Integrand size = 130, antiderivative size = 30 \[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx=\frac {(x+5 x \log (3)) \log \left (x+\frac {x}{2+x-\log (x)}+\log (x)\right )}{2 x} \] Output:
1/2*(5*x*ln(3)+x)*ln(ln(x)+x/(2-ln(x)+x)+x)/x
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx=\frac {1}{2} (1+\log (243)) \left (-\log (2+x-\log (x))+\log \left (3 x+x^2+2 \log (x)-\log ^2(x)\right )\right ) \] Input:
Integrate[(4 + 11*x + 5*x^2 + x^3 + (20 + 55*x + 25*x^2 + 5*x^3)*Log[3] + (-4 - 7*x - 2*x^2 + (-20 - 35*x - 10*x^2)*Log[3])*Log[x] + (1 + x + (5 + 5 *x)*Log[3])*Log[x]^2)/(12*x^2 + 10*x^3 + 2*x^4 + (8*x - 2*x^2 - 2*x^3)*Log [x] + (-8*x - 2*x^2)*Log[x]^2 + 2*x*Log[x]^3),x]
Output:
((1 + Log[243])*(-Log[2 + x - Log[x]] + Log[3*x + x^2 + 2*Log[x] - Log[x]^ 2]))/2
Time = 1.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {7292, 27, 7293, 2009}
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.
\(\displaystyle \int \frac {x^3+5 x^2+\left (-2 x^2+\left (-10 x^2-35 x-20\right ) \log (3)-7 x-4\right ) \log (x)+\left (5 x^3+25 x^2+55 x+20\right ) \log (3)+11 x+(x+(5 x+5) \log (3)+1) \log ^2(x)+4}{2 x^4+10 x^3+12 x^2+\left (-2 x^2-8 x\right ) \log ^2(x)+\left (-2 x^3-2 x^2+8 x\right ) \log (x)+2 x \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {(1+\log (243)) \left (x^3+5 x^2-2 x^2 \log (x)+11 x+x \log ^2(x)+\log ^2(x)-7 x \log (x)-4 \log (x)+4\right )}{2 x \left (x^3+5 x^2-x^2 \log (x)+6 x+\log ^3(x)-x \log ^2(x)-4 \log ^2(x)-x \log (x)+4 \log (x)\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} (1+\log (243)) \int \frac {x^3-2 \log (x) x^2+5 x^2+\log ^2(x) x-7 \log (x) x+11 x+\log ^2(x)-4 \log (x)+4}{x \left (x^3-\log (x) x^2+5 x^2-\log ^2(x) x-\log (x) x+6 x+\log ^3(x)-4 \log ^2(x)+4 \log (x)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} (1+\log (243)) \int \left (\frac {1-x}{x (x-\log (x)+2)}+\frac {2 x^2+3 x-2 \log (x)+2}{x \left (x^2+3 x-\log ^2(x)+2 \log (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} (1+\log (243)) \left (\log \left (x^2+3 x-\log ^2(x)+2 \log (x)\right )-\log (x-\log (x)+2)\right )\) |
Input:
Int[(4 + 11*x + 5*x^2 + x^3 + (20 + 55*x + 25*x^2 + 5*x^3)*Log[3] + (-4 - 7*x - 2*x^2 + (-20 - 35*x - 10*x^2)*Log[3])*Log[x] + (1 + x + (5 + 5*x)*Lo g[3])*Log[x]^2)/(12*x^2 + 10*x^3 + 2*x^4 + (8*x - 2*x^2 - 2*x^3)*Log[x] + (-8*x - 2*x^2)*Log[x]^2 + 2*x*Log[x]^3),x]
Output:
