Integrand size = 166, antiderivative size = 29 \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx=-4+\frac {(-2+(-4+x) x) \log \left (2 \left (1-x+\frac {x}{\log (4+x)}\right )\right )}{x} \] Output:
((-4+x)*x-2)/x*ln(2+2*x/ln(4+x)-2*x)-4
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx=\left (-\frac {2}{x}+x\right ) \log \left (2-2 x+\frac {2 x}{\log (4+x)}\right )+4 (\log (\log (4+x))-\log (-x+(-1+x) \log (4+x))) \] Input:
Integrate[(-2*x^2 - 4*x^3 + x^4 + (8*x + 18*x^2 - x^4)*Log[4 + x] + (-8*x - 18*x^2 + x^4)*Log[4 + x]^2 + ((-8*x - 2*x^2 - 4*x^3 - x^4)*Log[4 + x] + (-8 + 6*x - 2*x^2 + 3*x^3 + x^4)*Log[4 + x]^2)*Log[(2*x + (2 - 2*x)*Log[4 + x])/Log[4 + x]])/((-4*x^3 - x^4)*Log[4 + x] + (-4*x^2 + 3*x^3 + x^4)*Log [4 + x]^2),x]
Output:
(-2/x + x)*Log[2 - 2*x + (2*x)/Log[4 + x]] + 4*(Log[Log[4 + x]] - Log[-x + (-1 + x)*Log[4 + 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.
| \(\displaystyle \int \frac {x^4-4 x^3-2 x^2+\left (x^4-18 x^2-8 x\right ) \log ^2(x+4)+\left (-x^4+18 x^2+8 x\right ) \log (x+4)+\left (\left (x^4+3 x^3-2 x^2+6 x-8\right ) \log ^2(x+4)+\left (-x^4-4 x^3-2 x^2-8 x\right ) \log (x+4)\right ) \log \left (\frac {2 x+(2-2 x) \log (x+4)}{\log (x+4)}\right )}{\left (-x^4-4 x^3\right ) \log (x+4)+\left (x^4+3 x^3-4 x^2\right ) \log ^2(x+4)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-x^4+4 x^3+2 x^2-\left (x^4-18 x^2-8 x\right ) \log ^2(x+4)-\left (-x^4+18 x^2+8 x\right ) \log (x+4)-\left (\left (x^4+3 x^3-2 x^2+6 x-8\right ) \log ^2(x+4)+\left (-x^4-4 x^3-2 x^2-8 x\right ) \log (x+4)\right ) \log \left (\frac {2 x+(2-2 x) \log (x+4)}{\log (x+4)}\right )}{x^2 (x+4) \log (x+4) (x+x (-\log (x+4))+\log (x+4))}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2}{(x+4) \log (x+4) (-x+x \log (x+4)-\log (x+4))}+\frac {-x^2+4 x+2}{x (-x+x \log (x+4)-\log (x+4))}+\frac {\left (x^2-4 x-2\right ) \log (x+4)}{x (-x+x \log (x+4)-\log (x+4))}+\frac {\left (x^2+2\right ) \log \left (-2 x+\frac {2 x}{\log (x+4)}+2\right )}{x^2}-\frac {4 x}{(x+4) \log (x+4) (-x+x \log (x+4)-\log (x+4))}-\frac {2}{(x+4) \log (x+4) (-x+x \log (x+4)-\log (x+4))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {\log \left (\frac {2 x}{\log (x+4)}-2 x+2\right )}{x^2}dx+\frac {1}{2} \int \frac {1}{x \log (x+4)}dx-8 \int \frac {1}{\log (x+4) x-x-\log (x+4)}dx-5 \int \frac {1}{(x-1) (\log (x+4) x-x-\log (x+4))}dx+\frac {5}{2} \int \frac {1}{x (\log (x+4) x-x-\log (x+4))}dx+\int \frac {x}{\log (x+4) x-x-\log (x+4)}dx+\frac {75}{2} \int \frac {1}{(x+4) (\log (x+4) x-x-\log (x+4))}dx+\int \log \left (\frac {2 x}{\log (x+4)}-2 x+2\right )dx-\operatorname {LogIntegral}(x+4)+x-5 \log (1-x)+2 \log (x)+\frac {15}{2} \log (\log (x+4))\) |
Input:
Int[(-2*x^2 - 4*x^3 + x^4 + (8*x + 18*x^2 - x^4)*Log[4 + x] + (-8*x - 18*x ^2 + x^4)*Log[4 + x]^2 + ((-8*x - 2*x^2 - 4*x^3 - x^4)*Log[4 + x] + (-8 + 6*x - 2*x^2 + 3*x^3 + x^4)*Log[4 + x]^2)*Log[(2*x + (2 - 2*x)*Log[4 + x])/ Log[4 + x]])/((-4*x^3 - x^4)*Log[4 + x] + (-4*x^2 + 3*x^3 + x^4)*Log[4 + x ]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(28)=56\).
