Integrand size = 59, antiderivative size = 28 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=-\frac {4}{x}+\frac {3+x^4 \log ^2(x)}{\frac {5}{x}-x} \] Output:
(3+x^4*ln(x)^2)/(5/x-x)-4/x
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=\frac {20-7 x^2-x^6 \log ^2(x)}{x \left (-5+x^2\right )} \] Input:
Integrate[(100 - 25*x^2 + 7*x^4 + (10*x^6 - 2*x^8)*Log[x] + (25*x^6 - 3*x^ 8)*Log[x]^2)/(25*x^2 - 10*x^4 + x^6),x]
Output:
(20 - 7*x^2 - x^6*Log[x]^2)/(x*(-5 + x^2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.07 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {2026, 1380, 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.
\(\displaystyle \int \frac {7 x^4-25 x^2+\left (25 x^6-3 x^8\right ) \log ^2(x)+\left (10 x^6-2 x^8\right ) \log (x)+100}{x^6-10 x^4+25 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {7 x^4-25 x^2+\left (25 x^6-3 x^8\right ) \log ^2(x)+\left (10 x^6-2 x^8\right ) \log (x)+100}{x^2 \left (x^4-10 x^2+25\right )}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int \frac {7 x^4-25 x^2+\left (25 x^6-3 x^8\right ) \log ^2(x)+2 \left (5 x^6-x^8\right ) \log (x)+100}{x^2 \left (5-x^2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {7 x^4-25 x^2+100}{\left (x^2-5\right )^2 x^2}-\frac {\left (3 x^2-25\right ) x^4 \log ^2(x)}{\left (x^2-5\right )^2}-\frac {2 x^4 \log (x)}{x^2-5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 10 \sqrt {5} \text {arctanh}\left (\frac {x}{\sqrt {5}}\right ) \log (x)-5 \sqrt {5} \operatorname {PolyLog}\left (2,\frac {x}{\sqrt {5}}\right )-5 \sqrt {5} \operatorname {PolyLog}\left (2,1-\frac {x}{\sqrt {5}}\right )+x^3 \left (-\log ^2(x)\right )+\frac {3 x}{5-x^2}-\frac {4}{x}+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (\sqrt {5}-x\right )}+\frac {5 \sqrt {5} x \log ^2(x)}{2 \left (x+\sqrt {5}\right )}-5 x \log ^2(x)+\frac {5}{2} \sqrt {5} \log (5) \log \left (\sqrt {5}-x\right )-5 \sqrt {5} \log (x) \log \left (\frac {x}{\sqrt {5}}+1\right )\) |
Input:
Int[(100 - 25*x^2 + 7*x^4 + (10*x^6 - 2*x^8)*Log[x] + (25*x^6 - 3*x^8)*Log [x]^2)/(25*x^2 - 10*x^4 + x^6),x]
Output:
-4/x + (3*x)/(5 - x^2) + (5*Sqrt[5]*Log[5]*Log[Sqrt[5] - x])/2 + 10*Sqrt[5 ]*ArcTanh[x/Sqrt[5]]*Log[x] - 5*x*Log[x]^2 + (5*Sqrt[5]*x*Log[x]^2)/(2*(Sq rt[5] - x)) - x^3*Log[x]^2 + (5*Sqrt[5]*x*Log[x]^2)/(2*(Sqrt[5] + x)) - 5* Sqrt[5]*Log[x]*Log[1 + x/Sqrt[5]] - 5*Sqrt[5]*PolyLog[2, x/Sqrt[5]] - 5*Sq rt[5]*PolyLog[2, 1 - x/Sqrt[5]]
