Integrand size = 132, antiderivative size = 27 \[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\log \left (\log \left (x-x^3-\log \left (5 \left (4+\left (x-\frac {x^2}{4}\right )^2\right )\right )\right )\right ) \] Output:
ln(ln(x-ln(20+5*(x-1/4*x^2)^2)-x^3))
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\log \left (\log \left (x-x^3-\log \left (\frac {5}{16} \left (64+16 x^2-8 x^3+x^4\right )\right )\right )\right ) \] Input:
Integrate[(-64 + 32*x + 152*x^2 + 12*x^3 + 47*x^4 - 24*x^5 + 3*x^6)/((-64* x + 48*x^3 + 8*x^4 + 15*x^5 - 8*x^6 + x^7 + (64 + 16*x^2 - 8*x^3 + x^4)*Lo g[(320 + 80*x^2 - 40*x^3 + 5*x^4)/16])*Log[x - x^3 - Log[(320 + 80*x^2 - 4 0*x^3 + 5*x^4)/16]]),x]
Output:
Log[Log[x - x^3 - Log[(5*(64 + 16*x^2 - 8*x^3 + x^4))/16]]]
Time = 0.45 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {7235}
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.
\(\displaystyle \int \frac {3 x^6-24 x^5+47 x^4+12 x^3+152 x^2+32 x-64}{\left (x^7-8 x^6+15 x^5+8 x^4+48 x^3+\left (x^4-8 x^3+16 x^2+64\right ) \log \left (\frac {1}{16} \left (5 x^4-40 x^3+80 x^2+320\right )\right )-64 x\right ) \log \left (-x^3-\log \left (\frac {1}{16} \left (5 x^4-40 x^3+80 x^2+320\right )\right )+x\right )} \, dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (\log \left (-x^3-\log \left (\frac {5}{16} \left (x^4-8 x^3+16 x^2+64\right )\right )+x\right )\right )\) |
Input:
Int[(-64 + 32*x + 152*x^2 + 12*x^3 + 47*x^4 - 24*x^5 + 3*x^6)/((-64*x + 48 *x^3 + 8*x^4 + 15*x^5 - 8*x^6 + x^7 + (64 + 16*x^2 - 8*x^3 + x^4)*Log[(320 + 80*x^2 - 40*x^3 + 5*x^4)/16])*Log[x - x^3 - Log[(320 + 80*x^2 - 40*x^3 + 5*x^4)/16]]),x]
Output:
Log[Log[x - x^3 - Log[(5*(64 + 16*x^2 - 8*x^3 + x^4))/16]]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 9.76 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\ln \left (\ln \left (-\ln \left (\frac {5}{16} x^{4}-\frac {5}{2} x^{3}+5 x^{2}+20\right )-x^{3}+x \right )\right )\) | \(30\) |
parallelrisch | \(\ln \left (\ln \left (-\ln \left (\frac {5}{16} x^{4}-\frac {5}{2} x^{3}+5 x^{2}+20\right )-x^{3}+x \right )\right )\) | \(30\) |
Input:
int((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+64)*ln (5/16*x^4-5/2*x^3+5*x^2+20)+x^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/ln(-ln(5/1 6*x^4-5/2*x^3+5*x^2+20)-x^3+x),x,method=_RETURNVERBOSE)
Output:
ln(ln(-ln(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x))
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\log \left (\log \left (-x^{3} + x - \log \left (\frac {5}{16} \, x^{4} - \frac {5}{2} \, x^{3} + 5 \, x^{2} + 20\right )\right )\right ) \] Input:
integrate((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+ 64)*log(5/16*x^4-5/2*x^3+5*x^2+20)+x^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/log (-log(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x, algorithm="fricas")
Output:
log(log(-x^3 + x - log(5/16*x^4 - 5/2*x^3 + 5*x^2 + 20)))
Time = 0.71 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\log {\left (\log {\left (- x^{3} + x - \log {\left (\frac {5 x^{4}}{16} - \frac {5 x^{3}}{2} + 5 x^{2} + 20 \right )} \right )} \right )} \] Input:
