Integrand size = 109, antiderivative size = 21 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\log ((2-2 x) x)+\frac {1}{\left (3-\log ^2\left (x^2\right )\right )^2} \] Output:
ln((2-2*x)*x)+1/(3-ln(x^2)^2)^2
Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\log (1-x)+\log (x)+\frac {1}{\left (-3+\log ^2\left (x^2\right )\right )^2} \] Input:
Integrate[(27 - 54*x + (8 - 8*x)*Log[x^2] + (-27 + 54*x)*Log[x^2]^2 + (9 - 18*x)*Log[x^2]^4 + (-1 + 2*x)*Log[x^2]^6)/(27*x - 27*x^2 + (-27*x + 27*x^ 2)*Log[x^2]^2 + (9*x - 9*x^2)*Log[x^2]^4 + (-x + x^2)*Log[x^2]^6),x]
Output:
Log[1 - x] + Log[x] + (-3 + Log[x^2]^2)^(-2)
Time = 0.79 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {7292, 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.
\(\displaystyle \int \frac {(2 x-1) \log ^6\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(54 x-27) \log ^2\left (x^2\right )+(8-8 x) \log \left (x^2\right )-54 x+27}{-27 x^2+\left (x^2-x\right ) \log ^6\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (27 x^2-27 x\right ) \log ^2\left (x^2\right )+27 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {(2 x-1) \log ^6\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(54 x-27) \log ^2\left (x^2\right )+(8-8 x) \log \left (x^2\right )-54 x+27}{(1-x) x \left (3-\log ^2\left (x^2\right )\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x-1}{(x-1) x}-\frac {8 \log \left (x^2\right )}{x \left (\log ^2\left (x^2\right )-3\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{\left (3-\log ^2\left (x^2\right )\right )^2}+\log (1-x)+\log (x)\) |
Input:
Int[(27 - 54*x + (8 - 8*x)*Log[x^2] + (-27 + 54*x)*Log[x^2]^2 + (9 - 18*x) *Log[x^2]^4 + (-1 + 2*x)*Log[x^2]^6)/(27*x - 27*x^2 + (-27*x + 27*x^2)*Log [x^2]^2 + (9*x - 9*x^2)*Log[x^2]^4 + (-x + x^2)*Log[x^2]^6),x]
Output:
Log[1 - x] + Log[x] + (3 - Log[x^2]^2)^(-2)
Time = 1.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\ln \left (x^{2}-x \right )+\frac {1}{{\left (\ln \left (x^{2}\right )^{2}-3\right )}^{2}}\) | \(20\) |
norman | \(\frac {-3 \ln \left (x^{2}\right )^{3}+\frac {\ln \left (x^{2}\right )^{5}}{2}+\frac {9 \ln \left (x^{2}\right )}{2}+1}{{\left (\ln \left (x^{2}\right )^{2}-3\right )}^{2}}+\ln \left (-1+x \right )\) | \(41\) |
parallelrisch | \(\frac {2+2 \ln \left (-1+x \right ) \ln \left (x^{2}\right )^{4}+\ln \left (x^{2}\right )^{5}-12 \ln \left (-1+x \right ) \ln \left (x^{2}\right )^{2}-6 \ln \left (x^{2}\right )^{3}+18 \ln \left (-1+x \right )+9 \ln \left (x^{2}\right )}{2 \ln \left (x^{2}\right )^{4}-12 \ln \left (x^{2}\right )^{2}+18}\) | \(73\) |
Input:
int(((-1+2*x)*ln(x^2)^6+(-18*x+9)*ln(x^2)^4+(54*x-27)*ln(x^2)^2+(-8*x+8)*l n(x^2)-54*x+27)/((x^2-x)*ln(x^2)^6+(-9*x^2+9*x)*ln(x^2)^4+(27*x^2-27*x)*ln (x^2)^2-27*x^2+27*x),x,method=_RETURNVERBOSE)
Output:
ln(x^2-x)+1/(ln(x^2)^2-3)^2
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (18) = 36\).
