Integrand size = 229, antiderivative size = 33 \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=\log \left (\frac {1-x+\frac {x}{\log (x)}+\frac {3 x^2}{1+\log \left (\frac {3+x}{4}\right )}}{x}\right ) \] Output:
ln((1+3*x^2/(1+ln(3/4+1/4*x))+x/ln(x)-x)/x)
Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(33)=66\).
Time = 28.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.24 \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=-\log (x)-\log (\log (x))-\log \left (1+\log \left (\frac {3+x}{4}\right )\right )+\log \left (x+\log (x)-x \log (x)+3 x^2 \log (x)+x \log \left (\frac {3+x}{4}\right )+\log (x) \log \left (\frac {3+x}{4}\right )-x \log (x) \log \left (\frac {3+x}{4}\right )\right ) \] Input:
Integrate[(3*x + x^2 + (6*x + 2*x^2)*Log[(3 + x)/4] + (3*x + x^2)*Log[(3 + x)/4]^2 + Log[x]^2*(3 + x - 9*x^2 + (6 + 2*x - 9*x^2 - 3*x^3)*Log[(3 + x) /4] + (3 + x)*Log[(3 + x)/4]^2))/(Log[x]*(-3*x^2 - x^3 + (-6*x^2 - 2*x^3)* Log[(3 + x)/4] + (-3*x^2 - x^3)*Log[(3 + x)/4]^2) + Log[x]^2*(-3*x + 2*x^2 - 8*x^3 - 3*x^4 + (-6*x + 4*x^2 - 7*x^3 - 3*x^4)*Log[(3 + x)/4] + (-3*x + 2*x^2 + x^3)*Log[(3 + x)/4]^2)),x]
Output:
-Log[x] - Log[Log[x]] - Log[1 + Log[(3 + x)/4]] + Log[x + Log[x] - x*Log[x ] + 3*x^2*Log[x] + x*Log[(3 + x)/4] + Log[x]*Log[(3 + x)/4] - x*Log[x]*Log [(3 + x)/4]]
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.
| \(\displaystyle \int \frac {x^2+\left (x^2+3 x\right ) \log ^2\left (\frac {x+3}{4}\right )+\left (2 x^2+6 x\right ) \log \left (\frac {x+3}{4}\right )+\log ^2(x) \left (-9 x^2+\left (-3 x^3-9 x^2+2 x+6\right ) \log \left (\frac {x+3}{4}\right )+x+(x+3) \log ^2\left (\frac {x+3}{4}\right )+3\right )+3 x}{\left (-x^3-3 x^2+\left (-x^3-3 x^2\right ) \log ^2\left (\frac {x+3}{4}\right )+\left (-2 x^3-6 x^2\right ) \log \left (\frac {x+3}{4}\right )\right ) \log (x)+\left (-3 x^4-8 x^3+2 x^2+\left (x^3+2 x^2-3 x\right ) \log ^2\left (\frac {x+3}{4}\right )+\left (-3 x^4-7 x^3+4 x^2-6 x\right ) \log \left (\frac {x+3}{4}\right )-3 x\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-x^2-\left (x^2+3 x\right ) \log ^2\left (\frac {x+3}{4}\right )-\left (2 x^2+6 x\right ) \log \left (\frac {x+3}{4}\right )-\log ^2(x) \left (-9 x^2+\left (-3 x^3-9 x^2+2 x+6\right ) \log \left (\frac {x+3}{4}\right )+x+(x+3) \log ^2\left (\frac {x+3}{4}\right )+3\right )-3 x}{x (x+3) \log (x) \left (\log \left (\frac {x+3}{4}\right )+1\right ) \left (3 x^2 \log (x)+x-x \log (x)-x \log (x) \log \left (\frac {x+3}{4}\right )+x \log \left (\frac {x+3}{4}\right )+\log (x)+\log (x) \log \left (\frac {x+3}{4}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-3 x^3+3 x^3 \log ^2(x)-3 x^3 \log (x)-10 x^2+2 x^2 \log ^2(x)-7 x^2 \log (x)-16 x \log ^2(x)-\log ^2(x)-2 x \log (x)}{(x+3) (-x+x \log (x)-\log (x)) \left (3 x^2 \log (x)+x-x \log (x)-x \log (x) \log \left (\frac {x+3}{4}\right )+x \log \left (\frac {x+3}{4}\right )+\log (x)+\log (x) \log \left (\frac {x+3}{4}\right )\right )}+\frac {x+\log ^2(x)}{x \log (x) (-x+x \log (x)-\log (x))}-\frac {1}{(x+3) \left (\log \left (\frac {x+3}{4}\right )+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {-3 x^3+3 x^3 \log ^2(x)-3 x^3 \log (x)-10 x^2+2 x^2 \log ^2(x)-7 x^2 \log (x)-16 x \log ^2(x)-\log ^2(x)-2 x \log (x)}{(x+3) (-x+x \log (x)-\log (x)) \left (3 x^2 \log (x)+x-x \log (x)-x \log (x) \log \left (\frac {x+3}{4}\right )+x \log \left (\frac {x+3}{4}\right )+\log (x)+\log (x) \log \left (\frac {x+3}{4}\right )\right )}+\frac {x+\log ^2(x)}{x \log (x) (-x+x \log (x)-\log (x))}-\frac {1}{(x+3) \left (\log \left (\frac {x+3}{4}\right )+1\right )}\right )dx\) |
Input:
Int[(3*x + x^2 + (6*x + 2*x^2)*Log[(3 + x)/4] + (3*x + x^2)*Log[(3 + x)/4] ^2 + Log[x]^2*(3 + x - 9*x^2 + (6 + 2*x - 9*x^2 - 3*x^3)*Log[(3 + x)/4] + (3 + x)*Log[(3 + x)/4]^2))/(Log[x]*(-3*x^2 - x^3 + (-6*x^2 - 2*x^3)*Log[(3 + x)/4] + (-3*x^2 - x^3)*Log[(3 + x)/4]^2) + Log[x]^2*(-3*x + 2*x^2 - 8*x ^3 - 3*x^4 + (-6*x + 4*x^2 - 7*x^3 - 3*x^4)*Log[(3 + x)/4] + (-3*x + 2*x^2 + x^3)*Log[(3 + x)/4]^2)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(31)=62\).
Time = 0.16 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.97
\[\ln \left (\ln \left (x \right )-1\right )-\ln \left (\ln \left (x \right )\right )-\ln \left (x \right )+\ln \left (x -\frac {\ln \left (x \right )}{\ln \left (x \right )-1}\right )+\ln \left (\ln \left (3+x \right )-\frac {2 x \ln \left (2\right ) \ln \left (x \right )+3 x^{2} \ln \left (x \right )-2 \ln \left (2\right ) \ln \left (x \right )-2 x \ln \left (2\right )-x \ln \left (x \right )+\ln \left (x \right )+x}{x \ln \left (x \right )-\ln \left (x \right )-x}\right )-\ln \left (-2 \ln \left (2\right )+\ln \left (3+x \right )+1\right )\]
Input:
int((((3+x)*ln(3/4+1/4*x)^2+(-3*x^3-9*x^2+2*x+6)*ln(3/4+1/4*x)-9*x^2+x+3)* ln(x)^2+(x^2+3*x)*ln(3/4+1/4*x)^2+(2*x^2+6*x)*ln(3/4+1/4*x)+x^2+3*x)/(((x^ 3+2*x^2-3*x)*ln(3/4+1/4*x)^2+(-3*x^4-7*x^3+4*x^2-6*x)*ln(3/4+1/4*x)-3*x^4- 8*x^3+2*x^2-3*x)*ln(x)^2+((-x^3-3*x^2)*ln(3/4+1/4*x)^2+(-2*x^3-6*x^2)*ln(3 /4+1/4*x)-x^3-3*x^2)*ln(x)),x)
Output:
ln(ln(x)-1)-ln(ln(x))-ln(x)+ln(x-ln(x)/(ln(x)-1))+ln(ln(3+x)-(2*x*ln(2)*ln (x)+3*x^2*ln(x)-2*ln(2)*ln(x)-2*x*ln(2)-x*ln(x)+ln(x)+x)/(x*ln(x)-ln(x)-x) )-ln(-2*ln(2)+ln(3+x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (31) = 62\).
