Integrand size = 57, antiderivative size = 18 \[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=\log \left (30 x \left (x^2+\frac {2}{-x+\log (4)}\right )\right ) \] Output:
ln(30*(2/(2*ln(2)-x)+x^2)*x)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.06 \[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=\log (x)+\text {RootSum}\left [\log (16)-2 \text {$\#$1}+\log ^2(4) \text {$\#$1}^2-2 \log (4) \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})+\log ^2(4) \log (x-\text {$\#$1}) \text {$\#$1}-2 \log (4) \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^3}{-1+\log ^2(4) \text {$\#$1}-3 \log (4) \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(3*x^4 + (2 - 6*x^3)*Log[4] + 3*x^2*Log[4]^2)/(-2*x^2 + x^5 + (2 *x - 2*x^4)*Log[4] + x^3*Log[4]^2),x]
Output:
Log[x] + RootSum[Log[16] - 2*#1 + Log[4]^2*#1^2 - 2*Log[4]*#1^3 + #1^4 & , (Log[x - #1] + Log[4]^2*Log[x - #1]*#1 - 2*Log[4]*Log[x - #1]*#1^2 + Log[ x - #1]*#1^3)/(-1 + Log[4]^2*#1 - 3*Log[4]*#1^2 + 2*#1^3) & ]
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.
\(\displaystyle \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)-2 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{x \left (x^4-2 x^3 \log (4)+x^2 \log ^2(4)-2 x+\log (16)\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-3 x^4+6 x^3 \log (4)-3 x^2 \log ^2(4)-\log (16)}{x \left (-x^4+2 x^3 \log (4)-x^2 \log ^2(4)+2 x-\log (16)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (x^3-2 x^2 \log (4)+x \log ^2(4)+1\right )}{x^4-2 x^3 \log (4)+x^2 \log ^2(4)-2 x+\log (16)}+\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {1}{x^4-2 \log (4) x^3+\log ^2(4) x^2-2 x+\log (16)}dx+\log ^2(4) \int \frac {x}{x^4-2 \log (4) x^3+\log ^2(4) x^2-2 x+\log (16)}dx-\log (4) \int \frac {x^2}{x^4-2 \log (4) x^3+\log ^2(4) x^2-2 x+\log (16)}dx+\frac {1}{2} \log \left (x^4-2 x^3 \log (4)+x^2 \log ^2(4)-2 x+\log (16)\right )+\log (x)\) |
Input:
Int[(3*x^4 + (2 - 6*x^3)*Log[4] + 3*x^2*Log[4]^2)/(-2*x^2 + x^5 + (2*x - 2 *x^4)*Log[4] + x^3*Log[4]^2),x]
Output:
$Aborted
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44
method | result | size |
default | \(\ln \left (x \right )-\ln \left (x -2 \ln \left (2\right )\right )+\ln \left (-2 x^{2} \ln \left (2\right )+x^{3}-2\right )\) | \(26\) |
risch | \(-\ln \left (x -2 \ln \left (2\right )\right )+\ln \left (-2 x^{3} \ln \left (2\right )+x^{4}-2 x \right )\) | \(26\) |
parallelrisch | \(\ln \left (x \right )-\ln \left (x -2 \ln \left (2\right )\right )+\ln \left (-2 x^{2} \ln \left (2\right )+x^{3}-2\right )\) | \(26\) |
norman | \(-\ln \left (2 \ln \left (2\right )-x \right )+\ln \left (x \right )+\ln \left (2 x^{2} \ln \left (2\right )-x^{3}+2\right )\) | \(30\) |
Input:
int((12*x^2*ln(2)^2+2*(-6*x^3+2)*ln(2)+3*x^4)/(4*x^3*ln(2)^2+2*(-2*x^4+2*x )*ln(2)+x^5-2*x^2),x,method=_RETURNVERBOSE)