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {\log \left (x^{2} - x\right ) \log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2} - x\right ) \log \left (x^{2}\right )^{2} + 9 \, \log \left (x^{2} - x\right ) + 1}{\log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2}\right )^{2} + 9} \] Input:
integrate(((-1+2*x)*log(x^2)^6+(-18*x+9)*log(x^2)^4+(54*x-27)*log(x^2)^2+( -8*x+8)*log(x^2)-54*x+27)/((x^2-x)*log(x^2)^6+(-9*x^2+9*x)*log(x^2)^4+(27* x^2-27*x)*log(x^2)^2-27*x^2+27*x),x, algorithm="fricas")
Output:
(log(x^2 - x)*log(x^2)^4 - 6*log(x^2 - x)*log(x^2)^2 + 9*log(x^2 - x) + 1) /(log(x^2)^4 - 6*log(x^2)^2 + 9)
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\log {\left (x^{2} - x \right )} + \frac {1}{\log {\left (x^{2} \right )}^{4} - 6 \log {\left (x^{2} \right )}^{2} + 9} \] Input:
integrate(((-1+2*x)*ln(x**2)**6+(-18*x+9)*ln(x**2)**4+(54*x-27)*ln(x**2)** 2+(-8*x+8)*ln(x**2)-54*x+27)/((x**2-x)*ln(x**2)**6+(-9*x**2+9*x)*ln(x**2)* *4+(27*x**2-27*x)*ln(x**2)**2-27*x**2+27*x),x)
Output:
log(x**2 - x) + 1/(log(x**2)**4 - 6*log(x**2)**2 + 9)
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {1}{16 \, \log \left (x\right )^{4} - 24 \, \log \left (x\right )^{2} + 9} + \log \left (x - 1\right ) + \log \left (x\right ) \] Input:
integrate(((-1+2*x)*log(x^2)^6+(-18*x+9)*log(x^2)^4+(54*x-27)*log(x^2)^2+( -8*x+8)*log(x^2)-54*x+27)/((x^2-x)*log(x^2)^6+(-9*x^2+9*x)*log(x^2)^4+(27* x^2-27*x)*log(x^2)^2-27*x^2+27*x),x, algorithm="maxima")
Output:
1/(16*log(x)^4 - 24*log(x)^2 + 9) + log(x - 1) + log(x)
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {1}{\log \left (x^{2}\right )^{4} - 6 \, \log \left (x^{2}\right )^{2} + 9} + \log \left (x - 1\right ) + \log \left (x\right ) \] Input:
integrate(((-1+2*x)*log(x^2)^6+(-18*x+9)*log(x^2)^4+(54*x-27)*log(x^2)^2+( -8*x+8)*log(x^2)-54*x+27)/((x^2-x)*log(x^2)^6+(-9*x^2+9*x)*log(x^2)^4+(27* x^2-27*x)*log(x^2)^2-27*x^2+27*x),x, algorithm="giac")
Output:
1/(log(x^2)^4 - 6*log(x^2)^2 + 9) + log(x - 1) + log(x)
Time = 1.91 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\ln \left (x\,\left (x-1\right )\right )+\frac {1}{{\left ({\ln \left (x^2\right )}^2-3\right )}^2} \] Input:
int((54*x - log(x^2)^6*(2*x - 1) + log(x^2)^4*(18*x - 9) - log(x^2)^2*(54* x - 27) + log(x^2)*(8*x - 8) - 27)/(log(x^2)^6*(x - x^2) - 27*x - log(x^2) ^4*(9*x - 9*x^2) + log(x^2)^2*(27*x - 27*x^2) + 27*x^2),x)
Output:
log(x*(x - 1)) + 1/(log(x^2)^2 - 3)^2
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.48 \[ \int \frac {27-54 x+(8-8 x) \log \left (x^2\right )+(-27+54 x) \log ^2\left (x^2\right )+(9-18 x) \log ^4\left (x^2\right )+(-1+2 x) \log ^6\left (x^2\right )}{27 x-27 x^2+\left (-27 x+27 x^2\right ) \log ^2\left (x^2\right )+\left (9 x-9 x^2\right ) \log ^4\left (x^2\right )+\left (-x+x^2\right ) \log ^6\left (x^2\right )} \, dx=\frac {\mathrm {log}\left (x^{2}\right )^{4} \mathrm {log}\left (x -1\right )+\mathrm {log}\left (x^{2}\right )^{4} \mathrm {log}\left (x \right )-6 \mathrm {log}\left (x^{2}\right )^{2} \mathrm {log}\left (x -1\right )-6 \mathrm {log}\left (x^{2}\right )^{2} \mathrm {log}\left (x \right )+9 \,\mathrm {log}\left (x -1\right )+9 \,\mathrm {log}\left (x \right )+1}{\mathrm {log}\left (x^{2}\right )^{4}-6 \mathrm {log}\left (x^{2}\right )^{2}+9} \] Input:
int(((-1+2*x)*log(x^2)^6+(-18*x+9)*log(x^2)^4+(54*x-27)*log(x^2)^2+(-8*x+8 )*log(x^2)-54*x+27)/((x^2-x)*log(x^2)^6+(-9*x^2+9*x)*log(x^2)^4+(27*x^2-27 *x)*log(x^2)^2-27*x^2+27*x),x)
Output:
(log(x**2)**4*log(x - 1) + log(x**2)**4*log(x) - 6*log(x**2)**2*log(x - 1) - 6*log(x**2)**2*log(x) + 9*log(x - 1) + 9*log(x) + 1)/(log(x**2)**4 - 6* log(x**2)**2 + 9)