Time = 7.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86
| method | result | size |
| parallelrisch | \(-\frac {-2 x^{2} \ln \left (\frac {\left (2-2 x \right ) \ln \left (4+x \right )+2 x}{\ln \left (4+x \right )}\right )+8 x \ln \left (\frac {\left (2-2 x \right ) \ln \left (4+x \right )+2 x}{\ln \left (4+x \right )}\right )+4 \ln \left (\frac {\left (2-2 x \right ) \ln \left (4+x \right )+2 x}{\ln \left (4+x \right )}\right )}{2 x}\) | \(83\) |
| risch | \(\frac {\left (x^{2}-2\right ) \ln \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{x}+\frac {-2 i \pi \,x^{2} \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}+2 x^{2} \ln \left (2\right )-8 \ln \left (-1+x \right ) x +4 \ln \left (\ln \left (4+x \right )\right )-4 \ln \left (2\right )-4 i \pi +8 \ln \left (\ln \left (4+x \right )\right ) x -8 \ln \left (\ln \left (4+x \right )-\frac {x}{-1+x}\right ) x -2 x^{2} \ln \left (\ln \left (4+x \right )\right )+i \pi \,x^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}-i \pi \,x^{2} \operatorname {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )+i \pi \,x^{2} \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{3}-2 i \pi \,\operatorname {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}+i \pi \,x^{2} \operatorname {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}+2 i \pi \,x^{2}+2 i \pi \,\operatorname {csgn}\left (i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )-2 i \pi \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{3}+4 i \pi \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\left (\ln \left (4+x \right )-1\right ) x -\ln \left (4+x \right )\right )}{\ln \left (4+x \right )}\right )^{2}}{2 x}\) | \(541\) |
Input:
int((((x^4+3*x^3-2*x^2+6*x-8)*ln(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*ln(4+x))*ln (((2-2*x)*ln(4+x)+2*x)/ln(4+x))+(x^4-18*x^2-8*x)*ln(4+x)^2+(-x^4+18*x^2+8* x)*ln(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*ln(4+x)^2+(-x^4-4*x^3)*ln(4 +x)),x,method=_RETURNVERBOSE)
Output:
-1/2*(-2*x^2*ln(((2-2*x)*ln(4+x)+2*x)/ln(4+x))+8*x*ln(((2-2*x)*ln(4+x)+2*x )/ln(4+x))+4*ln(((2-2*x)*ln(4+x)+2*x)/ln(4+x)))/x
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx=\frac {{\left (x^{2} - 4 \, x - 2\right )} \log \left (-\frac {2 \, {\left ({\left (x - 1\right )} \log \left (x + 4\right ) - x\right )}}{\log \left (x + 4\right )}\right )}{x} \] Input:
integrate((((x^4+3*x^3-2*x^2+6*x-8)*log(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*log( 4+x))*log(((2-2*x)*log(4+x)+2*x)/log(4+x))+(x^4-18*x^2-8*x)*log(4+x)^2+(-x ^4+18*x^2+8*x)*log(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*log(4+x)^2+(-x ^4-4*x^3)*log(4+x)),x, algorithm="fricas")
Output:
(x^2 - 4*x - 2)*log(-2*((x - 1)*log(x + 4) - x)/log(x + 4))/x
Exception generated. \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((((x**4+3*x**3-2*x**2+6*x-8)*ln(4+x)**2+(-x**4-4*x**3-2*x**2-8*x )*ln(4+x))*ln(((2-2*x)*ln(4+x)+2*x)/ln(4+x))+(x**4-18*x**2-8*x)*ln(4+x)**2 +(-x**4+18*x**2+8*x)*ln(4+x)+x**4-4*x**3-2*x**2)/((x**4+3*x**3-4*x**2)*ln( 4+x)**2+(-x**4-4*x**3)*ln(4+x)),x)
Output:
Exception raised: PolynomialError >> 1/(x**3 + 2*x**2 - 7*x + 4) contains an element of the set of generators.
