Integrand size = 72, antiderivative size = 25 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {\left (1+x+\left (2+\frac {1}{x^2}\right ) x\right ) \left (3+x^2\right ) \log (x)}{(1-x)^2} \] Output:
ln(x)*(x^2+3)/(1-x)^2*(x+(1/x^2+2)*x+1)
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {\left (3+3 x+10 x^2+x^3+3 x^4\right ) \log (x)}{(-1+x)^2 x} \] Input:
Integrate[(-3 - 7*x^2 + 9*x^3 - 2*x^4 + 3*x^5 + (3 - 9*x - 16*x^2 - 12*x^3 - 9*x^4 + 3*x^5)*Log[x])/(-x^2 + 3*x^3 - 3*x^4 + x^5),x]
Output:
((3 + 3*x + 10*x^2 + x^3 + 3*x^4)*Log[x])/((-1 + x)^2*x)
Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(25)=50\).
Time = 0.66 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2026, 2007, 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.
\(\displaystyle \int \frac {3 x^5-2 x^4+9 x^3-7 x^2+\left (3 x^5-9 x^4-12 x^3-16 x^2-9 x+3\right ) \log (x)-3}{x^5-3 x^4+3 x^3-x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {3 x^5-2 x^4+9 x^3-7 x^2+\left (3 x^5-9 x^4-12 x^3-16 x^2-9 x+3\right ) \log (x)-3}{x^2 \left (x^3-3 x^2+3 x-1\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {3 x^5-2 x^4+9 x^3-7 x^2+\left (3 x^5-9 x^4-12 x^3-16 x^2-9 x+3\right ) \log (x)-3}{(x-1)^3 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x^3}{(x-1)^3}-\frac {2 x^2}{(x-1)^3}-\frac {3}{(x-1)^3 x^2}+\frac {\left (3 x^5-9 x^4-12 x^3-16 x^2-9 x+3\right ) \log (x)}{(x-1)^3 x^2}+\frac {9 x}{(x-1)^3}-\frac {7}{(x-1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9 x^2}{2 (1-x)^2}-\frac {9}{1-x}+\frac {9}{2 (1-x)^2}-\frac {18 x \log (x)}{1-x}+3 x \log (x)+\frac {20 \log (x)}{(1-x)^2}-11 \log (x)+\frac {3 \log (x)}{x}\) |
Input:
Int[(-3 - 7*x^2 + 9*x^3 - 2*x^4 + 3*x^5 + (3 - 9*x - 16*x^2 - 12*x^3 - 9*x ^4 + 3*x^5)*Log[x])/(-x^2 + 3*x^3 - 3*x^4 + x^5),x]
Output:
9/(2*(1 - x)^2) - 9/(1 - x) - (9*x^2)/(2*(1 - x)^2) - 11*Log[x] + (20*Log[ x])/(1 - x)^2 + (3*Log[x])/x + 3*x*Log[x] - (18*x*Log[x])/(1 - x)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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 = 0.90 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60
method | result | size |
norman | \(\frac {x^{3} \ln \left (x \right )+3 x \ln \left (x \right )+10 x^{2} \ln \left (x \right )+3 x^{4} \ln \left (x \right )+3 \ln \left (x \right )}{x \left (-1+x \right )^{2}}\) | \(40\) |
default | \(9 \ln \left (x \right )+3 x \ln \left (x \right )+\frac {18 x \ln \left (x \right )}{-1+x}-\frac {20 \ln \left (x \right ) x \left (-2+x \right )}{\left (-1+x \right )^{2}}+\frac {3 \ln \left (x \right )}{x}\) | \(41\) |
