x)
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.00 \[ \int \frac {10 x+(-5+5 x) \log \left (-x^2\right )+x^3 \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )+\log (x) \left (-10-2 x^2 \log ^2\left (-x^2\right )\right )}{x^3 \log ^2\left (-x^2\right )-2 x^2 \log (x) \log ^2\left (-x^2\right )+x \log ^2(x) \log ^2\left (-x^2\right )} \, dx=\frac {\mathrm {log}\left (-x^{2}\right ) \mathrm {log}\left (x \right ) x -\mathrm {log}\left (-x^{2}\right ) x^{2}+5}{\mathrm {log}\left (-x^{2}\right ) \left (\mathrm {log}\left (x \right )-x \right )} \] Input:
int((x*log(-x^2)^2*log(x)^2+(-2*x^2*log(-x^2)^2-10)*log(x)+x^3*log(-x^2)^2 +(5*x-5)*log(-x^2)+10*x)/(x*log(-x^2)^2*log(x)^2-2*x^2*log(-x^2)^2*log(x)+ x^3*log(-x^2)^2),x)
Output:
(log( - x**2)*log(x)*x - log( - x**2)*x**2 + 5)/(log( - x**2)*(log(x) - x) )