Integrand size = 52, antiderivative size = 31 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=\frac {\frac {4}{x}-x^2-\log \left (\frac {\log (18)}{x}\right )}{x \left (5+x^2\right )} \] Output:
(4/x-x^2-ln(ln(18)/x))/(x^2+5)/x
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 6.06 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=\frac {1}{50} \left (\frac {40}{x^2}-\frac {40}{5+x^2}-\frac {50 x}{5+x^2}-10 \sqrt {5} \arctan \left (\frac {x}{\sqrt {5}}\right )-i \sqrt {5} \log \left (\sqrt {5}+i x\right )+i \sqrt {5} \log \left (i \sqrt {5}+x\right )+4 i \sqrt {5} \log \left (1-\frac {i x}{\sqrt {5}}\right )-4 i \sqrt {5} \log \left (1+\frac {i x}{\sqrt {5}}\right )+\frac {5 i \log \left (\frac {\log (18)}{x}\right )}{\sqrt {5}+i x}-\frac {10 \log \left (\frac {\log (18)}{x}\right )}{x}+\frac {5 \log \left (\frac {\log (18)}{x}\right )}{i \sqrt {5}+x}\right ) \] Input:
Integrate[(-40 + 5*x - 16*x^2 - 4*x^3 + x^5 + (5*x + 3*x^3)*Log[Log[18]/x] )/(25*x^3 + 10*x^5 + x^7),x]
Output:
(40/x^2 - 40/(5 + x^2) - (50*x)/(5 + x^2) - 10*Sqrt[5]*ArcTan[x/Sqrt[5]] - I*Sqrt[5]*Log[Sqrt[5] + I*x] + I*Sqrt[5]*Log[I*Sqrt[5] + x] + (4*I)*Sqrt[ 5]*Log[1 - (I*x)/Sqrt[5]] - (4*I)*Sqrt[5]*Log[1 + (I*x)/Sqrt[5]] + ((5*I)* Log[Log[18]/x])/(Sqrt[5] + I*x) - (10*Log[Log[18]/x])/x + (5*Log[Log[18]/x ])/(I*Sqrt[5] + x))/50
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(31)=62\).
Time = 0.70 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {2026, 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.
| \(\displaystyle \int \frac {x^5-4 x^3+\left (3 x^3+5 x\right ) \log \left (\frac {\log (18)}{x}\right )-16 x^2+5 x-40}{x^7+10 x^5+25 x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {x^5-4 x^3+\left (3 x^3+5 x\right ) \log \left (\frac {\log (18)}{x}\right )-16 x^2+5 x-40}{x^3 \left (x^4+10 x^2+25\right )}dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \int -\frac {-x^5+4 x^3-\left (3 x^3+5 x\right ) \log \left (\frac {\log (18)}{x}\right )+16 x^2-5 x+40}{x^3 \left (x^2+5\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {-x^5+4 x^3+16 x^2-5 x-\left (3 x^3+5 x\right ) \log \left (\frac {\log (18)}{x}\right )+40}{x^3 \left (x^2+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (-\frac {x^2}{\left (x^2+5\right )^2}+\frac {4}{\left (x^2+5\right )^2}+\frac {16}{\left (x^2+5\right )^2 x}-\frac {\left (3 x^2+5\right ) \log \left (\frac {\log (18)}{x}\right )}{\left (x^2+5\right )^2 x^2}-\frac {5}{\left (x^2+5\right )^2 x^2}+\frac {40}{\left (x^2+5\right )^2 x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {9 x}{10 \left (x^2+5\right )}-\frac {4}{5 \left (x^2+5\right )}+\frac {1}{2 \left (x^2+5\right ) x}+\frac {4}{5 x^2}+\frac {x \log \left (\frac {\log (18)}{x}\right )}{5 \left (x^2+5\right )}-\frac {1}{10 x}-\frac {\log \left (\frac {\log (18)}{x}\right )}{5 x}\) |
Input:
Int[(-40 + 5*x - 16*x^2 - 4*x^3 + x^5 + (5*x + 3*x^3)*Log[Log[18]/x])/(25* x^3 + 10*x^5 + x^7),x]
Output:
4/(5*x^2) - 1/(10*x) - 4/(5*(5 + x^2)) + 1/(2*x*(5 + x^2)) - (9*x)/(10*(5 + x^2)) - Log[Log[18]/x]/(5*x) + (x*Log[Log[18]/x])/(5*(5 + x^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]
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 = 1.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
| method | result | size |
| norman | \(\frac {4-x^{3}-\ln \left (\frac {\ln \left (18\right )}{x}\right ) x}{x^{2} \left (x^{2}+5\right )}\) | \(29\) |
| parallelrisch | \(\frac {4-x^{3}-\ln \left (\frac {\ln \left (18\right )}{x}\right ) x}{x^{2} \left (x^{2}+5\right )}\) | \(29\) |
| risch | \(-\frac {\ln \left (\frac {\ln \left (2\right )+2 \ln \left (3\right )}{x}\right )}{x \left (x^{2}+5\right )}-\frac {x^{3}-4}{x^{2} \left (x^{2}+5\right )}\) | \(43\) |