parts | \(9 \ln \left (x \right )+3 x \ln \left (x \right )+\frac {18 x \ln \left (x \right )}{-1+x}-\frac {20 \ln \left (x \right ) x \left (-2+x \right )}{\left (-1+x \right )^{2}}+\frac {3 \ln \left (x \right )}{x}\) | \(41\) |
risch | \(\frac {\left (3 x^{4}-6 x^{3}+24 x^{2}-4 x +3\right ) \ln \left (x \right )}{x \left (x^{2}-2 x +1\right )}+7 \ln \left (x \right )\) | \(42\) |
parallelrisch | \(\frac {x^{3} \ln \left (x \right )+3 x \ln \left (x \right )+10 x^{2} \ln \left (x \right )+3 x^{4} \ln \left (x \right )+3 \ln \left (x \right )}{x \left (x^{2}-2 x +1\right )}\) | \(45\) |
orering | \(\frac {x \left (-1+x \right ) \left (9 x^{9}-45 x^{8}-85 x^{7}-343 x^{6}-484 x^{5}-340 x^{4}-225 x^{3}-99 x^{2}-15 x +27\right ) \left (\left (3 x^{5}-9 x^{4}-12 x^{3}-16 x^{2}-9 x +3\right ) \ln \left (x \right )+3 x^{5}-2 x^{4}+9 x^{3}-7 x^{2}-3\right )}{\left (9 x^{10}-75 x^{9}-117 x^{8}+144 x^{7}+208 x^{6}+480 x^{5}+124 x^{4}+252 x^{3}-153 x^{2}-81 x +9\right ) \left (x^{5}-3 x^{4}+3 x^{3}-x^{2}\right )}-\frac {\left (9 x^{8}+85 x^{6}+44 x^{5}+132 x^{4}+52 x^{3}+45 x^{2}+24 x +9\right ) \left (-1+x \right )^{2} x^{2} \left (\frac {\left (15 x^{4}-36 x^{3}-36 x^{2}-32 x -9\right ) \ln \left (x \right )+\frac {3 x^{5}-9 x^{4}-12 x^{3}-16 x^{2}-9 x +3}{x}+15 x^{4}-8 x^{3}+27 x^{2}-14 x}{x^{5}-3 x^{4}+3 x^{3}-x^{2}}-\frac {\left (\left (3 x^{5}-9 x^{4}-12 x^{3}-16 x^{2}-9 x +3\right ) \ln \left (x \right )+3 x^{5}-2 x^{4}+9 x^{3}-7 x^{2}-3\right ) \left (5 x^{4}-12 x^{3}+9 x^{2}-2 x \right )}{\left (x^{5}-3 x^{4}+3 x^{3}-x^{2}\right )^{2}}\right )}{9 x^{10}-75 x^{9}-117 x^{8}+144 x^{7}+208 x^{6}+480 x^{5}+124 x^{4}+252 x^{3}-153 x^{2}-81 x +9}\) | \(458\) |
Input:
int(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*ln(x)+3*x^5-2*x^4+9*x^3-7*x^2-3)/(x ^5-3*x^4+3*x^3-x^2),x,method=_RETURNVERBOSE)
Output:
(x^3*ln(x)+3*x*ln(x)+10*x^2*ln(x)+3*x^4*ln(x)+3*ln(x))/x/(-1+x)^2
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {{\left (3 \, x^{4} + x^{3} + 10 \, x^{2} + 3 \, x + 3\right )} \log \left (x\right )}{x^{3} - 2 \, x^{2} + x} \] Input:
integrate(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*log(x)+3*x^5-2*x^4+9*x^3-7*x^ 2-3)/(x^5-3*x^4+3*x^3-x^2),x, algorithm="fricas")
Output:
(3*x^4 + x^3 + 10*x^2 + 3*x + 3)*log(x)/(x^3 - 2*x^2 + x)
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx=7 \log {\left (x \right )} + \frac {\left (3 x^{4} - 6 x^{3} + 24 x^{2} - 4 x + 3\right ) \log {\left (x \right )}}{x^{3} - 2 x^{2} + x} \] Input:
integrate(((3*x**5-9*x**4-12*x**3-16*x**2-9*x+3)*ln(x)+3*x**5-2*x**4+9*x** 3-7*x**2-3)/(x**5-3*x**4+3*x**3-x**2),x)
Output:
7*log(x) + (3*x**4 - 6*x**3 + 24*x**2 - 4*x + 3)*log(x)/(x**3 - 2*x**2 + x )