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]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 1.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {-5 x^{6} \ln \left (x \right )^{2}+100-35 x^{2}}{5 x \left (x^{2}-5\right )}\) | \(29\) |
risch | \(-\frac {x^{5} \ln \left (x \right )^{2}}{x^{2}-5}-\frac {7 x^{2}-20}{x \left (x^{2}-5\right )}\) | \(37\) |
orering | \(\frac {x \left (19 x^{8}-305 x^{6}-56 x^{4}-3040 x^{2}+12600\right ) \left (\left (-3 x^{8}+25 x^{6}\right ) \ln \left (x \right )^{2}+\left (-2 x^{8}+10 x^{6}\right ) \ln \left (x \right )+7 x^{4}-25 x^{2}+100\right )}{\left (27 x^{8}-625 x^{6}-504 x^{4}+2720 x^{2}-45000\right ) \left (x^{6}-10 x^{4}+25 x^{2}\right )}-\frac {2 x^{2} \left (3 x^{8}-30 x^{6}+56 x^{4}-1120 x^{2}+5400\right ) \left (\frac {\left (-24 x^{7}+150 x^{5}\right ) \ln \left (x \right )^{2}+\frac {2 \left (-3 x^{8}+25 x^{6}\right ) \ln \left (x \right )}{x}+\left (-16 x^{7}+60 x^{5}\right ) \ln \left (x \right )+\frac {-2 x^{8}+10 x^{6}}{x}+28 x^{3}-50 x}{x^{6}-10 x^{4}+25 x^{2}}-\frac {\left (\left (-3 x^{8}+25 x^{6}\right ) \ln \left (x \right )^{2}+\left (-2 x^{8}+10 x^{6}\right ) \ln \left (x \right )+7 x^{4}-25 x^{2}+100\right ) \left (6 x^{5}-40 x^{3}+50 x \right )}{\left (x^{6}-10 x^{4}+25 x^{2}\right )^{2}}\right )}{27 x^{8}-625 x^{6}-504 x^{4}+2720 x^{2}-45000}+\frac {x^{3} \left (x^{6}+56 x^{2}-360\right ) \left (x^{2}-5\right ) \left (\frac {\left (-168 x^{6}+750 x^{4}\right ) \ln \left (x \right )^{2}+\frac {4 \left (-24 x^{7}+150 x^{5}\right ) \ln \left (x \right )}{x}+\frac {-6 x^{8}+50 x^{6}}{x^{2}}-\frac {2 \left (-3 x^{8}+25 x^{6}\right ) \ln \left (x \right )}{x^{2}}+\left (-112 x^{6}+300 x^{4}\right ) \ln \left (x \right )+\frac {-32 x^{7}+120 x^{5}}{x}-\frac {-2 x^{8}+10 x^{6}}{x^{2}}+84 x^{2}-50}{x^{6}-10 x^{4}+25 x^{2}}-\frac {2 \left (\left (-24 x^{7}+150 x^{5}\right ) \ln \left (x \right )^{2}+\frac {2 \left (-3 x^{8}+25 x^{6}\right ) \ln \left (x \right )}{x}+\left (-16 x^{7}+60 x^{5}\right ) \ln \left (x \right )+\frac {-2 x^{8}+10 x^{6}}{x}+28 x^{3}-50 x \right ) \left (6 x^{5}-40 x^{3}+50 x \right )}{\left (x^{6}-10 x^{4}+25 x^{2}\right )^{2}}+\frac {2 \left (\left (-3 x^{8}+25 x^{6}\right ) \ln \left (x \right )^{2}+\left (-2 x^{8}+10 x^{6}\right ) \ln \left (x \right )+7 x^{4}-25 x^{2}+100\right ) \left (6 x^{5}-40 x^{3}+50 x \right )^{2}}{\left (x^{6}-10 x^{4}+25 x^{2}\right )^{3}}-\frac {\left (\left (-3 x^{8}+25 x^{6}\right ) \ln \left (x \right )^{2}+\left (-2 x^{8}+10 x^{6}\right ) \ln \left (x \right )+7 x^{4}-25 x^{2}+100\right ) \left (30 x^{4}-120 x^{2}+50\right )}{\left (x^{6}-10 x^{4}+25 x^{2}\right )^{2}}\right )}{27 x^{8}-625 x^{6}-504 x^{4}+2720 x^{2}-45000}\) | \(757\) |
