Integrand size = 71, antiderivative size = 29 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=-3+x+x \log (x) \left (x^2-\frac {-4+x-\log (x)}{-1+x^2}+\log (x)\right ) \] Output:
x-3+(x^2+ln(x)-(-ln(x)+x-4)/(x^2-1))*x*ln(x)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {x \left (-1+x^2\right )+\left (x \left (4-x-x^2+x^4\right )+\left (-1+x^2\right ) \log (1-x)\right ) \log (x)+x^3 \log ^2(x)}{-1+x^2}+\operatorname {PolyLog}(2,1-x)+\operatorname {PolyLog}(2,x) \] Input:
Integrate[(-3 + x + 3*x^2 - x^3 - x^4 + x^6 + (-4 + 2*x - 3*x^2 - 4*x^4 + 3*x^6)*Log[x] + (-3*x^2 + x^4)*Log[x]^2)/(1 - 2*x^2 + x^4),x]
Output:
(x*(-1 + x^2) + (x*(4 - x - x^2 + x^4) + (-1 + x^2)*Log[1 - x])*Log[x] + x ^3*Log[x]^2)/(-1 + x^2) + PolyLog[2, 1 - x] + PolyLog[2, x]
Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(29)=58\).
Time = 0.70 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1380, 25, 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.
\(\displaystyle \int \frac {x^6-x^4-x^3+3 x^2+\left (x^4-3 x^2\right ) \log ^2(x)+\left (3 x^6-4 x^4-3 x^2+2 x-4\right ) \log (x)+x-3}{x^4-2 x^2+1} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int -\frac {-x^6+x^4+x^3-3 x^2+\left (3 x^2-x^4\right ) \log ^2(x)+\left (-3 x^6+4 x^4+3 x^2-2 x+4\right ) \log (x)-x+3}{\left (1-x^2\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {-x^6+x^4+x^3-3 x^2-x+\left (3 x^2-x^4\right ) \log ^2(x)+\left (-3 x^6+4 x^4+3 x^2-2 x+4\right ) \log (x)+3}{\left (1-x^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\int \left (-\frac {x^6}{\left (x^2-1\right )^2}+\frac {x^4}{\left (x^2-1\right )^2}+\frac {x^3}{\left (x^2-1\right )^2}-\frac {\left (x^2-3\right ) \log ^2(x) x^2}{\left (x^2-1\right )^2}-\frac {3 x^2}{\left (x^2-1\right )^2}-\frac {x}{\left (x^2-1\right )^2}-\frac {\left (3 x^6-4 x^4-3 x^2+2 x-4\right ) \log (x)}{\left (x^2-1\right )^2}+\frac {3}{\left (x^2-1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \text {arctanh}(x)+\frac {x^3}{2}+x^3 \log (x)-\frac {1}{2} \log \left (1-x^2\right )+\frac {x^5}{2 \left (1-x^2\right )}-\frac {x^3}{2 \left (1-x^2\right )}+x-\frac {x \log ^2(x)}{2 (1-x)}-\frac {x \log ^2(x)}{2 (x+1)}+x \log ^2(x)-\frac {3 x \log (x)}{2 (1-x)}-\frac {5 x \log (x)}{2 (x+1)}-\frac {3}{2} \log (1-x)+\frac {5}{2} \log (x+1)\) |
Input:
Int[(-3 + x + 3*x^2 - x^3 - x^4 + x^6 + (-4 + 2*x - 3*x^2 - 4*x^4 + 3*x^6) *Log[x] + (-3*x^2 + x^4)*Log[x]^2)/(1 - 2*x^2 + x^4),x]
Output:
x + x^3/2 - x^3/(2*(1 - x^2)) + x^5/(2*(1 - x^2)) - 4*ArcTanh[x] - (3*Log[ 1 - x])/2 - (3*x*Log[x])/(2*(1 - x)) + x^3*Log[x] - (5*x*Log[x])/(2*(1 + x )) + x*Log[x]^2 - (x*Log[x]^2)/(2*(1 - x)) - (x*Log[x]^2)/(2*(1 + x)) + (5 *Log[1 + x])/2 - Log[1 - x^2]/2
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 2.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55
method | result | size |
risch | \(\frac {x^{3} \ln \left (x \right )^{2}}{x^{2}-1}+\frac {\left (x^{5}-x^{3}+4 x -1\right ) \ln \left (x \right )}{x^{2}-1}+x -\ln \left (x \right )\) | \(45\) |
norman | \(\frac {x^{3}+x^{3} \ln \left (x \right )^{2}+x^{5} \ln \left (x \right )-x +4 x \ln \left (x \right )-x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )}{x^{2}-1}\) | \(49\) |