| orering | \(-\frac {x \left (10 x^{5}+3 x^{3}+32 x^{2}-15 x +80\right ) \left (\left (3 x^{3}+5 x \right ) \ln \left (\frac {\ln \left (18\right )}{x}\right )+x^{5}-4 x^{3}-16 x^{2}+5 x -40\right )}{\left (6 x^{5}-19 x^{3}+48 x^{2}-5 x +40\right ) \left (x^{7}+10 x^{5}+25 x^{3}\right )}-\frac {\left (2 x^{3}-x +4\right ) x^{2} \left (x^{2}+5\right ) \left (\frac {\left (9 x^{2}+5\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right )-\frac {3 x^{3}+5 x}{x}+5 x^{4}-12 x^{2}-32 x +5}{x^{7}+10 x^{5}+25 x^{3}}-\frac {\left (\left (3 x^{3}+5 x \right ) \ln \left (\frac {\ln \left (18\right )}{x}\right )+x^{5}-4 x^{3}-16 x^{2}+5 x -40\right ) \left (7 x^{6}+50 x^{4}+75 x^{2}\right )}{\left (x^{7}+10 x^{5}+25 x^{3}\right )^{2}}\right )}{6 x^{5}-19 x^{3}+48 x^{2}-5 x +40}\) | \(271\) |
| parts | \(\frac {4}{5 x^{2}}+\frac {-x -\frac {4}{5}}{x^{2}+5}-\frac {\ln \left (\frac {\ln \left (18\right )}{x}\right )}{5 x}+\frac {i \sqrt {5}\, \ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{50}-\frac {i \sqrt {5}\, \ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{50}-\frac {i \ln \left (18\right )^{2} \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right )}+\frac {i \ln \left (18\right )^{2} \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \ln \left (18\right )^{2}+\frac {250 \ln \left (18\right )^{2}}{x^{2}}}-\frac {i \ln \left (18\right )^{2} \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{10 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right ) x^{2}}+\frac {i \ln \left (18\right )^{2} \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{10 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right ) x^{2}}+\frac {\ln \left (18\right )^{2} \ln \left (\frac {\ln \left (18\right )}{x}\right )}{5 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right ) x}\) | \(358\) |
| derivativedivides | \(-\ln \left (18\right ) \left (\frac {-\frac {4 \ln \left (18\right )^{2}}{5 x^{2}}+\frac {\ln \left (18\right )^{2}}{5 x}+\frac {2 \ln \left (18\right )^{3} \left (\frac {\frac {5 \ln \left (18\right )}{2 x}-\frac {2 \ln \left (18\right )}{5}}{\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}}-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {5}}{x}\right )}{10 \ln \left (18\right )}\right )}{5}}{\ln \left (18\right )^{3}}+\frac {\ln \left (\frac {\ln \left (18\right )}{x}\right )}{5 \ln \left (18\right ) x}-\frac {1}{5 x \ln \left (18\right )}-\frac {i \sqrt {5}\, \ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \ln \left (18\right )}+\frac {i \sqrt {5}\, \ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \ln \left (18\right )}+\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {5}}{x}\right )}{25 \ln \left (18\right )}+\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \ln \left (18\right )^{2}+\frac {250 \ln \left (18\right )^{2}}{x^{2}}}-\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right )}+\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{10 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right ) x^{2}}-\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{10 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right ) x^{2}}-\frac {\ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (18\right )}{5 x \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right )}\right )\) | \(448\) |
| default | \(-\ln \left (18\right ) \left (\frac {-\frac {4 \ln \left (18\right )^{2}}{5 x^{2}}+\frac {\ln \left (18\right )^{2}}{5 x}+\frac {2 \ln \left (18\right )^{3} \left (\frac {\frac {5 \ln \left (18\right )}{2 x}-\frac {2 \ln \left (18\right )}{5}}{\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}}-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {5}}{x}\right )}{10 \ln \left (18\right )}\right )}{5}}{\ln \left (18\right )^{3}}+\frac {\ln \left (\frac {\ln \left (18\right )}{x}\right )}{5 \ln \left (18\right ) x}-\frac {1}{5 x \ln \left (18\right )}-\frac {i \sqrt {5}\, \ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \ln \left (18\right )}+\frac {i \sqrt {5}\, \ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \ln \left (18\right )}+\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {5}}{x}\right )}{25 \ln \left (18\right )}+\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \ln \left (18\right )^{2}+\frac {250 \ln \left (18\right )^{2}}{x^{2}}}-\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{50 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right )}+\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}+\ln \left (18\right )}{\ln \left (18\right )}\right )}{10 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right ) x^{2}}-\frac {i \ln \left (18\right ) \ln \left (\frac {\ln \left (18\right )}{x}\right ) \sqrt {5}\, \ln \left (-\frac {\frac {i \ln \left (18\right ) \sqrt {5}}{x}-\ln \left (18\right )}{\ln \left (18\right )}\right )}{10 \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right ) x^{2}}-\frac {\ln \left (\frac {\ln \left (18\right )}{x}\right ) \ln \left (18\right )}{5 x \left (\ln \left (18\right )^{2}+\frac {5 \ln \left (18\right )^{2}}{x^{2}}\right )}\right )\) | \(448\) |
Input:
int(((3*x^3+5*x)*ln(ln(18)/x)+x^5-4*x^3-16*x^2+5*x-40)/(x^7+10*x^5+25*x^3) ,x,method=_RETURNVERBOSE)
Output:
(4-x^3-ln(ln(18)/x)*x)/x^2/(x^2+5)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=-\frac {x^{3} + x \log \left (\frac {\log \left (18\right )}{x}\right ) - 4}{x^{4} + 5 \, x^{2}} \] Input:
integrate(((3*x^3+5*x)*log(log(18)/x)+x^5-4*x^3-16*x^2+5*x-40)/(x^7+10*x^5 +25*x^3),x, algorithm="fricas")
Output:
-(x^3 + x*log(log(18)/x) - 4)/(x^4 + 5*x^2)
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=\frac {4 - x^{3}}{x^{4} + 5 x^{2}} - \frac {\log {\left (\frac {\log {\left (18 \right )}}{x} \right )}}{x^{3} + 5 x} \] Input:
integrate(((3*x**3+5*x)*ln(ln(18)/x)+x**5-4*x**3-16*x**2+5*x-40)/(x**7+10* x**5+25*x**3),x)
Output:
(4 - x**3)/(x**4 + 5*x**2) - log(log(18)/x)/(x**3 + 5*x)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=-\frac {35 \, x^{2} + 2 \, {\left (16 \, x^{3} + 80 \, x - 25\right )} \log \left (x\right ) + 80 \, x + 50 \, \log \left (2 \, \log \left (3\right ) + \log \left (2\right )\right ) - 50}{50 \, {\left (x^{3} + 5 \, x\right )}} - \frac {3 \, x^{2} + 10}{10 \, {\left (x^{3} + 5 \, x\right )}} + \frac {4 \, {\left (2 \, x^{2} + 5\right )}}{5 \, {\left (x^{4} + 5 \, x^{2}\right )}} + \frac {16}{25} \, \log \left (x\right ) \] Input:
integrate(((3*x^3+5*x)*log(log(18)/x)+x^5-4*x^3-16*x^2+5*x-40)/(x^7+10*x^5 +25*x^3),x, algorithm="maxima")
Output:
-1/50*(35*x^2 + 2*(16*x^3 + 80*x - 25)*log(x) + 80*x + 50*log(2*log(3) + l og(2)) - 50)/(x^3 + 5*x) - 1/10*(3*x^2 + 10)/(x^3 + 5*x) + 4/5*(2*x^2 + 5) /(x^4 + 5*x^2) + 16/25*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (28) = 56\).
Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=\frac {5 \, {\left (\frac {\log \left (18\right )^{3}}{{\left (\log \left (18\right )^{2} + \frac {5 \, \log \left (18\right )^{2}}{x^{2}}\right )} x} - \frac {\log \left (18\right )}{x}\right )} \log \left (\frac {\log \left (18\right )}{x}\right ) + \frac {4 \, \log \left (18\right )^{3} - \frac {25 \, \log \left (18\right )^{3}}{x}}{\log \left (18\right )^{2} + \frac {5 \, \log \left (18\right )^{2}}{x^{2}}} + \frac {20 \, \log \left (18\right )}{x^{2}}}{25 \, \log \left (18\right )} \] Input:
integrate(((3*x^3+5*x)*log(log(18)/x)+x^5-4*x^3-16*x^2+5*x-40)/(x^7+10*x^5 +25*x^3),x, algorithm="giac")
Output:
1/25*(5*(log(18)^3/((log(18)^2 + 5*log(18)^2/x^2)*x) - log(18)/x)*log(log( 18)/x) + (4*log(18)^3 - 25*log(18)^3/x)/(log(18)^2 + 5*log(18)^2/x^2) + 20 *log(18)/x^2)/log(18)
Time = 2.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=-\frac {x^3+x\,\ln \left (\frac {\ln \left (18\right )}{x}\right )-4}{x^4+5\,x^2} \] Input:
int((5*x - 16*x^2 - 4*x^3 + x^5 + log(log(18)/x)*(5*x + 3*x^3) - 40)/(25*x ^3 + 10*x^5 + x^7),x)
Output:
-(x^3 + x*log(log(18)/x) - 4)/(5*x^2 + x^4)
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-40+5 x-16 x^2-4 x^3+x^5+\left (5 x+3 x^3\right ) \log \left (\frac {\log (18)}{x}\right )}{25 x^3+10 x^5+x^7} \, dx=\frac {-\mathrm {log}\left (\frac {\mathrm {log}\left (18\right )}{x}\right ) x -x^{3}+4}{x^{2} \left (x^{2}+5\right )} \] Input:
int(((3*x^3+5*x)*log(log(18)/x)+x^5-4*x^3-16*x^2+5*x-40)/(x^7+10*x^5+25*x^ 3),x)
Output:
( - log(log(18)/x)*x - x**3 + 4)/(x**2*(x**2 + 5))