Input:
int(((-3*x^8+25*x^6)*ln(x)^2+(-2*x^8+10*x^6)*ln(x)+7*x^4-25*x^2+100)/(x^6- 10*x^4+25*x^2),x,method=_RETURNVERBOSE)
Output:
1/5/x*(-5*x^6*ln(x)^2+100-35*x^2)/(x^2-5)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=-\frac {x^{6} \log \left (x\right )^{2} + 7 \, x^{2} - 20}{x^{3} - 5 \, x} \] Input:
integrate(((-3*x^8+25*x^6)*log(x)^2+(-2*x^8+10*x^6)*log(x)+7*x^4-25*x^2+10 0)/(x^6-10*x^4+25*x^2),x, algorithm="fricas")
Output:
-(x^6*log(x)^2 + 7*x^2 - 20)/(x^3 - 5*x)
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=- \frac {x^{5} \log {\left (x \right )}^{2}}{x^{2} - 5} + \frac {20 - 7 x^{2}}{x^{3} - 5 x} \] Input:
integrate(((-3*x**8+25*x**6)*ln(x)**2+(-2*x**8+10*x**6)*ln(x)+7*x**4-25*x* *2+100)/(x**6-10*x**4+25*x**2),x)
Output:
-x**5*log(x)**2/(x**2 - 5) + (20 - 7*x**2)/(x**3 - 5*x)
Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=-\frac {x^{5} \log \left (x\right )^{2}}{x^{2} - 5} - \frac {2 \, {\left (3 \, x^{2} - 10\right )}}{x^{3} - 5 \, x} - \frac {x}{x^{2} - 5} \] Input:
integrate(((-3*x^8+25*x^6)*log(x)^2+(-2*x^8+10*x^6)*log(x)+7*x^4-25*x^2+10 0)/(x^6-10*x^4+25*x^2),x, algorithm="maxima")
Output:
-x^5*log(x)^2/(x^2 - 5) - 2*(3*x^2 - 10)/(x^3 - 5*x) - x/(x^2 - 5)
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=-{\left (x^{3} + 5 \, x + \frac {25 \, x}{x^{2} - 5}\right )} \log \left (x\right )^{2} - \frac {3 \, x}{x^{2} - 5} - \frac {4}{x} \] Input:
integrate(((-3*x^8+25*x^6)*log(x)^2+(-2*x^8+10*x^6)*log(x)+7*x^4-25*x^2+10 0)/(x^6-10*x^4+25*x^2),x, algorithm="giac")
Output:
-(x^3 + 5*x + 25*x/(x^2 - 5))*log(x)^2 - 3*x/(x^2 - 5) - 4/x
Time = 2.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=-\frac {x^6\,{\ln \left (x\right )}^2+7\,x^2-20}{x\,\left (x^2-5\right )} \] Input:
int((log(x)*(10*x^6 - 2*x^8) + log(x)^2*(25*x^6 - 3*x^8) - 25*x^2 + 7*x^4 + 100)/(25*x^2 - 10*x^4 + x^6),x)
Output:
-(x^6*log(x)^2 + 7*x^2 - 20)/(x*(x^2 - 5))
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {100-25 x^2+7 x^4+\left (10 x^6-2 x^8\right ) \log (x)+\left (25 x^6-3 x^8\right ) \log ^2(x)}{25 x^2-10 x^4+x^6} \, dx=\frac {-\mathrm {log}\left (x \right )^{2} x^{6}-7 x^{2}+20}{x \left (x^{2}-5\right )} \] Input:
int(((-3*x^8+25*x^6)*log(x)^2+(-2*x^8+10*x^6)*log(x)+7*x^4-25*x^2+100)/(x^ 6-10*x^4+25*x^2),x)
Output:
( - log(x)**2*x**6 - 7*x**2 + 20)/(x*(x**2 - 5))