Integrand size = 122, antiderivative size = 35 \[ \int \frac {4 x^5+\left (-4 x^3-2 x^5\right ) \log \left (4+4 x^2+x^4\right )+\left (-6-3 x^2\right ) \log ^2\left (4+4 x^2+x^4\right )}{\left (-2 x^4-x^6\right ) \log \left (4+4 x^2+x^4\right )+\left (6 x-2 x^2+3 x^3-x^4+\left (2 x^2+x^4\right ) \log (5)\right ) \log ^2\left (4+4 x^2+x^4\right )} \, dx=\log \left (-x-\frac {-3+x-x^2}{x}+\log (5)-\frac {x^2}{\log \left (\left (2+x^2\right )^2\right )}\right ) \]
Time = 3.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {4 x^5+\left (-4 x^3-2 x^5\right ) \log \left (4+4 x^2+x^4\right )+\left (-6-3 x^2\right ) \log ^2\left (4+4 x^2+x^4\right )}{\left (-2 x^4-x^6\right ) \log \left (4+4 x^2+x^4\right )+\left (6 x-2 x^2+3 x^3-x^4+\left (2 x^2+x^4\right ) \log (5)\right ) \log ^2\left (4+4 x^2+x^4\right )} \, dx=-\log (x)-\log \left (\log \left (\left (2+x^2\right )^2\right )\right )+\log \left (x^3-3 \log \left (\left (2+x^2\right )^2\right )+x \log \left (\left (2+x^2\right )^2\right )-x \log (5) \log \left (\left (2+x^2\right )^2\right )\right ) \]
Integrate[(4*x^5 + (-4*x^3 - 2*x^5)*Log[4 + 4*x^2 + x^4] + (-6 - 3*x^2)*Lo g[4 + 4*x^2 + x^4]^2)/((-2*x^4 - x^6)*Log[4 + 4*x^2 + x^4] + (6*x - 2*x^2 + 3*x^3 - x^4 + (2*x^2 + x^4)*Log[5])*Log[4 + 4*x^2 + x^4]^2),x]
-Log[x] - Log[Log[(2 + x^2)^2]] + Log[x^3 - 3*Log[(2 + x^2)^2] + x*Log[(2 + x^2)^2] - x*Log[5]*Log[(2 + x^2)^2]]
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.
| \(\displaystyle \int \frac {4 x^5+\left (-3 x^2-6\right ) \log ^2\left (x^4+4 x^2+4\right )+\left (-2 x^5-4 x^3\right ) \log \left (x^4+4 x^2+4\right )}{\left (-x^6-2 x^4\right ) \log \left (x^4+4 x^2+4\right )+\left (-x^4+3 x^3-2 x^2+\left (x^4+2 x^2\right ) \log (5)+6 x\right ) \log ^2\left (x^4+4 x^2+4\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-4 x^5-\left (-3 x^2-6\right ) \log ^2\left (x^4+4 x^2+4\right )-\left (-2 x^5-4 x^3\right ) \log \left (x^4+4 x^2+4\right )}{x \left (x^2+2\right ) \log \left (\left (x^2+2\right )^2\right ) \left (x^3+x (1-\log (5)) \log \left (\left (x^2+2\right )^2\right )-3 \log \left (\left (x^2+2\right )^2\right )\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {2 x^2}{x^3+x (1-\log (5)) \log \left (\left (x^2+2\right )^2\right )-3 \log \left (\left (x^2+2\right )^2\right )}+\frac {3 \log \left (\left (x^2+2\right )^2\right )}{x \left (x^3+x (1-\log (5)) \log \left (\left (x^2+2\right )^2\right )-3 \log \left (\left (x^2+2\right )^2\right )\right )}+\frac {4 x^4}{\left (-x^2-2\right ) \log \left (\left (x^2+2\right )^2\right ) \left (x^3+x (1-\log (5)) \log \left (\left (x^2+2\right )^2\right )-3 \log \left (\left (x^2+2\right )^2\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 (1-\log (5)) \int \frac {1}{-x^3-(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x+3 \log \left (\left (x^2+2\right )^2\right )}dx+\frac {9 \int \frac {1}{-x^3-(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x+3 \log \left (\left (x^2+2\right )^2\right )}dx}{(1-\log (5))^2}-2 i \sqrt {2} (1-\log (5)) \int \frac {1}{\left (i \sqrt {2}-x\right ) \left (x^3+(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x-3 \log \left (\left (x^2+2\right )^2\right )\right )}dx+6 \int \frac {1}{\left (i \sqrt {2}-x\right ) \left (x^3+(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x-3 \log \left (\left (x^2+2\right )^2\right )\right )}dx-\frac {3 \int \frac {x}{x^3+(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x-3 \log \left (\left (x^2+2\right )^2\right )}dx}{1-\log (5)}+2 \int \frac {x^2}{x^3+(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x-3 \log \left (\left (x^2+2\right )^2\right )}dx-2 i \sqrt {2} (1-\log (5)) \int \frac {1}{\left (x+i \sqrt {2}\right ) \left (x^3+(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x-3 \log \left (\left (x^2+2\right )^2\right )\right )}dx-6 \int \frac {1}{\left (x+i \sqrt {2}\right ) \left (x^3+(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x-3 \log \left (\left (x^2+2\right )^2\right )\right )}dx+\frac {27 \int \frac {1}{(3-x (1-\log (5))) \left (x^3+(1-\log (5)) \log \left (\left (x^2+2\right )^2\right ) x-3 \log \left (\left (x^2+2\right )^2\right )\right )}dx}{(1-\log (5))^2}-\log \left (\log \left (\left (x^2+2\right )^2\right )\right )-\log (x)+\log (3-x (1-\log (5)))\) |
Int[(4*x^5 + (-4*x^3 - 2*x^5)*Log[4 + 4*x^2 + x^4] + (-6 - 3*x^2)*Log[4 + 4*x^2 + x^4]^2)/((-2*x^4 - x^6)*Log[4 + 4*x^2 + x^4] + (6*x - 2*x^2 + 3*x^ 3 - x^4 + (2*x^2 + x^4)*Log[5])*Log[4 + 4*x^2 + x^4]^2),x]
3.6.38.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 4.45 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.71
| method | result | size |
| risch | \(-\ln \left (x \right )+\ln \left (\left (-\ln \left (5\right )+1\right ) x -3\right )+\ln \left (\ln \left (x^{4}+4 x^{2}+4\right )-\frac {x^{3}}{x \ln \left (5\right )-x +3}\right )-\ln \left (\ln \left (x^{4}+4 x^{2}+4\right )\right )\) | \(60\) |
| parallelrisch | \(-\ln \left (x \right )-\ln \left (\ln \left (x^{4}+4 x^{2}+4\right )\right )+\ln \left (-\ln \left (5\right ) \ln \left (x^{4}+4 x^{2}+4\right ) x +x^{3}+x \ln \left (x^{4}+4 x^{2}+4\right )-3 \ln \left (x^{4}+4 x^{2}+4\right )\right )\) | \(67\) |
| norman | \(-\ln \left (x \right )-\ln \left (\ln \left (x^{4}+4 x^{2}+4\right )\right )+\ln \left (\ln \left (5\right ) \ln \left (x^{4}+4 x^{2}+4\right ) x -x^{3}-x \ln \left (x^{4}+4 x^{2}+4\right )+3 \ln \left (x^{4}+4 x^{2}+4\right )\right )\) | \(69\) |
int(((-3*x^2-6)*ln(x^4+4*x^2+4)^2+(-2*x^5-4*x^3)*ln(x^4+4*x^2+4)+4*x^5)/(( (x^4+2*x^2)*ln(5)-x^4+3*x^3-2*x^2+6*x)*ln(x^4+4*x^2+4)^2+(-x^6-2*x^4)*ln(x ^4+4*x^2+4)),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (34) = 68\).