Output:
ln(x)-ln(x-2*ln(2))+ln(-2*x^2*ln(2)+x^3-2)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=\log \left (x^{4} - 2 \, x^{3} \log \left (2\right ) - 2 \, x\right ) - \log \left (x - 2 \, \log \left (2\right )\right ) \] Input:
integrate((12*x^2*log(2)^2+2*(-6*x^3+2)*log(2)+3*x^4)/(4*x^3*log(2)^2+2*(- 2*x^4+2*x)*log(2)+x^5-2*x^2),x, algorithm="fricas")
Output:
log(x^4 - 2*x^3*log(2) - 2*x) - log(x - 2*log(2))
Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=- \log {\left (x - 2 \log {\left (2 \right )} \right )} + \log {\left (x^{4} - 2 x^{3} \log {\left (2 \right )} - 2 x \right )} \] Input:
integrate((12*x**2*ln(2)**2+2*(-6*x**3+2)*ln(2)+3*x**4)/(4*x**3*ln(2)**2+2 *(-2*x**4+2*x)*ln(2)+x**5-2*x**2),x)
Output:
-log(x - 2*log(2)) + log(x**4 - 2*x**3*log(2) - 2*x)
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=\log \left (x^{3} - 2 \, x^{2} \log \left (2\right ) - 2\right ) - \log \left (x - 2 \, \log \left (2\right )\right ) + \log \left (x\right ) \] Input:
integrate((12*x^2*log(2)^2+2*(-6*x^3+2)*log(2)+3*x^4)/(4*x^3*log(2)^2+2*(- 2*x^4+2*x)*log(2)+x^5-2*x^2),x, algorithm="maxima")
Output:
log(x^3 - 2*x^2*log(2) - 2) - log(x - 2*log(2)) + log(x)
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=\log \left ({\left | x^{3} - 2 \, x^{2} \log \left (2\right ) - 2 \right |}\right ) - \log \left ({\left | x - 2 \, \log \left (2\right ) \right |}\right ) + \log \left ({\left | x \right |}\right ) \] Input:
integrate((12*x^2*log(2)^2+2*(-6*x^3+2)*log(2)+3*x^4)/(4*x^3*log(2)^2+2*(- 2*x^4+2*x)*log(2)+x^5-2*x^2),x, algorithm="giac")
Output:
log(abs(x^3 - 2*x^2*log(2) - 2)) - log(abs(x - 2*log(2))) + log(abs(x))
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=\ln \left (x^4-2\,\ln \left (2\right )\,x^3-2\,x\right )-\ln \left (x-\ln \left (4\right )\right ) \] Input:
int((12*x^2*log(2)^2 - 2*log(2)*(6*x^3 - 2) + 3*x^4)/(4*x^3*log(2)^2 + 2*l og(2)*(2*x - 2*x^4) - 2*x^2 + x^5),x)
Output:
log(x^4 - 2*x^3*log(2) - 2*x) - log(x - log(4))
\[ \int \frac {3 x^4+\left (2-6 x^3\right ) \log (4)+3 x^2 \log ^2(4)}{-2 x^2+x^5+\left (2 x-2 x^4\right ) \log (4)+x^3 \log ^2(4)} \, dx=-8 \left (\int \frac {1}{4 \mathrm {log}\left (2\right )^{2} x^{3}-4 \,\mathrm {log}\left (2\right ) x^{4}+4 \,\mathrm {log}\left (2\right ) x +x^{5}-2 x^{2}}d x \right ) \mathrm {log}\left (2\right )+6 \left (\int \frac {1}{4 \mathrm {log}\left (2\right )^{2} x^{2}-4 \,\mathrm {log}\left (2\right ) x^{3}+4 \,\mathrm {log}\left (2\right )+x^{4}-2 x}d x \right )+3 \,\mathrm {log}\left (x \right ) \] Input:
int((12*x^2*log(2)^2+2*(-6*x^3+2)*log(2)+3*x^4)/(4*x^3*log(2)^2+2*(-2*x^4+ 2*x)*log(2)+x^5-2*x^2),x)
Output:
- 8*int(1/(4*log(2)**2*x**3 - 4*log(2)*x**4 + 4*log(2)*x + x**5 - 2*x**2) ,x)*log(2) + 6*int(1/(4*log(2)**2*x**2 - 4*log(2)*x**3 + 4*log(2) + x**4 - 2*x),x) + 3*log(x)