Integrand size = 124, antiderivative size = 22 \[ \int \frac {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx=x^{-6/x} (-x+\log (9)) \log \left (-5+x^2+\log (x)\right ) \]
Time = 1.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx=-x^{-6/x} (x-\log (9)) \log \left (-5+x^2+\log (x)\right ) \]
Integrate[(-x^2 - 2*x^4 + (x + 2*x^3)*Log[9] + (-30*x + 5*x^2 + 6*x^3 - x^ 4 + (30 - 6*x^2)*Log[9] + (36*x - x^2 - 6*x^3 + (-36 + 6*x^2)*Log[9])*Log[ x] + (-6*x + 6*Log[9])*Log[x]^2)*Log[-5 + x^2 + Log[x]])/(x^(6/x)*(-5*x^2 + x^4 + x^2*Log[x])),x]
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.
\(\displaystyle \int \frac {x^{-6/x} \left (-2 x^4+\left (2 x^3+x\right ) \log (9)-x^2+\left (-x^4+6 x^3+5 x^2+\left (30-6 x^2\right ) \log (9)+\left (-6 x^3-x^2+\left (6 x^2-36\right ) \log (9)+36 x\right ) \log (x)-30 x+(6 \log (9)-6 x) \log ^2(x)\right ) \log \left (x^2+\log (x)-5\right )\right )}{x^4-5 x^2+x^2 \log (x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\left (x^{-\frac {6}{x}-2} \left (x^2-6 x+6 x \log (x)-6 \log (9) \log (x)+6 \log (9)\right ) \log \left (x^2+\log (x)-5\right )\right )-\frac {\left (2 x^2+1\right ) x^{-\frac {6}{x}-1} (x-\log (9))}{x^2+\log (x)-5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log (9) \int \frac {x^{-1-\frac {6}{x}}}{x^2+\log (x)-5}dx+2 \log (9) \int \frac {x^{1-\frac {6}{x}}}{x^2+\log (x)-5}dx-2 \int \frac {x^{2-\frac {6}{x}}}{x^2+\log (x)-5}dx-\int \frac {x^{-6/x}}{x^2+\log (x)-5}dx-6 \log (9) \int x^{-2-\frac {6}{x}} \log \left (x^2+\log (x)-5\right )dx+6 \int x^{-1-\frac {6}{x}} \log \left (x^2+\log (x)-5\right )dx-\int x^{-6/x} \log \left (x^2+\log (x)-5\right )dx+6 \log (9) \int x^{-2-\frac {6}{x}} \log (x) \log \left (x^2+\log (x)-5\right )dx-6 \int x^{-1-\frac {6}{x}} \log (x) \log \left (x^2+\log (x)-5\right )dx\) |
Int[(-x^2 - 2*x^4 + (x + 2*x^3)*Log[9] + (-30*x + 5*x^2 + 6*x^3 - x^4 + (3 0 - 6*x^2)*Log[9] + (36*x - x^2 - 6*x^3 + (-36 + 6*x^2)*Log[9])*Log[x] + ( -6*x + 6*Log[9])*Log[x]^2)*Log[-5 + x^2 + Log[x]])/(x^(6/x)*(-5*x^2 + x^4 + x^2*Log[x])),x]
3.30.75.3.1 Defintions of rubi rules used
Time = 16.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23
method | result | size |
risch | \(\ln \left (\ln \left (x \right )+x^{2}-5\right ) x^{-\frac {6}{x}} \left (2 \ln \left (3\right )-x \right )\) | \(27\) |
parallelrisch | \(\frac {\left (2 \ln \left (3\right ) \ln \left (\ln \left (x \right )+x^{2}-5\right ) x -\ln \left (\ln \left (x \right )+x^{2}-5\right ) x^{2}\right ) {\mathrm e}^{-\frac {6 \ln \left (x \right )}{x}}}{x}\) | \(42\) |