parallelrisch | \(\frac {x^{3}+x^{3} \ln \left (x \right )^{2}+x^{5} \ln \left (x \right )-x +4 x \ln \left (x \right )-x^{2} \ln \left (x \right )-x^{3} \ln \left (x \right )}{x^{2}-1}\) | \(49\) |
orering | \(\frac {x \left (54 x^{10}-210 x^{8}-27 x^{7}+222 x^{6}-561 x^{5}+2178 x^{4}-896 x^{3}+525 x^{2}-120 x -327\right ) \left (\left (x^{4}-3 x^{2}\right ) \ln \left (x \right )^{2}+\left (3 x^{6}-4 x^{4}-3 x^{2}+2 x -4\right ) \ln \left (x \right )+x^{6}-x^{4}-x^{3}+3 x^{2}+x -3\right )}{3 \left (18 x^{10}-134 x^{8}-9 x^{7}-38 x^{6}-169 x^{5}+454 x^{4}-38 x^{3}+110 x^{2}+30 x -96\right ) \left (x^{4}-2 x^{2}+1\right )}-\frac {x \left (48 x^{11}-24 x^{9}-384 x^{7}-197 x^{6}+2178 x^{5}-1054 x^{4}-456 x^{3}+711 x^{2}-462 x +156\right ) \left (\frac {\left (4 x^{3}-6 x \right ) \ln \left (x \right )^{2}+\frac {2 \left (x^{4}-3 x^{2}\right ) \ln \left (x \right )}{x}+\left (18 x^{5}-16 x^{3}-6 x +2\right ) \ln \left (x \right )+\frac {3 x^{6}-4 x^{4}-3 x^{2}+2 x -4}{x}+6 x^{5}-4 x^{3}-3 x^{2}+6 x +1}{x^{4}-2 x^{2}+1}-\frac {\left (\left (x^{4}-3 x^{2}\right ) \ln \left (x \right )^{2}+\left (3 x^{6}-4 x^{4}-3 x^{2}+2 x -4\right ) \ln \left (x \right )+x^{6}-x^{4}-x^{3}+3 x^{2}+x -3\right ) \left (4 x^{3}-4 x \right )}{\left (x^{4}-2 x^{2}+1\right )^{2}}\right )}{6 \left (18 x^{10}-134 x^{8}-9 x^{7}-38 x^{6}-169 x^{5}+454 x^{4}-38 x^{3}+110 x^{2}+30 x -96\right )}+\frac {\left (-1+x \right ) \left (1+x \right ) x^{2} \left (12 x^{9}+48 x^{7}-324 x^{5}+197 x^{4}-117 x^{3}+23 x^{2}+231 x -156\right ) \left (\frac {\left (12 x^{2}-6\right ) \ln \left (x \right )^{2}+\frac {4 \left (4 x^{3}-6 x \right ) \ln \left (x \right )}{x}+\frac {2 x^{4}-6 x^{2}}{x^{2}}-\frac {2 \left (x^{4}-3 x^{2}\right ) \ln \left (x \right )}{x^{2}}+\left (90 x^{4}-48 x^{2}-6\right ) \ln \left (x \right )+\frac {36 x^{5}-32 x^{3}-12 x +4}{x}-\frac {3 x^{6}-4 x^{4}-3 x^{2}+2 x -4}{x^{2}}+30 x^{4}-12 x^{2}-6 x +6}{x^{4}-2 x^{2}+1}-\frac {2 \left (\left (4 x^{3}-6 x \right ) \ln \left (x \right )^{2}+\frac {2 \left (x^{4}-3 x^{2}\right ) \ln \left (x \right )}{x}+\left (18 x^{5}-16 x^{3}-6 x +2\right ) \ln \left (x \right )+\frac {3 x^{6}-4 x^{4}-3 x^{2}+2 x -4}{x}+6 x^{5}-4 x^{3}-3 x^{2}+6 x +1\right ) \left (4 x^{3}-4 x \right )}{\left (x^{4}-2 x^{2}+1\right )^{2}}+\frac {2 \left (\left (x^{4}-3 x^{2}\right ) \ln \left (x \right )^{2}+\left (3 x^{6}-4 x^{4}-3 x^{2}+2 x -4\right ) \ln \left (x \right )+x^{6}-x^{4}-x^{3}+3 x^{2}+x -3\right ) \left (4 x^{3}-4 x \right )^{2}}{\left (x^{4}-2 x^{2}+1\right )^{3}}-\frac {\left (\left (x^{4}-3 x^{2}\right ) \ln \left (x \right )^{2}+\left (3 x^{6}-4 x^{4}-3 x^{2}+2 x -4\right ) \ln \left (x \right )+x^{6}-x^{4}-x^{3}+3 x^{2}+x -3\right ) \left (12 x^{2}-4\right )}{\left (x^{4}-2 x^{2}+1\right )^{2}}\right )}{108 x^{10}-804 x^{8}-54 x^{7}-228 x^{6}-1014 x^{5}+2724 x^{4}-228 x^{3}+660 x^{2}+180 x -576}\) | \(966\) |
Input:
int(((x^4-3*x^2)*ln(x)^2+(3*x^6-4*x^4-3*x^2+2*x-4)*ln(x)+x^6-x^4-x^3+3*x^2 +x-3)/(x^4-2*x^2+1),x,method=_RETURNVERBOSE)
Output:
x^3/(x^2-1)*ln(x)^2+(x^5-x^3+4*x-1)/(x^2-1)*ln(x)+x-ln(x)
