Integrand size = 52, antiderivative size = 30 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\log \left (x+\frac {e^7-\frac {1}{2 x}+x-\log \left (6 e^{-x}\right )}{\log (4)}\right ) \] Output:
ln(x+1/2*(x-1/2/x-ln(6/exp(x))+exp(7))/ln(2))
Time = 0.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=-\log (x)+\log \left (1-2 e^7 x-2 x^2-x^2 \log (16)+2 x \log \left (6 e^{-x}\right )\right ) \] Input:
Integrate[(-1 - 4*x^2 - 2*x^2*Log[4])/(x - 2*E^7*x^2 - 2*x^3 - 2*x^3*Log[4 ] + 2*x^2*Log[6/E^x]),x]
Output:
-Log[x] + Log[1 - 2*E^7*x - 2*x^2 - x^2*Log[16] + 2*x*Log[6/E^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.
\(\displaystyle \int \frac {-4 x^2-2 x^2 \log (4)-1}{-2 x^3-2 x^3 \log (4)-2 e^7 x^2+2 x^2 \log \left (6 e^{-x}\right )+x} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^2 (-4-2 \log (4))-1}{-2 x^3-2 x^3 \log (4)-2 e^7 x^2+2 x^2 \log \left (6 e^{-x}\right )+x}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^2 (-4-2 \log (4))-1}{x^3 (-2-2 \log (4))-2 e^7 x^2+2 x^2 \log \left (6 e^{-x}\right )+x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x (-2-\log (4))}{-2 x^2 (1+\log (4))-2 e^7 x+2 x \log \left (6 e^{-x}\right )+1}+\frac {1}{x \left (2 x^2 (1+\log (4))+2 e^7 x-2 x \log \left (6 e^{-x}\right )-1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{x \left (2 (1+\log (4)) x^2-2 \log \left (6 e^{-x}\right ) x+2 e^7 x-1\right )}dx-2 (2+\log (4)) \int \frac {x}{-2 (1+\log (4)) x^2+2 \log \left (6 e^{-x}\right ) x-2 e^7 x+1}dx\) |
Input:
Int[(-1 - 4*x^2 - 2*x^2*Log[4])/(x - 2*E^7*x^2 - 2*x^3 - 2*x^3*Log[4] + 2* x^2*Log[6/E^x]),x]
Output:
$Aborted
Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
method | result | size |
norman | \(-\ln \left (x \right )+\ln \left (4 x^{2} \ln \left (2\right )-2 x \ln \left (6 \,{\mathrm e}^{-x}\right )+2 x \,{\mathrm e}^{7}+2 x^{2}-1\right )\) | \(36\) |
parallelrisch | \(-\ln \left (x \right )+\ln \left (\frac {4 x^{2} \ln \left (2\right )-2 x \ln \left (6 \,{\mathrm e}^{-x}\right )+2 x \,{\mathrm e}^{7}+2 x^{2}-1}{4 \ln \left (2\right )+2}\right )\) | \(46\) |
risch | \(\ln \left (\ln \left ({\mathrm e}^{x}\right )+\frac {i \left (-4 i x^{2} \ln \left (2\right )+2 i x \ln \left (3\right )+2 i x \ln \left (2\right )-2 i x \,{\mathrm e}^{7}-2 i x^{2}+i\right )}{2 x}\right )\) | \(47\) |
default | \(-\ln \left (x \right )+\ln \left (4 x^{2} \ln \left (2\right )+2 x \,{\mathrm e}^{7}+4 x^{2}+2 x \left (\ln \left ({\mathrm e}^{x}\right )-x \right )-2 x \left (\ln \left (6 \,{\mathrm e}^{-x}\right )+\ln \left ({\mathrm e}^{x}\right )\right )-1\right )\) | \(50\) |
Input:
int((-4*x^2*ln(2)-4*x^2-1)/(2*x^2*ln(6/exp(x))-4*x^3*ln(2)-2*x^2*exp(7)-2* x^3+x),x,method=_RETURNVERBOSE)
Output:
-ln(x)+ln(4*x^2*ln(2)-2*x*ln(6/exp(x))+2*x*exp(7)+2*x^2-1)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\log \left (4 \, x^{2} \log \left (2\right ) + 4 \, x^{2} + 2 \, x e^{7} - 2 \, x \log \left (6\right ) - 1\right ) - \log \left (x\right ) \] Input:
integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2* exp(7)-2*x^3+x),x, algorithm="fricas")
Output:
log(4*x^2*log(2) + 4*x^2 + 2*x*e^7 - 2*x*log(6) - 1) - log(x)
Time = 0.64 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=- \log {\left (x \right )} + \log {\left (x^{2} + \frac {x \left (- \log {\left (6 \right )} + e^{7}\right )}{2 \log {\left (2 \right )} + 2} - \frac {1}{4 \log {\left (2 \right )} + 4} \right )} \] Input:
integrate((-4*x**2*ln(2)-4*x**2-1)/(2*x**2*ln(6/exp(x))-4*x**3*ln(2)-2*x** 2*exp(7)-2*x**3+x),x)
Output:
-log(x) + log(x**2 + x*(-log(6) + exp(7))/(2*log(2) + 2) - 1/(4*log(2) + 4 ))
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (31) = 62\).