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.42 \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=\log \left (x - 1\right ) - \log \left (x\right ) + \log \left (\frac {{\left (3 \, x^{2} - {\left (x - 1\right )} \log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) - x + 1\right )} \log \left (x\right ) + x \log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) + x}{3 \, x^{2} - {\left (x - 1\right )} \log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) - x + 1}\right ) + \log \left (-\frac {3 \, x^{2} - {\left (x - 1\right )} \log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) - x + 1}{x - 1}\right ) - \log \left (\log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) + 1\right ) - \log \left (\log \left (x\right )\right ) \] Input:
integrate((((3+x)*log(3/4+1/4*x)^2+(-3*x^3-9*x^2+2*x+6)*log(3/4+1/4*x)-9*x ^2+x+3)*log(x)^2+(x^2+3*x)*log(3/4+1/4*x)^2+(2*x^2+6*x)*log(3/4+1/4*x)+x^2 +3*x)/(((x^3+2*x^2-3*x)*log(3/4+1/4*x)^2+(-3*x^4-7*x^3+4*x^2-6*x)*log(3/4+ 1/4*x)-3*x^4-8*x^3+2*x^2-3*x)*log(x)^2+((-x^3-3*x^2)*log(3/4+1/4*x)^2+(-2* x^3-6*x^2)*log(3/4+1/4*x)-x^3-3*x^2)*log(x)),x, algorithm="fricas")
Output:
log(x - 1) - log(x) + log(((3*x^2 - (x - 1)*log(1/4*x + 3/4) - x + 1)*log( x) + x*log(1/4*x + 3/4) + x)/(3*x^2 - (x - 1)*log(1/4*x + 3/4) - x + 1)) + log(-(3*x^2 - (x - 1)*log(1/4*x + 3/4) - x + 1)/(x - 1)) - log(log(1/4*x + 3/4) + 1) - log(log(x))
Exception generated. \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((((3+x)*ln(3/4+1/4*x)**2+(-3*x**3-9*x**2+2*x+6)*ln(3/4+1/4*x)-9* x**2+x+3)*ln(x)**2+(x**2+3*x)*ln(3/4+1/4*x)**2+(2*x**2+6*x)*ln(3/4+1/4*x)+ x**2+3*x)/(((x**3+2*x**2-3*x)*ln(3/4+1/4*x)**2+(-3*x**4-7*x**3+4*x**2-6*x) *ln(3/4+1/4*x)-3*x**4-8*x**3+2*x**2-3*x)*ln(x)**2+((-x**3-3*x**2)*ln(3/4+1 /4*x)**2+(-2*x**3-6*x**2)*ln(3/4+1/4*x)-x**3-3*x**2)*ln(x)),x)
Output:
Exception raised: PolynomialError >> 1/(_t0**2*x**3 + _t0**2*x**2 - 5*_t0* *2*x + 3*_t0**2 - 2*_t0*x**3 - 4*_t0*x**2 + 6*_t0*x + x**3 + 3*x**2) conta ins an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (31) = 62\).
Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.18 \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=\log \left (x - 1\right ) - \log \left (x\right ) + \log \left (\frac {x {\left (2 \, \log \left (2\right ) - 1\right )} + {\left ({\left (x - 1\right )} \log \left (x\right ) - x\right )} \log \left (x + 3\right ) - {\left (3 \, x^{2} + x {\left (2 \, \log \left (2\right ) - 1\right )} - 2 \, \log \left (2\right ) + 1\right )} \log \left (x\right )}{{\left (x - 1\right )} \log \left (x\right ) - x}\right ) + \log \left (\frac {{\left (x - 1\right )} \log \left (x\right ) - x}{x - 1}\right ) - \log \left (-2 \, \log \left (2\right ) + \log \left (x + 3\right ) + 1\right ) - \log \left (\log \left (x\right )\right ) \] Input:
integrate((((3+x)*log(3/4+1/4*x)^2+(-3*x^3-9*x^2+2*x+6)*log(3/4+1/4*x)-9*x ^2+x+3)*log(x)^2+(x^2+3*x)*log(3/4+1/4*x)^2+(2*x^2+6*x)*log(3/4+1/4*x)+x^2 +3*x)/(((x^3+2*x^2-3*x)*log(3/4+1/4*x)^2+(-3*x^4-7*x^3+4*x^2-6*x)*log(3/4+ 1/4*x)-3*x^4-8*x^3+2*x^2-3*x)*log(x)^2+((-x^3-3*x^2)*log(3/4+1/4*x)^2+(-2* x^3-6*x^2)*log(3/4+1/4*x)-x^3-3*x^2)*log(x)),x, algorithm="maxima")
Output:
log(x - 1) - log(x) + log((x*(2*log(2) - 1) + ((x - 1)*log(x) - x)*log(x + 3) - (3*x^2 + x*(2*log(2) - 1) - 2*log(2) + 1)*log(x))/((x - 1)*log(x) - x)) + log(((x - 1)*log(x) - x)/(x - 1)) - log(-2*log(2) + log(x + 3) + 1) - log(log(x))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (31) = 62\).