int((((12*ln(3)-6*x)*ln(x)^2+(2*(6*x^2-36)*ln(3)-6*x^3-x^2+36*x)*ln(x)+2*( -6*x^2+30)*ln(3)-x^4+6*x^3+5*x^2-30*x)*ln(ln(x)+x^2-5)+2*(2*x^3+x)*ln(3)-2 *x^4-x^2)/(x^2*ln(x)+x^4-5*x^2)/exp(6*ln(x)/x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx=-\frac {{\left (x - 2 \, \log \left (3\right )\right )} \log \left (x^{2} + \log \left (x\right ) - 5\right )}{x^{\frac {6}{x}}} \]
integrate((((12*log(3)-6*x)*log(x)^2+(2*(6*x^2-36)*log(3)-6*x^3-x^2+36*x)* log(x)+2*(-6*x^2+30)*log(3)-x^4+6*x^3+5*x^2-30*x)*log(log(x)+x^2-5)+2*(2*x ^3+x)*log(3)-2*x^4-x^2)/(x^2*log(x)+x^4-5*x^2)/exp(6*log(x)/x),x, algorith m=\
Time = 16.40 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx=\left (- x \log {\left (x^{2} + \log {\left (x \right )} - 5 \right )} + 2 \log {\left (3 \right )} \log {\left (x^{2} + \log {\left (x \right )} - 5 \right )}\right ) e^{- \frac {6 \log {\left (x \right )}}{x}} \]
integrate((((12*ln(3)-6*x)*ln(x)**2+(2*(6*x**2-36)*ln(3)-6*x**3-x**2+36*x) *ln(x)+2*(-6*x**2+30)*ln(3)-x**4+6*x**3+5*x**2-30*x)*ln(ln(x)+x**2-5)+2*(2 *x**3+x)*ln(3)-2*x**4-x**2)/(x**2*ln(x)+x**4-5*x**2)/exp(6*ln(x)/x),x)
Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx=-\frac {{\left (x - 2 \, \log \left (3\right )\right )} \log \left (x^{2} + \log \left (x\right ) - 5\right )}{x^{\frac {6}{x}}} \]
integrate((((12*log(3)-6*x)*log(x)^2+(2*(6*x^2-36)*log(3)-6*x^3-x^2+36*x)* log(x)+2*(-6*x^2+30)*log(3)-x^4+6*x^3+5*x^2-30*x)*log(log(x)+x^2-5)+2*(2*x ^3+x)*log(3)-2*x^4-x^2)/(x^2*log(x)+x^4-5*x^2)/exp(6*log(x)/x),x, algorith m=\
Time = 1.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx=-\frac {x \log \left (x^{2} + \log \left (x\right ) - 5\right )}{x^{\frac {6}{x}}} + \frac {2 \, \log \left (3\right ) \log \left (x^{2} + \log \left (x\right ) - 5\right )}{x^{\frac {6}{x}}} \]
integrate((((12*log(3)-6*x)*log(x)^2+(2*(6*x^2-36)*log(3)-6*x^3-x^2+36*x)* log(x)+2*(-6*x^2+30)*log(3)-x^4+6*x^3+5*x^2-30*x)*log(log(x)+x^2-5)+2*(2*x ^3+x)*log(3)-2*x^4-x^2)/(x^2*log(x)+x^4-5*x^2)/exp(6*log(x)/x),x, algorith m=\
Time = 9.75 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {x^{-6/x} \left (-x^2-2 x^4+\left (x+2 x^3\right ) \log (9)+\left (-30 x+5 x^2+6 x^3-x^4+\left (30-6 x^2\right ) \log (9)+\left (36 x-x^2-6 x^3+\left (-36+6 x^2\right ) \log (9)\right ) \log (x)+(-6 x+6 \log (9)) \log ^2(x)\right ) \log \left (-5+x^2+\log (x)\right )\right )}{-5 x^2+x^4+x^2 \log (x)} \, dx=-\frac {\ln \left (\ln \left (x\right )+x^2-5\right )\,\left (x-\ln \left (9\right )\right )}{x^{6/x}} \]