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {4 x^5+\left (-4 x^3-2 x^5\right ) \log \left (4+4 x^2+x^4\right )+\left (-6-3 x^2\right ) \log ^2\left (4+4 x^2+x^4\right )}{\left (-2 x^4-x^6\right ) \log \left (4+4 x^2+x^4\right )+\left (6 x-2 x^2+3 x^3-x^4+\left (2 x^2+x^4\right ) \log (5)\right ) \log ^2\left (4+4 x^2+x^4\right )} \, dx=\log \left (x \log \left (5\right ) - x + 3\right ) - \log \left (x\right ) + \log \left (-\frac {x^{3} - {\left (x \log \left (5\right ) - x + 3\right )} \log \left (x^{4} + 4 \, x^{2} + 4\right )}{x \log \left (5\right ) - x + 3}\right ) - \log \left (\log \left (x^{4} + 4 \, x^{2} + 4\right )\right ) \]
integrate(((-3*x^2-6)*log(x^4+4*x^2+4)^2+(-2*x^5-4*x^3)*log(x^4+4*x^2+4)+4 *x^5)/(((x^4+2*x^2)*log(5)-x^4+3*x^3-2*x^2+6*x)*log(x^4+4*x^2+4)^2+(-x^6-2 *x^4)*log(x^4+4*x^2+4)),x, algorithm=\
log(x*log(5) - x + 3) - log(x) + log(-(x^3 - (x*log(5) - x + 3)*log(x^4 + 4*x^2 + 4))/(x*log(5) - x + 3)) - log(log(x^4 + 4*x^2 + 4))
Exception generated. \[ \int \frac {4 x^5+\left (-4 x^3-2 x^5\right ) \log \left (4+4 x^2+x^4\right )+\left (-6-3 x^2\right ) \log ^2\left (4+4 x^2+x^4\right )}{\left (-2 x^4-x^6\right ) \log \left (4+4 x^2+x^4\right )+\left (6 x-2 x^2+3 x^3-x^4+\left (2 x^2+x^4\right ) \log (5)\right ) \log ^2\left (4+4 x^2+x^4\right )} \, dx=\text {Exception raised: PolynomialError} \]
integrate(((-3*x**2-6)*ln(x**4+4*x**2+4)**2+(-2*x**5-4*x**3)*ln(x**4+4*x** 2+4)+4*x**5)/(((x**4+2*x**2)*ln(5)-x**4+3*x**3-2*x**2+6*x)*ln(x**4+4*x**2+ 4)**2+(-x**6-2*x**4)*ln(x**4+4*x**2+4)),x)
Exception raised: PolynomialError >> 1/(-2*x**4*log(5) + x**4 + x**4*log(5 )**2 - 6*x**3 + 6*x**3*log(5) - 4*x**2*log(5) + 2*x**2*log(5)**2 + 11*x**2 - 12*x + 12*x*log(5) + 18) contains an element of the set of generators.
Time = 0.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {4 x^5+\left (-4 x^3-2 x^5\right ) \log \left (4+4 x^2+x^4\right )+\left (-6-3 x^2\right ) \log ^2\left (4+4 x^2+x^4\right )}{\left (-2 x^4-x^6\right ) \log \left (4+4 x^2+x^4\right )+\left (6 x-2 x^2+3 x^3-x^4+\left (2 x^2+x^4\right ) \log (5)\right ) \log ^2\left (4+4 x^2+x^4\right )} \, dx=\log \left (x {\left (\log \left (5\right ) - 1\right )} + 3\right ) - \log \left (x\right ) + \log \left (-\frac {x^{3} - 2 \, {\left (x {\left (\log \left (5\right ) - 1\right )} + 3\right )} \log \left (x^{2} + 2\right )}{2 \, {\left (x {\left (\log \left (5\right ) - 1\right )} + 3\right )}}\right ) - \log \left (\log \left (x^{2} + 2\right )\right ) \]
integrate(((-3*x^2-6)*log(x^4+4*x^2+4)^2+(-2*x^5-4*x^3)*log(x^4+4*x^2+4)+4 *x^5)/(((x^4+2*x^2)*log(5)-x^4+3*x^3-2*x^2+6*x)*log(x^4+4*x^2+4)^2+(-x^6-2 *x^4)*log(x^4+4*x^2+4)),x, algorithm=\
log(x*(log(5) - 1) + 3) - log(x) + log(-1/2*(x^3 - 2*(x*(log(5) - 1) + 3)* log(x^2 + 2))/(x*(log(5) - 1) + 3)) - log(log(x^2 + 2))