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.00 \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx=\frac {-2 i \, \pi + {\left (i \, \pi + \log \left (2\right )\right )} x^{2} + {\left (x^{2} - 2\right )} \log \left (x {\left (\log \left (x + 4\right ) - 1\right )} - \log \left (x + 4\right )\right ) - {\left (x^{2} - 4 \, x - 2\right )} \log \left (\log \left (x + 4\right )\right ) - 2 \, \log \left (2\right )}{x} - 4 \, \log \left (x - 1\right ) - 4 \, \log \left (\frac {{\left (x - 1\right )} \log \left (x + 4\right ) - x}{x - 1}\right ) \] Input:
integrate((((x^4+3*x^3-2*x^2+6*x-8)*log(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*log( 4+x))*log(((2-2*x)*log(4+x)+2*x)/log(4+x))+(x^4-18*x^2-8*x)*log(4+x)^2+(-x ^4+18*x^2+8*x)*log(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*log(4+x)^2+(-x ^4-4*x^3)*log(4+x)),x, algorithm="maxima")
Output:
(-2*I*pi + (I*pi + log(2))*x^2 + (x^2 - 2)*log(x*(log(x + 4) - 1) - log(x + 4)) - (x^2 - 4*x - 2)*log(log(x + 4)) - 2*log(2))/x - 4*log(x - 1) - 4*l og(((x - 1)*log(x + 4) - x)/(x - 1))
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (28) = 56\).
Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.31 \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx={\left (x - \frac {2}{x}\right )} \log \left (-2 \, x \log \left (x + 4\right ) + 2 \, x + 2 \, \log \left (x + 4\right )\right ) - {\left (x - \frac {2}{x}\right )} \log \left (\log \left (x + 4\right )\right ) - 4 \, \log \left (x \log \left (x + 4\right ) - x - \log \left (x + 4\right )\right ) + 4 \, \log \left (\log \left (x + 4\right )\right ) \] Input:
integrate((((x^4+3*x^3-2*x^2+6*x-8)*log(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*log( 4+x))*log(((2-2*x)*log(4+x)+2*x)/log(4+x))+(x^4-18*x^2-8*x)*log(4+x)^2+(-x ^4+18*x^2+8*x)*log(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*log(4+x)^2+(-x ^4-4*x^3)*log(4+x)),x, algorithm="giac")
Output:
(x - 2/x)*log(-2*x*log(x + 4) + 2*x + 2*log(x + 4)) - (x - 2/x)*log(log(x + 4)) - 4*log(x*log(x + 4) - x - log(x + 4)) + 4*log(log(x + 4))
