Integrand size = 90, antiderivative size = 21 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=-x^2-\log \left (-4+\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right ) \] Output:
-x^2-ln(ln(4*x/ln(3+x)^2)-4)
Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=-x^2-\log \left (4-\log \left (\frac {4 x}{\log ^2(3+x)}\right )\right ) \] Input:
Integrate[(2*x + (-3 - x + 24*x^2 + 8*x^3)*Log[3 + x] + (-6*x^2 - 2*x^3)*L og[3 + x]*Log[(4*x)/Log[3 + x]^2])/((-12*x - 4*x^2)*Log[3 + x] + (3*x + x^ 2)*Log[3 + x]*Log[(4*x)/Log[3 + x]^2]),x]
Output:
-x^2 - Log[4 - Log[(4*x)/Log[3 + x]^2]]
Time = 1.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, 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.
\(\displaystyle \int \frac {\left (-2 x^3-6 x^2\right ) \log (x+3) \log \left (\frac {4 x}{\log ^2(x+3)}\right )+\left (8 x^3+24 x^2-x-3\right ) \log (x+3)+2 x}{\left (x^2+3 x\right ) \log \left (\frac {4 x}{\log ^2(x+3)}\right ) \log (x+3)+\left (-4 x^2-12 x\right ) \log (x+3)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (-2 x^3-6 x^2\right ) \log (x+3) \log \left (\frac {4 x}{\log ^2(x+3)}\right )-\left (8 x^3+24 x^2-x-3\right ) \log (x+3)-2 x}{x (x+3) \log (x+3) \left (4-\log \left (\frac {4 x}{\log ^2(x+3)}\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x+x (-\log (x+3))-3 \log (x+3)}{x (x+3) \log (x+3) \left (\log \left (\frac {4 x}{\log ^2(x+3)}\right )-4\right )}-2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x^2-\log \left (4-\log \left (\frac {4 x}{\log ^2(x+3)}\right )\right )\) |
Input:
Int[(2*x + (-3 - x + 24*x^2 + 8*x^3)*Log[3 + x] + (-6*x^2 - 2*x^3)*Log[3 + x]*Log[(4*x)/Log[3 + x]^2])/((-12*x - 4*x^2)*Log[3 + x] + (3*x + x^2)*Log [3 + x]*Log[(4*x)/Log[3 + x]^2]),x]
Output:
-x^2 - Log[4 - Log[(4*x)/Log[3 + x]^2]]
Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05
method | result | size |
parallelrisch | \(-x^{2}-\ln \left (\ln \left (\frac {4 x}{\ln \left (3+x \right )^{2}}\right )-4\right )\) | \(22\) |
risch | \(-x^{2}-\ln \left (\ln \left (\ln \left (3+x \right )\right )+\frac {i \left (\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (3+x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (3+x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{2}-\pi \operatorname {csgn}\left (i \ln \left (3+x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (3+x \right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \ln \left (3+x \right )\right ) \operatorname {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{2}-\pi \operatorname {csgn}\left (i \ln \left (3+x \right )^{2}\right )^{3}+\pi \operatorname {csgn}\left (\frac {i x}{\ln \left (3+x \right )^{2}}\right )^{3}+4 i \ln \left (2\right )+2 i \ln \left (x \right )-8 i\right )}{4}\right )\) | \(183\) |
Input:
int(((-2*x^3-6*x^2)*ln(3+x)*ln(4*x/ln(3+x)^2)+(8*x^3+24*x^2-x-3)*ln(3+x)+2 *x)/((x^2+3*x)*ln(3+x)*ln(4*x/ln(3+x)^2)+(-4*x^2-12*x)*ln(3+x)),x,method=_ RETURNVERBOSE)
Output:
-x^2-ln(ln(4*x/ln(3+x)^2)-4)
Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=-x^{2} - \log \left (\log \left (\frac {4 \, x}{\log \left (x + 3\right )^{2}}\right ) - 4\right ) \] Input:
integrate(((-2*x^3-6*x^2)*log(3+x)*log(4*x/log(3+x)^2)+(8*x^3+24*x^2-x-3)* log(3+x)+2*x)/((x^2+3*x)*log(3+x)*log(4*x/log(3+x)^2)+(-4*x^2-12*x)*log(3+ x)),x, algorithm="fricas")
Output:
-x^2 - log(log(4*x/log(x + 3)^2) - 4)
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=- x^{2} - \log {\left (\log {\left (\frac {4 x}{\log {\left (x + 3 \right )}^{2}} \right )} - 4 \right )} \] Input:
integrate(((-2*x**3-6*x**2)*ln(3+x)*ln(4*x/ln(3+x)**2)+(8*x**3+24*x**2-x-3 )*ln(3+x)+2*x)/((x**2+3*x)*ln(3+x)*ln(4*x/ln(3+x)**2)+(-4*x**2-12*x)*ln(3+ x)),x)
Output:
-x**2 - log(log(4*x/log(x + 3)**2) - 4)
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=-x^{2} - \log \left (-\log \left (2\right ) - \frac {1}{2} \, \log \left (x\right ) + \log \left (\log \left (x + 3\right )\right ) + 2\right ) \] Input:
integrate(((-2*x^3-6*x^2)*log(3+x)*log(4*x/log(3+x)^2)+(8*x^3+24*x^2-x-3)* log(3+x)+2*x)/((x^2+3*x)*log(3+x)*log(4*x/log(3+x)^2)+(-4*x^2-12*x)*log(3+ x)),x, algorithm="maxima")
Output:
-x^2 - log(-log(2) - 1/2*log(x) + log(log(x + 3)) + 2)
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=-x^{2} - \log \left (\log \left (\log \left (x + 3\right )^{2}\right ) - \log \left (4 \, x\right ) + 4\right ) \] Input:
integrate(((-2*x^3-6*x^2)*log(3+x)*log(4*x/log(3+x)^2)+(8*x^3+24*x^2-x-3)* log(3+x)+2*x)/((x^2+3*x)*log(3+x)*log(4*x/log(3+x)^2)+(-4*x^2-12*x)*log(3+ x)),x, algorithm="giac")
Output:
-x^2 - log(log(log(x + 3)^2) - log(4*x) + 4)
Time = 4.54 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=-\ln \left (\ln \left (\frac {4\,x}{{\ln \left (x+3\right )}^2}\right )-4\right )-x^2 \] Input:
int((log(x + 3)*(x - 24*x^2 - 8*x^3 + 3) - 2*x + log(x + 3)*log((4*x)/log( x + 3)^2)*(6*x^2 + 2*x^3))/(log(x + 3)*(12*x + 4*x^2) - log(x + 3)*log((4* x)/log(x + 3)^2)*(3*x + x^2)),x)
Output:
- log(log((4*x)/log(x + 3)^2) - 4) - x^2
Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {2 x+\left (-3-x+24 x^2+8 x^3\right ) \log (3+x)+\left (-6 x^2-2 x^3\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )}{\left (-12 x-4 x^2\right ) \log (3+x)+\left (3 x+x^2\right ) \log (3+x) \log \left (\frac {4 x}{\log ^2(3+x)}\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\frac {4 x}{\mathrm {log}\left (x +3\right )^{2}}\right )-4\right )-x^{2} \] Input:
int(((-2*x^3-6*x^2)*log(3+x)*log(4*x/log(3+x)^2)+(8*x^3+24*x^2-x-3)*log(3+ x)+2*x)/((x^2+3*x)*log(3+x)*log(4*x/log(3+x)^2)+(-4*x^2-12*x)*log(3+x)),x)
Output:
- (log(log((4*x)/log(x + 3)**2) - 4) + x**2)