((1 + Log[243])*(-Log[2 + x - Log[x]] + Log[3*x + x^2 + 2*Log[x] - Log[x]^ 2]))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.70 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40
method | result | size |
norman | \(\left (-\frac {5 \ln \left (3\right )}{2}-\frac {1}{2}\right ) \ln \left (2-\ln \left (x \right )+x \right )+\left (\frac {5 \ln \left (3\right )}{2}+\frac {1}{2}\right ) \ln \left (x^{2}-\ln \left (x \right )^{2}+3 x +2 \ln \left (x \right )\right )\) | \(42\) |
default | \(-\frac {\ln \left (-x -2+\ln \left (x \right )\right )}{2}+\frac {\ln \left (\ln \left (x \right )^{2}-x^{2}-2 \ln \left (x \right )-3 x \right )}{2}+\frac {5 \ln \left (3\right ) \left (-\ln \left (-x -2+\ln \left (x \right )\right )+\ln \left (\ln \left (x \right )^{2}-x^{2}-2 \ln \left (x \right )-3 x \right )\right )}{2}\) | \(65\) |
risch | \(-\frac {5 \ln \left (-x -2+\ln \left (x \right )\right ) \ln \left (3\right )}{2}-\frac {\ln \left (-x -2+\ln \left (x \right )\right )}{2}+\frac {5 \ln \left (\ln \left (x \right )^{2}-x^{2}-2 \ln \left (x \right )-3 x \right ) \ln \left (3\right )}{2}+\frac {\ln \left (\ln \left (x \right )^{2}-x^{2}-2 \ln \left (x \right )-3 x \right )}{2}\) | \(66\) |
parallelrisch | \(-\frac {5 \ln \left (2-\ln \left (x \right )+x \right ) \ln \left (3\right )}{2}-\frac {\ln \left (2-\ln \left (x \right )+x \right )}{2}+\frac {5 \ln \left (x^{2}-\ln \left (x \right )^{2}+3 x +2 \ln \left (x \right )\right ) \ln \left (3\right )}{2}+\frac {\ln \left (x^{2}-\ln \left (x \right )^{2}+3 x +2 \ln \left (x \right )\right )}{2}\) | \(66\) |
Input:
int((((5*x+5)*ln(3)+x+1)*ln(x)^2+((-10*x^2-35*x-20)*ln(3)-2*x^2-7*x-4)*ln( x)+(5*x^3+25*x^2+55*x+20)*ln(3)+x^3+5*x^2+11*x+4)/(2*x*ln(x)^3+(-2*x^2-8*x )*ln(x)^2+(-2*x^3-2*x^2+8*x)*ln(x)+2*x^4+10*x^3+12*x^2),x,method=_RETURNVE RBOSE)
Output:
(-5/2*ln(3)-1/2)*ln(2-ln(x)+x)+(5/2*ln(3)+1/2)*ln(x^2-ln(x)^2+3*x+2*ln(x))
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx=\frac {1}{2} \, {\left (5 \, \log \left (3\right ) + 1\right )} \log \left (-x^{2} + \log \left (x\right )^{2} - 3 \, x - 2 \, \log \left (x\right )\right ) - \frac {1}{2} \, {\left (5 \, \log \left (3\right ) + 1\right )} \log \left (-x + \log \left (x\right ) - 2\right ) \] Input:
integrate((((5*x+5)*log(3)+x+1)*log(x)^2+((-10*x^2-35*x-20)*log(3)-2*x^2-7 *x-4)*log(x)+(5*x^3+25*x^2+55*x+20)*log(3)+x^3+5*x^2+11*x+4)/(2*x*log(x)^3 +(-2*x^2-8*x)*log(x)^2+(-2*x^3-2*x^2+8*x)*log(x)+2*x^4+10*x^3+12*x^2),x, a lgorithm="fricas")