Time = 3.39 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.55 \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx=4\,\ln \left (x-1\right )-4\,\ln \left (\frac {8\,x+8\,\ln \left (x+4\right )-8\,x\,\ln \left (x+4\right )}{x^2+3\,x-4}\right )-4\,\ln \left (x^2-x+5\right )+4\,\ln \left (\frac {\ln \left (x+4\right )\,\left (x^2-x+5\right )}{{\left (x-1\right )}^2\,\left (x+4\right )}\right )+x\,\ln \left (\frac {2\,x+2\,\ln \left (x+4\right )-2\,x\,\ln \left (x+4\right )}{\ln \left (x+4\right )}\right )-\frac {2\,\ln \left (\frac {2\,x+2\,\ln \left (x+4\right )-2\,x\,\ln \left (x+4\right )}{\ln \left (x+4\right )}\right )}{x} \] Input:
int((log(x + 4)^2*(8*x + 18*x^2 - x^4) - log(x + 4)*(8*x + 18*x^2 - x^4) + 2*x^2 + 4*x^3 - x^4 + log((2*x - log(x + 4)*(2*x - 2))/log(x + 4))*(log(x + 4)*(8*x + 2*x^2 + 4*x^3 + x^4) - log(x + 4)^2*(6*x - 2*x^2 + 3*x^3 + x^ 4 - 8)))/(log(x + 4)*(4*x^3 + x^4) - log(x + 4)^2*(3*x^3 - 4*x^2 + x^4)),x )
Output:
4*log(x - 1) - 4*log((8*x + 8*log(x + 4) - 8*x*log(x + 4))/(3*x + x^2 - 4) ) - 4*log(x^2 - x + 5) + 4*log((log(x + 4)*(x^2 - x + 5))/((x - 1)^2*(x + 4))) + x*log((2*x + 2*log(x + 4) - 2*x*log(x + 4))/log(x + 4)) - (2*log((2 *x + 2*log(x + 4) - 2*x*log(x + 4))/log(x + 4)))/x
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.07 \[ \int \frac {-2 x^2-4 x^3+x^4+\left (8 x+18 x^2-x^4\right ) \log (4+x)+\left (-8 x-18 x^2+x^4\right ) \log ^2(4+x)+\left (\left (-8 x-2 x^2-4 x^3-x^4\right ) \log (4+x)+\left (-8+6 x-2 x^2+3 x^3+x^4\right ) \log ^2(4+x)\right ) \log \left (\frac {2 x+(2-2 x) \log (4+x)}{\log (4+x)}\right )}{\left (-4 x^3-x^4\right ) \log (4+x)+\left (-4 x^2+3 x^3+x^4\right ) \log ^2(4+x)} \, dx=\frac {4 \,\mathrm {log}\left (\mathrm {log}\left (x +4\right )\right ) x -4 \,\mathrm {log}\left (\mathrm {log}\left (x +4\right ) x -\mathrm {log}\left (x +4\right )-x \right ) x +\mathrm {log}\left (\frac {-2 \,\mathrm {log}\left (x +4\right ) x +2 \,\mathrm {log}\left (x +4\right )+2 x}{\mathrm {log}\left (x +4\right )}\right ) x^{2}-2 \,\mathrm {log}\left (\frac {-2 \,\mathrm {log}\left (x +4\right ) x +2 \,\mathrm {log}\left (x +4\right )+2 x}{\mathrm {log}\left (x +4\right )}\right )}{x} \] Input:
int((((x^4+3*x^3-2*x^2+6*x-8)*log(4+x)^2+(-x^4-4*x^3-2*x^2-8*x)*log(4+x))* log(((2-2*x)*log(4+x)+2*x)/log(4+x))+(x^4-18*x^2-8*x)*log(4+x)^2+(-x^4+18* x^2+8*x)*log(4+x)+x^4-4*x^3-2*x^2)/((x^4+3*x^3-4*x^2)*log(4+x)^2+(-x^4-4*x ^3)*log(4+x)),x)
Output:
(4*log(log(x + 4))*x - 4*log(log(x + 4)*x - log(x + 4) - x)*x + log(( - 2* log(x + 4)*x + 2*log(x + 4) + 2*x)/log(x + 4))*x**2 - 2*log(( - 2*log(x + 4)*x + 2*log(x + 4) + 2*x)/log(x + 4)))/x