Time = 0.12 (sec) , antiderivative size = 579, normalized size of antiderivative = 19.30 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\text {Too large to display} \] Input:
integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2* exp(7)-2*x^3+x),x, algorithm="maxima")
Output:
-1/2*((e^7 - log(3) - log(2))*log((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2 ))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - l og(3) - log(2))/(4*x*(log(2) + 1) + sqrt(-2*(e^7 - log(2))*log(3) + log(3) ^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))/( sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4)*(log(2) + 1)) - log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3) - lo g(2)) - 1)/(log(2) + 1))*log(2) + 1/2*(e^7 - log(3) - log(2))*log((4*x*(lo g(2) + 1) - sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1) + sqrt(- 2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2)))/sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) - 1/2*(e^7 - log(3) - log(2))*lo g((4*x*(log(2) + 1) - sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4) + e^7 - log(3) - log(2))/(4*x*(log(2) + 1 ) + sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2) ^2 + e^14 + 4) + e^7 - log(3) - log(2)))/(sqrt(-2*(e^7 - log(2))*log(3) + log(3)^2 - 2*(e^7 - 2)*log(2) + log(2)^2 + e^14 + 4)*(log(2) + 1)) + 1/2*l og(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3) - log(2)) - 1)/(log(2) + 1) + 1/ 2*log(4*x^2*(log(2) + 1) + 2*x*(e^7 - log(3) - log(2)) - 1) - log(x)
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\log \left ({\left | 4 \, x^{2} \log \left (2\right ) + 4 \, x^{2} + 2 \, x e^{7} - 2 \, x \log \left (6\right ) - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \] Input:
integrate((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2* exp(7)-2*x^3+x),x, algorithm="giac")
Output:
log(abs(4*x^2*log(2) + 4*x^2 + 2*x*e^7 - 2*x*log(6) - 1)) - log(abs(x))
Time = 0.54 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\ln \left (4\,x\,\ln \left (6\right )-4\,x\,{\mathrm {e}}^7-8\,x^2\,\ln \left (2\right )-8\,x^2+2\right )-\ln \left (x\right ) \] Input:
int((4*x^2*log(2) + 4*x^2 + 1)/(2*x^2*exp(7) - 2*x^2*log(6*exp(-x)) - x + 4*x^3*log(2) + 2*x^3),x)
Output:
log(4*x*log(6) - 4*x*exp(7) - 8*x^2*log(2) - 8*x^2 + 2) - log(x)
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-1-4 x^2-2 x^2 \log (4)}{x-2 e^7 x^2-2 x^3-2 x^3 \log (4)+2 x^2 \log \left (6 e^{-x}\right )} \, dx=\mathrm {log}\left (2 \,\mathrm {log}\left (\frac {6}{e^{x}}\right ) x -4 \,\mathrm {log}\left (2\right ) x^{2}-2 e^{7} x -2 x^{2}+1\right )-\mathrm {log}\left (x \right ) \] Input:
int((-4*x^2*log(2)-4*x^2-1)/(2*x^2*log(6/exp(x))-4*x^3*log(2)-2*x^2*exp(7) -2*x^3+x),x)
Output:
log(2*log(6/e**x)*x - 4*log(2)*x**2 - 2*e**7*x - 2*x**2 + 1) - log(x)