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {x^{3} \log \left (x\right )^{2} + x^{3} + {\left (x^{5} - x^{3} - x^{2} + 4 \, x\right )} \log \left (x\right ) - x}{x^{2} - 1} \] Input:
integrate(((x^4-3*x^2)*log(x)^2+(3*x^6-4*x^4-3*x^2+2*x-4)*log(x)+x^6-x^4-x ^3+3*x^2+x-3)/(x^4-2*x^2+1),x, algorithm="fricas")
Output:
(x^3*log(x)^2 + x^3 + (x^5 - x^3 - x^2 + 4*x)*log(x) - x)/(x^2 - 1)
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {x^{3} \log {\left (x \right )}^{2}}{x^{2} - 1} + x - \log {\left (x \right )} + \frac {\left (x^{5} - x^{3} + 4 x - 1\right ) \log {\left (x \right )}}{x^{2} - 1} \] Input:
integrate(((x**4-3*x**2)*ln(x)**2+(3*x**6-4*x**4-3*x**2+2*x-4)*ln(x)+x**6- x**4-x**3+3*x**2+x-3)/(x**4-2*x**2+1),x)
Output:
x**3*log(x)**2/(x**2 - 1) + x - log(x) + (x**5 - x**3 + 4*x - 1)*log(x)/(x **2 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.59 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {1}{3} \, x^{3} + x - \frac {x^{5} - 3 \, x^{3} \log \left (x\right )^{2} - x^{3} - 3 \, {\left (x^{5} - x^{3} + 4 \, x - 1\right )} \log \left (x\right )}{3 \, {\left (x^{2} - 1\right )}} - \frac {1}{2} \, \log \left (x^{2} - 1\right ) + \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \log \left (x\right ) \] Input:
integrate(((x^4-3*x^2)*log(x)^2+(3*x^6-4*x^4-3*x^2+2*x-4)*log(x)+x^6-x^4-x ^3+3*x^2+x-3)/(x^4-2*x^2+1),x, algorithm="maxima")
Output:
1/3*x^3 + x - 1/3*(x^5 - 3*x^3*log(x)^2 - x^3 - 3*(x^5 - x^3 + 4*x - 1)*lo g(x))/(x^2 - 1) - 1/2*log(x^2 - 1) + 1/2*log(x + 1) + 1/2*log(x - 1) - log (x)
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx={\left (x + \frac {x}{x^{2} - 1}\right )} \log \left (x\right )^{2} + {\left (x^{3} + \frac {4 \, x - 1}{x^{2} - 1}\right )} \log \left (x\right ) + x - \log \left (x\right ) \] Input:
integrate(((x^4-3*x^2)*log(x)^2+(3*x^6-4*x^4-3*x^2+2*x-4)*log(x)+x^6-x^4-x ^3+3*x^2+x-3)/(x^4-2*x^2+1),x, algorithm="giac")
Output:
(x + x/(x^2 - 1))*log(x)^2 + (x^3 + (4*x - 1)/(x^2 - 1))*log(x) + x - log( x)
Time = 2.95 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=x^3\,\ln \left (x\right )-\ln \left (x\right )+x\,\left ({\ln \left (x\right )}^2+1\right )-\frac {\ln \left (x\right )-x\,\left ({\ln \left (x\right )}^2+4\,\ln \left (x\right )\right )}{x^2-1} \] Input:
int(-(log(x)*(3*x^2 - 2*x + 4*x^4 - 3*x^6 + 4) - x + log(x)^2*(3*x^2 - x^4 ) - 3*x^2 + x^3 + x^4 - x^6 + 3)/(x^4 - 2*x^2 + 1),x)
Output:
x^3*log(x) - log(x) + x*(log(x)^2 + 1) - (log(x) - x*(4*log(x) + log(x)^2) )/(x^2 - 1)
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-3+x+3 x^2-x^3-x^4+x^6+\left (-4+2 x-3 x^2-4 x^4+3 x^6\right ) \log (x)+\left (-3 x^2+x^4\right ) \log ^2(x)}{1-2 x^2+x^4} \, dx=\frac {x \left (\mathrm {log}\left (x \right )^{2} x^{2}+\mathrm {log}\left (x \right ) x^{4}-\mathrm {log}\left (x \right ) x^{2}-\mathrm {log}\left (x \right ) x +4 \,\mathrm {log}\left (x \right )+x^{2}-1\right )}{x^{2}-1} \] Input:
int(((x^4-3*x^2)*log(x)^2+(3*x^6-4*x^4-3*x^2+2*x-4)*log(x)+x^6-x^4-x^3+3*x ^2+x-3)/(x^4-2*x^2+1),x)
Output:
(x*(log(x)**2*x**2 + log(x)*x**4 - log(x)*x**2 - log(x)*x + 4*log(x) + x** 2 - 1))/(x**2 - 1)