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.64 \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=\log \left (-3 \, {\left (x + 3\right )}^{2} \log \left (x\right ) + {\left (x + 3\right )} \log \left (x\right ) \log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) + 19 \, {\left (x + 3\right )} \log \left (x\right ) - {\left (x + 3\right )} \log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) - 4 \, \log \left (x\right ) \log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) - x - 31 \, \log \left (x\right ) + 3 \, \log \left (\frac {1}{4} \, x + \frac {3}{4}\right )\right ) - \log \left (x\right ) - \log \left (\log \left (\frac {1}{4} \, x + \frac {3}{4}\right ) + 1\right ) - \log \left (\log \left (x\right )\right ) \] Input:
integrate((((3+x)*log(3/4+1/4*x)^2+(-3*x^3-9*x^2+2*x+6)*log(3/4+1/4*x)-9*x ^2+x+3)*log(x)^2+(x^2+3*x)*log(3/4+1/4*x)^2+(2*x^2+6*x)*log(3/4+1/4*x)+x^2 +3*x)/(((x^3+2*x^2-3*x)*log(3/4+1/4*x)^2+(-3*x^4-7*x^3+4*x^2-6*x)*log(3/4+ 1/4*x)-3*x^4-8*x^3+2*x^2-3*x)*log(x)^2+((-x^3-3*x^2)*log(3/4+1/4*x)^2+(-2* x^3-6*x^2)*log(3/4+1/4*x)-x^3-3*x^2)*log(x)),x, algorithm="giac")
Output:
log(-3*(x + 3)^2*log(x) + (x + 3)*log(x)*log(1/4*x + 3/4) + 19*(x + 3)*log (x) - (x + 3)*log(1/4*x + 3/4) - 4*log(x)*log(1/4*x + 3/4) - x - 31*log(x) + 3*log(1/4*x + 3/4)) - log(x) - log(log(1/4*x + 3/4) + 1) - log(log(x))
Timed out. \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=\int -\frac {3\,x+\ln \left (\frac {x}{4}+\frac {3}{4}\right )\,\left (2\,x^2+6\,x\right )+{\ln \left (\frac {x}{4}+\frac {3}{4}\right )}^2\,\left (x^2+3\,x\right )+{\ln \left (x\right )}^2\,\left (x+{\ln \left (\frac {x}{4}+\frac {3}{4}\right )}^2\,\left (x+3\right )+\ln \left (\frac {x}{4}+\frac {3}{4}\right )\,\left (-3\,x^3-9\,x^2+2\,x+6\right )-9\,x^2+3\right )+x^2}{\left (3\,x-{\ln \left (\frac {x}{4}+\frac {3}{4}\right )}^2\,\left (x^3+2\,x^2-3\,x\right )-2\,x^2+8\,x^3+3\,x^4+\ln \left (\frac {x}{4}+\frac {3}{4}\right )\,\left (3\,x^4+7\,x^3-4\,x^2+6\,x\right )\right )\,{\ln \left (x\right )}^2+\left (\ln \left (\frac {x}{4}+\frac {3}{4}\right )\,\left (2\,x^3+6\,x^2\right )+{\ln \left (\frac {x}{4}+\frac {3}{4}\right )}^2\,\left (x^3+3\,x^2\right )+3\,x^2+x^3\right )\,\ln \left (x\right )} \,d x \] Input:
int(-(3*x + log(x/4 + 3/4)*(6*x + 2*x^2) + log(x/4 + 3/4)^2*(3*x + x^2) + log(x)^2*(x + log(x/4 + 3/4)^2*(x + 3) + log(x/4 + 3/4)*(2*x - 9*x^2 - 3*x ^3 + 6) - 9*x^2 + 3) + x^2)/(log(x)^2*(3*x - log(x/4 + 3/4)^2*(2*x^2 - 3*x + x^3) - 2*x^2 + 8*x^3 + 3*x^4 + log(x/4 + 3/4)*(6*x - 4*x^2 + 7*x^3 + 3* x^4)) + log(x)*(log(x/4 + 3/4)*(6*x^2 + 2*x^3) + log(x/4 + 3/4)^2*(3*x^2 + x^3) + 3*x^2 + x^3)),x)
Output:
int(-(3*x + log(x/4 + 3/4)*(6*x + 2*x^2) + log(x/4 + 3/4)^2*(3*x + x^2) + log(x)^2*(x + log(x/4 + 3/4)^2*(x + 3) + log(x/4 + 3/4)*(2*x - 9*x^2 - 3*x ^3 + 6) - 9*x^2 + 3) + x^2)/(log(x)^2*(3*x - log(x/4 + 3/4)^2*(2*x^2 - 3*x + x^3) - 2*x^2 + 8*x^3 + 3*x^4 + log(x/4 + 3/4)*(6*x - 4*x^2 + 7*x^3 + 3* x^4)) + log(x)*(log(x/4 + 3/4)*(6*x^2 + 2*x^3) + log(x/4 + 3/4)^2*(3*x^2 + x^3) + 3*x^2 + x^3)), x)
Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.61 \[ \int \frac {3 x+x^2+\left (6 x+2 x^2\right ) \log \left (\frac {3+x}{4}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {3+x}{4}\right )+\log ^2(x) \left (3+x-9 x^2+\left (6+2 x-9 x^2-3 x^3\right ) \log \left (\frac {3+x}{4}\right )+(3+x) \log ^2\left (\frac {3+x}{4}\right )\right )}{\log (x) \left (-3 x^2-x^3+\left (-6 x^2-2 x^3\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x^2-x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )+\log ^2(x) \left (-3 x+2 x^2-8 x^3-3 x^4+\left (-6 x+4 x^2-7 x^3-3 x^4\right ) \log \left (\frac {3+x}{4}\right )+\left (-3 x+2 x^2+x^3\right ) \log ^2\left (\frac {3+x}{4}\right )\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-2 \,\mathrm {log}\left (\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right )+1\right )+\mathrm {log}\left (\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right )^{2} \mathrm {log}\left (x \right ) x -\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right )^{2} \mathrm {log}\left (x \right )-\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right )^{2} x -3 \,\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right ) \mathrm {log}\left (x \right ) x^{2}+2 \,\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right ) \mathrm {log}\left (x \right ) x -2 \,\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right ) \mathrm {log}\left (x \right )-2 \,\mathrm {log}\left (\frac {x}{4}+\frac {3}{4}\right ) x -3 \,\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (x \right ) x -\mathrm {log}\left (x \right )-x \right )-\mathrm {log}\left (x \right ) \] Input:
int((((3+x)*log(3/4+1/4*x)^2+(-3*x^3-9*x^2+2*x+6)*log(3/4+1/4*x)-9*x^2+x+3 )*log(x)^2+(x^2+3*x)*log(3/4+1/4*x)^2+(2*x^2+6*x)*log(3/4+1/4*x)+x^2+3*x)/ (((x^3+2*x^2-3*x)*log(3/4+1/4*x)^2+(-3*x^4-7*x^3+4*x^2-6*x)*log(3/4+1/4*x) -3*x^4-8*x^3+2*x^2-3*x)*log(x)^2+((-x^3-3*x^2)*log(3/4+1/4*x)^2+(-2*x^3-6* x^2)*log(3/4+1/4*x)-x^3-3*x^2)*log(x)),x)
Output:
- log(log(x)) - 2*log(log((x + 3)/4) + 1) + log(log((x + 3)/4)**2*log(x)* x - log((x + 3)/4)**2*log(x) - log((x + 3)/4)**2*x - 3*log((x + 3)/4)*log( x)*x**2 + 2*log((x + 3)/4)*log(x)*x - 2*log((x + 3)/4)*log(x) - 2*log((x + 3)/4)*x - 3*log(x)*x**2 + log(x)*x - log(x) - x) - log(x)