Output:
1/2*(5*log(3) + 1)*log(-x^2 + log(x)^2 - 3*x - 2*log(x)) - 1/2*(5*log(3) + 1)*log(-x + log(x) - 2)
Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx=- \frac {\left (1 + 5 \log {\left (3 \right )}\right ) \log {\left (- x + \log {\left (x \right )} - 2 \right )}}{2} + \frac {\left (1 + 5 \log {\left (3 \right )}\right ) \log {\left (- x^{2} - 3 x + \log {\left (x \right )}^{2} - 2 \log {\left (x \right )} \right )}}{2} \] Input:
integrate((((5*x+5)*ln(3)+x+1)*ln(x)**2+((-10*x**2-35*x-20)*ln(3)-2*x**2-7 *x-4)*ln(x)+(5*x**3+25*x**2+55*x+20)*ln(3)+x**3+5*x**2+11*x+4)/(2*x*ln(x)* *3+(-2*x**2-8*x)*ln(x)**2+(-2*x**3-2*x**2+8*x)*ln(x)+2*x**4+10*x**3+12*x** 2),x)
Output:
-(1 + 5*log(3))*log(-x + log(x) - 2)/2 + (1 + 5*log(3))*log(-x**2 - 3*x + log(x)**2 - 2*log(x))/2
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx=\frac {1}{2} \, {\left (5 \, \log \left (3\right ) + 1\right )} \log \left (-x^{2} + \log \left (x\right )^{2} - 3 \, x - 2 \, \log \left (x\right )\right ) - \frac {1}{2} \, {\left (5 \, \log \left (3\right ) + 1\right )} \log \left (-x + \log \left (x\right ) - 2\right ) \] Input:
integrate((((5*x+5)*log(3)+x+1)*log(x)^2+((-10*x^2-35*x-20)*log(3)-2*x^2-7 *x-4)*log(x)+(5*x^3+25*x^2+55*x+20)*log(3)+x^3+5*x^2+11*x+4)/(2*x*log(x)^3 +(-2*x^2-8*x)*log(x)^2+(-2*x^3-2*x^2+8*x)*log(x)+2*x^4+10*x^3+12*x^2),x, a lgorithm="maxima")
Output:
1/2*(5*log(3) + 1)*log(-x^2 + log(x)^2 - 3*x - 2*log(x)) - 1/2*(5*log(3) + 1)*log(-x + log(x) - 2)
Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx=\frac {1}{2} \, {\left (5 \, \log \left (3\right ) + 1\right )} \log \left (x^{2} - \log \left (x\right )^{2} + 3 \, x + 2 \, \log \left (x\right )\right ) - \frac {1}{2} \, {\left (5 \, \log \left (3\right ) + 1\right )} \log \left (x - \log \left (x\right ) + 2\right ) \] Input:
integrate((((5*x+5)*log(3)+x+1)*log(x)^2+((-10*x^2-35*x-20)*log(3)-2*x^2-7 *x-4)*log(x)+(5*x^3+25*x^2+55*x+20)*log(3)+x^3+5*x^2+11*x+4)/(2*x*log(x)^3 +(-2*x^2-8*x)*log(x)^2+(-2*x^3-2*x^2+8*x)*log(x)+2*x^4+10*x^3+12*x^2),x, a lgorithm="giac")
Output:
1/2*(5*log(3) + 1)*log(x^2 - log(x)^2 + 3*x + 2*log(x)) - 1/2*(5*log(3) + 1)*log(x - log(x) + 2)
Time = 1.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx=\left (\frac {\ln \left (243\right )}{2}+\frac {1}{2}\right )\,\left (\ln \left (x^2+3\,x-{\ln \left (x\right )}^2+2\,\ln \left (x\right )\right )-\ln \left (x-\ln \left (x\right )+2\right )\right ) \] Input:
int((11*x + log(3)*(55*x + 25*x^2 + 5*x^3 + 20) + log(x)^2*(x + log(3)*(5* x + 5) + 1) + 5*x^2 + x^3 - log(x)*(7*x + log(3)*(35*x + 10*x^2 + 20) + 2* x^2 + 4) + 4)/(2*x*log(x)^3 - log(x)^2*(8*x + 2*x^2) + 12*x^2 + 10*x^3 + 2 *x^4 - log(x)*(2*x^2 - 8*x + 2*x^3)),x)
Output:
(log(243)/2 + 1/2)*(log(3*x + 2*log(x) - log(x)^2 + x^2) - log(x - log(x) + 2))
\[ \int \frac {4+11 x+5 x^2+x^3+\left (20+55 x+25 x^2+5 x^3\right ) \log (3)+\left (-4-7 x-2 x^2+\left (-20-35 x-10 x^2\right ) \log (3)\right ) \log (x)+(1+x+(5+5 x) \log (3)) \log ^2(x)}{12 x^2+10 x^3+2 x^4+\left (8 x-2 x^2-2 x^3\right ) \log (x)+\left (-8 x-2 x^2\right ) \log ^2(x)+2 x \log ^3(x)} \, dx =\text {Too large to display} \] Input:
int((((5*x+5)*log(3)+x+1)*log(x)^2+((-10*x^2-35*x-20)*log(3)-2*x^2-7*x-4)* log(x)+(5*x^3+25*x^2+55*x+20)*log(3)+x^3+5*x^2+11*x+4)/(2*x*log(x)^3+(-2*x ^2-8*x)*log(x)^2+(-2*x^3-2*x^2+8*x)*log(x)+2*x^4+10*x^3+12*x^2),x)
Output:
(5*int(log(x)**2/(log(x)**3 - log(x)**2*x - 4*log(x)**2 - log(x)*x**2 - lo g(x)*x + 4*log(x) + x**3 + 5*x**2 + 6*x),x)*log(3) + int(log(x)**2/(log(x) **3 - log(x)**2*x - 4*log(x)**2 - log(x)*x**2 - log(x)*x + 4*log(x) + x**3 + 5*x**2 + 6*x),x) + 5*int(log(x)**2/(log(x)**3*x - log(x)**2*x**2 - 4*lo g(x)**2*x - log(x)*x**3 - log(x)*x**2 + 4*log(x)*x + x**4 + 5*x**3 + 6*x** 2),x)*log(3) + int(log(x)**2/(log(x)**3*x - log(x)**2*x**2 - 4*log(x)**2*x - log(x)*x**3 - log(x)*x**2 + 4*log(x)*x + x**4 + 5*x**3 + 6*x**2),x) + 5 *int(x**2/(log(x)**3 - log(x)**2*x - 4*log(x)**2 - log(x)*x**2 - log(x)*x + 4*log(x) + x**3 + 5*x**2 + 6*x),x)*log(3) + int(x**2/(log(x)**3 - log(x) **2*x - 4*log(x)**2 - log(x)*x**2 - log(x)*x + 4*log(x) + x**3 + 5*x**2 + 6*x),x) - 35*int(log(x)/(log(x)**3 - log(x)**2*x - 4*log(x)**2 - log(x)*x* *2 - log(x)*x + 4*log(x) + x**3 + 5*x**2 + 6*x),x)*log(3) - 7*int(log(x)/( log(x)**3 - log(x)**2*x - 4*log(x)**2 - log(x)*x**2 - log(x)*x + 4*log(x) + x**3 + 5*x**2 + 6*x),x) - 20*int(log(x)/(log(x)**3*x - log(x)**2*x**2 - 4*log(x)**2*x - log(x)*x**3 - log(x)*x**2 + 4*log(x)*x + x**4 + 5*x**3 + 6 *x**2),x)*log(3) - 4*int(log(x)/(log(x)**3*x - log(x)**2*x**2 - 4*log(x)** 2*x - log(x)*x**3 - log(x)*x**2 + 4*log(x)*x + x**4 + 5*x**3 + 6*x**2),x) - 10*int((log(x)*x)/(log(x)**3 - log(x)**2*x - 4*log(x)**2 - log(x)*x**2 - log(x)*x + 4*log(x) + x**3 + 5*x**2 + 6*x),x)*log(3) - 2*int((log(x)*x)/( log(x)**3 - log(x)**2*x - 4*log(x)**2 - log(x)*x**2 - log(x)*x + 4*log(...