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.94 \[ \int \frac {4 x^5+\left (-4 x^3-2 x^5\right ) \log \left (4+4 x^2+x^4\right )+\left (-6-3 x^2\right ) \log ^2\left (4+4 x^2+x^4\right )}{\left (-2 x^4-x^6\right ) \log \left (4+4 x^2+x^4\right )+\left (6 x-2 x^2+3 x^3-x^4+\left (2 x^2+x^4\right ) \log (5)\right ) \log ^2\left (4+4 x^2+x^4\right )} \, dx=\log \left (-x^{3} + x \log \left (5\right ) \log \left (x^{4} + 4 \, x^{2} + 4\right ) - x \log \left (x^{4} + 4 \, x^{2} + 4\right ) + 3 \, \log \left (x^{4} + 4 \, x^{2} + 4\right )\right ) - \log \left (x\right ) - \log \left (\log \left (x^{4} + 4 \, x^{2} + 4\right )\right ) \]
integrate(((-3*x^2-6)*log(x^4+4*x^2+4)^2+(-2*x^5-4*x^3)*log(x^4+4*x^2+4)+4 *x^5)/(((x^4+2*x^2)*log(5)-x^4+3*x^3-2*x^2+6*x)*log(x^4+4*x^2+4)^2+(-x^6-2 *x^4)*log(x^4+4*x^2+4)),x, algorithm=\
log(-x^3 + x*log(5)*log(x^4 + 4*x^2 + 4) - x*log(x^4 + 4*x^2 + 4) + 3*log( x^4 + 4*x^2 + 4)) - log(x) - log(log(x^4 + 4*x^2 + 4))
Time = 12.98 (sec) , antiderivative size = 27623, normalized size of antiderivative = 789.23 \[ \int \frac {4 x^5+\left (-4 x^3-2 x^5\right ) \log \left (4+4 x^2+x^4\right )+\left (-6-3 x^2\right ) \log ^2\left (4+4 x^2+x^4\right )}{\left (-2 x^4-x^6\right ) \log \left (4+4 x^2+x^4\right )+\left (6 x-2 x^2+3 x^3-x^4+\left (2 x^2+x^4\right ) \log (5)\right ) \log ^2\left (4+4 x^2+x^4\right )} \, dx=\text {Too large to display} \]
int((log(4*x^2 + x^4 + 4)*(4*x^3 + 2*x^5) + log(4*x^2 + x^4 + 4)^2*(3*x^2 + 6) - 4*x^5)/(log(4*x^2 + x^4 + 4)*(2*x^4 + x^6) - log(4*x^2 + x^4 + 4)^2 *(6*x + log(5)*(2*x^2 + x^4) - 2*x^2 + 3*x^3 - x^4)),x)
log(x*(9*log((x^2 + 2)^2) + x^2*log((x^2 + 2)^2) - x^4*log(5) - 3*x^3 + x^ 4 - 6*x*log((x^2 + 2)^2) + x^2*log((x^2 + 2)^2)*log(5)^2 + 6*x*log((x^2 + 2)^2)*log(5) - 2*x^2*log((x^2 + 2)^2)*log(5))) - log(x*(72*log(4*x^2 + x^4 + 4) + 12*x^2*log(4*x^2 + x^4 + 4) - 9*x^3*log(4*x^2 + x^4 + 4) + 2*x^4*l og(4*x^2 + x^4 + 4) - 4*x^4*log(5) - 66*x*log(4*x^2 + x^4 + 4) - 12*x^3 + 4*x^4 + 48*x*log(5)*log(4*x^2 + x^4 + 4) - 20*x^2*log(5)*log(4*x^2 + x^4 + 4) - 2*x^4*log(5)*log(4*x^2 + x^4 + 4) + 8*x^2*log(5)^2*log(4*x^2 + x^4 + 4)) - x*(9*x^3*log(4*x^2 + x^4 + 4) - 4*x^2*log(4*x^2 + x^4 + 4) - 2*x^4* log(4*x^2 + x^4 + 4) - 4*x^4*log(5) + 18*x*log(4*x^2 + x^4 + 4) - 12*x^3 + 4*x^4 + 4*x^2*log(5)*log(4*x^2 + x^4 + 4) + 2*x^4*log(5)*log(4*x^2 + x^4 + 4))) - log(x) + symsum(log(33043620105792*log(625) - 191446983005184*log (5) - root(12476089554*z^5*log(5)*log(625) - 4950250980*z^5*log(5)^2*log(6 25)^2 + 484358724*z^5*log(5)^4*log(625)^3 - 239636880*z^5*log(5)^2*log(625 )^4 - 94885776*z^5*log(5)^5*log(625)^3 + 88216560*z^5*log(5)^7*log(625)^2 - 54064008*z^5*log(5)^6*log(625)^2 - 25406208*z^5*log(5)^8*log(625)^2 + 19 994256*z^5*log(5)^6*log(625)^3 + 19789920*z^5*log(5)^3*log(625)^4 + 800000 0*z^5*log(5)^9*log(625)^2 - 4233600*z^5*log(5)^7*log(625)^3 - 3421440*z^5* log(5)^4*log(625)^4 + 559872*z^5*log(5)^5*log(625)^4 - 450048*z^5*log(5)^1 0*log(625)^2 + 179712*z^5*log(5)^8*log(625)^3 - 33024*z^5*log(5)^11*log(62 5)^2 + 2048*z^5*log(5)^12*log(625)^2 + 1215973440*z^5*log(5)^6*log(625)...