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.08 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx=3 \, x - \frac {3 \, x^{4} - 6 \, x^{3} - 20 \, x^{2} - {\left (3 \, x^{4} - 6 \, x^{3} + 24 \, x^{2} - 4 \, x + 3\right )} \log \left (x\right ) + 26 \, x - 3}{x^{3} - 2 \, x^{2} + x} - \frac {3 \, {\left (6 \, x^{2} - 9 \, x + 2\right )}}{2 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} - \frac {3 \, {\left (6 \, x - 5\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {4 \, x - 3}{x^{2} - 2 \, x + 1} - \frac {9 \, {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {7}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + 7 \, \log \left (x\right ) \] Input:
integrate(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*log(x)+3*x^5-2*x^4+9*x^3-7*x^ 2-3)/(x^5-3*x^4+3*x^3-x^2),x, algorithm="maxima")
Output:
3*x - (3*x^4 - 6*x^3 - 20*x^2 - (3*x^4 - 6*x^3 + 24*x^2 - 4*x + 3)*log(x) + 26*x - 3)/(x^3 - 2*x^2 + x) - 3/2*(6*x^2 - 9*x + 2)/(x^3 - 2*x^2 + x) - 3/2*(6*x - 5)/(x^2 - 2*x + 1) + (4*x - 3)/(x^2 - 2*x + 1) - 9/2*(2*x - 1)/ (x^2 - 2*x + 1) + 7/2/(x^2 - 2*x + 1) + 7*log(x)
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx={\left (3 \, x + \frac {2 \, {\left (9 \, x + 1\right )}}{x^{2} - 2 \, x + 1} + \frac {3}{x}\right )} \log \left (x\right ) + 7 \, \log \left (x\right ) \] Input:
integrate(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*log(x)+3*x^5-2*x^4+9*x^3-7*x^ 2-3)/(x^5-3*x^4+3*x^3-x^2),x, algorithm="giac")
Output:
(3*x + 2*(9*x + 1)/(x^2 - 2*x + 1) + 3/x)*log(x) + 7*log(x)
Time = 3.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {\ln \left (x\right )\,\left (3\,x^4+x^3+10\,x^2+3\,x+3\right )}{x\,{\left (x-1\right )}^2} \] Input:
int((log(x)*(9*x + 16*x^2 + 12*x^3 + 9*x^4 - 3*x^5 - 3) + 7*x^2 - 9*x^3 + 2*x^4 - 3*x^5 + 3)/(x^2 - 3*x^3 + 3*x^4 - x^5),x)
Output:
(log(x)*(3*x + 10*x^2 + x^3 + 3*x^4 + 3))/(x*(x - 1)^2)
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int \frac {-3-7 x^2+9 x^3-2 x^4+3 x^5+\left (3-9 x-16 x^2-12 x^3-9 x^4+3 x^5\right ) \log (x)}{-x^2+3 x^3-3 x^4+x^5} \, dx=\frac {3 \,\mathrm {log}\left (x \right ) x^{4}+\mathrm {log}\left (x \right ) x^{3}+10 \,\mathrm {log}\left (x \right ) x^{2}+3 \,\mathrm {log}\left (x \right ) x +3 \,\mathrm {log}\left (x \right )-13 x^{3}+26 x^{2}-13 x}{x \left (x^{2}-2 x +1\right )} \] Input:
int(((3*x^5-9*x^4-12*x^3-16*x^2-9*x+3)*log(x)+3*x^5-2*x^4+9*x^3-7*x^2-3)/( x^5-3*x^4+3*x^3-x^2),x)
Output:
(3*log(x)*x**4 + log(x)*x**3 + 10*log(x)*x**2 + 3*log(x)*x + 3*log(x) - 13 *x**3 + 26*x**2 - 13*x)/(x*(x**2 - 2*x + 1))