integrate((3*x**6-24*x**5+47*x**4+12*x**3+152*x**2+32*x-64)/((x**4-8*x**3+ 16*x**2+64)*ln(5/16*x**4-5/2*x**3+5*x**2+20)+x**7-8*x**6+15*x**5+8*x**4+48 *x**3-64*x)/ln(-ln(5/16*x**4-5/2*x**3+5*x**2+20)-x**3+x),x)
Output:
log(log(-x**3 + x - log(5*x**4/16 - 5*x**3/2 + 5*x**2 + 20)))
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\log \left (\log \left (-x^{3} + x - \log \left (5\right ) + 4 \, \log \left (2\right ) - \log \left (x^{4} - 8 \, x^{3} + 16 \, x^{2} + 64\right )\right )\right ) \] Input:
integrate((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+ 64)*log(5/16*x^4-5/2*x^3+5*x^2+20)+x^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/log (-log(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x, algorithm="maxima")
Output:
log(log(-x^3 + x - log(5) + 4*log(2) - log(x^4 - 8*x^3 + 16*x^2 + 64)))
\[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\int { \frac {3 \, x^{6} - 24 \, x^{5} + 47 \, x^{4} + 12 \, x^{3} + 152 \, x^{2} + 32 \, x - 64}{{\left (x^{7} - 8 \, x^{6} + 15 \, x^{5} + 8 \, x^{4} + 48 \, x^{3} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2} + 64\right )} \log \left (\frac {5}{16} \, x^{4} - \frac {5}{2} \, x^{3} + 5 \, x^{2} + 20\right ) - 64 \, x\right )} \log \left (-x^{3} + x - \log \left (\frac {5}{16} \, x^{4} - \frac {5}{2} \, x^{3} + 5 \, x^{2} + 20\right )\right )} \,d x } \] Input:
integrate((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+ 64)*log(5/16*x^4-5/2*x^3+5*x^2+20)+x^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/log (-log(5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x, algorithm="giac")
Output:
integrate((3*x^6 - 24*x^5 + 47*x^4 + 12*x^3 + 152*x^2 + 32*x - 64)/((x^7 - 8*x^6 + 15*x^5 + 8*x^4 + 48*x^3 + (x^4 - 8*x^3 + 16*x^2 + 64)*log(5/16*x^ 4 - 5/2*x^3 + 5*x^2 + 20) - 64*x)*log(-x^3 + x - log(5/16*x^4 - 5/2*x^3 + 5*x^2 + 20))), x)
Time = 2.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\ln \left (\ln \left (x-\ln \left (\frac {5\,x^4}{16}-\frac {5\,x^3}{2}+5\,x^2+20\right )-x^3\right )\right ) \] Input:
int((32*x + 152*x^2 + 12*x^3 + 47*x^4 - 24*x^5 + 3*x^6 - 64)/(log(x - log( 5*x^2 - (5*x^3)/2 + (5*x^4)/16 + 20) - x^3)*(log(5*x^2 - (5*x^3)/2 + (5*x^ 4)/16 + 20)*(16*x^2 - 8*x^3 + x^4 + 64) - 64*x + 48*x^3 + 8*x^4 + 15*x^5 - 8*x^6 + x^7)),x)
Output:
log(log(x - log(5*x^2 - (5*x^3)/2 + (5*x^4)/16 + 20) - x^3))
Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {-64+32 x+152 x^2+12 x^3+47 x^4-24 x^5+3 x^6}{\left (-64 x+48 x^3+8 x^4+15 x^5-8 x^6+x^7+\left (64+16 x^2-8 x^3+x^4\right ) \log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right ) \log \left (x-x^3-\log \left (\frac {1}{16} \left (320+80 x^2-40 x^3+5 x^4\right )\right )\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (-\mathrm {log}\left (\frac {5}{16} x^{4}-\frac {5}{2} x^{3}+5 x^{2}+20\right )-x^{3}+x \right )\right ) \] Input:
int((3*x^6-24*x^5+47*x^4+12*x^3+152*x^2+32*x-64)/((x^4-8*x^3+16*x^2+64)*lo g(5/16*x^4-5/2*x^3+5*x^2+20)+x^7-8*x^6+15*x^5+8*x^4+48*x^3-64*x)/log(-log( 5/16*x^4-5/2*x^3+5*x^2+20)-x^3+x),x)
Output:
log(log( - log((5*x**4 - 40*x**3 + 80*x**2 + 320)/16) - x**3 + x))