Integrand size = 22, antiderivative size = 63 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=-\frac {2}{1+x}-\frac {7+5 x}{3 \left (1+x+x^2\right )}-\frac {25 \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\log (1+x)+\frac {1}{2} \log \left (1+x+x^2\right ) \]
-2/(1+x)+1/3*(-7-5*x)/(x^2+x+1)-ln(1+x)+1/2*ln(x^2+x+1)-25/9*arctan(1/3*(1 +2*x)*3^(1/2))*3^(1/2)
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=-\frac {2}{1+x}-\frac {7+5 x}{3 \left (1+x+x^2\right )}-\frac {25 \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\log (1+x)+\frac {1}{2} \log \left (1+x+x^2\right ) \]
-2/(1 + x) - (7 + 5*x)/(3*(1 + x + x^2)) - (25*ArcTan[(1 + 2*x)/Sqrt[3]])/ (3*Sqrt[3]) - Log[1 + x] + Log[1 + x + x^2]/2
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2177, 25, 2159, 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.
\(\displaystyle \int \frac {x^2-3 x-2}{(x+1)^2 \left (x^2+x+1\right )^2} \, dx\) |
\(\Big \downarrow \) 2177 |
\(\displaystyle \frac {1}{3} \int -\frac {5 x^2+19 x+8}{(x+1)^2 \left (x^2+x+1\right )}dx-\frac {5 x+7}{3 \left (x^2+x+1\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {5 x^2+19 x+8}{(x+1)^2 \left (x^2+x+1\right )}dx-\frac {5 x+7}{3 \left (x^2+x+1\right )}\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {11-3 x}{x^2+x+1}+\frac {3}{x+1}-\frac {6}{(x+1)^2}\right )dx-\frac {5 x+7}{3 \left (x^2+x+1\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {25 \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (x^2+x+1\right )-\frac {6}{x+1}-3 \log (x+1)\right )-\frac {5 x+7}{3 \left (x^2+x+1\right )}\) |
-1/3*(7 + 5*x)/(1 + x + x^2) + (-6/(1 + x) - (25*ArcTan[(1 + 2*x)/Sqrt[3]] )/Sqrt[3] - 3*Log[1 + x] + (3*Log[1 + x + x^2])/2)/3
3.2.82.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x + c* x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + (2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^ m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Qx)/(d + e*x )^m - ((2*p + 3)*(2*c*R - b*S))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a* e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Time = 0.47 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {-\frac {5 x}{3}-\frac {7}{3}}{x^{2}+x +1}+\frac {\ln \left (x^{2}+x +1\right )}{2}-\frac {25 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}-\frac {2}{1+x}-\ln \left (1+x \right )\) | \(54\) |
risch | \(\frac {-\frac {11}{3} x^{2}-6 x -\frac {13}{3}}{\left (x^{2}+x +1\right ) \left (1+x \right )}+\frac {\ln \left (4 x^{2}+4 x +4\right )}{2}-\frac {25 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}-\ln \left (1+x \right )\) | \(61\) |
(-5/3*x-7/3)/(x^2+x+1)+1/2*ln(x^2+x+1)-25/9*arctan(1/3*(1+2*x)*3^(1/2))*3^ (1/2)-2/(1+x)-ln(1+x)
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.54 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=-\frac {50 \, \sqrt {3} {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + 66 \, x^{2} - 9 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} \log \left (x^{2} + x + 1\right ) + 18 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + 108 \, x + 78}{18 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}} \]
-1/18*(50*sqrt(3)*(x^3 + 2*x^2 + 2*x + 1)*arctan(1/3*sqrt(3)*(2*x + 1)) + 66*x^2 - 9*(x^3 + 2*x^2 + 2*x + 1)*log(x^2 + x + 1) + 18*(x^3 + 2*x^2 + 2* x + 1)*log(x + 1) + 108*x + 78)/(x^3 + 2*x^2 + 2*x + 1)
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=\frac {- 11 x^{2} - 18 x - 13}{3 x^{3} + 6 x^{2} + 6 x + 3} - \log {\left (x + 1 \right )} + \frac {\log {\left (x^{2} + x + 1 \right )}}{2} - \frac {25 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{9} \]
(-11*x**2 - 18*x - 13)/(3*x**3 + 6*x**2 + 6*x + 3) - log(x + 1) + log(x**2 + x + 1)/2 - 25*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=-\frac {25}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {11 \, x^{2} + 18 \, x + 13}{3 \, {\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}} + \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left (x + 1\right ) \]
-25/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/3*(11*x^2 + 18*x + 13)/(x^ 3 + 2*x^2 + 2*x + 1) + 1/2*log(x^2 + x + 1) - log(x + 1)
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=-\frac {25}{9} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (\frac {2}{x + 1} - 1\right )}\right ) + \frac {\frac {7}{x + 1} - 2}{3 \, {\left (\frac {1}{x + 1} - \frac {1}{{\left (x + 1\right )}^{2}} - 1\right )}} - \frac {2}{x + 1} + \frac {1}{2} \, \log \left (-\frac {1}{x + 1} + \frac {1}{{\left (x + 1\right )}^{2}} + 1\right ) \]
-25/9*sqrt(3)*arctan(-1/3*sqrt(3)*(2/(x + 1) - 1)) + 1/3*(7/(x + 1) - 2)/( 1/(x + 1) - 1/(x + 1)^2 - 1) - 2/(x + 1) + 1/2*log(-1/(x + 1) + 1/(x + 1)^ 2 + 1)
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.16 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=-\ln \left (x+1\right )-\frac {\frac {11\,x^2}{3}+6\,x+\frac {13}{3}}{x^3+2\,x^2+2\,x+1}+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,25{}\mathrm {i}}{18}\right )-\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,25{}\mathrm {i}}{18}\right ) \]
log(x - (3^(1/2)*1i)/2 + 1/2)*((3^(1/2)*25i)/18 + 1/2) - (6*x + (11*x^2)/3 + 13/3)/(2*x + 2*x^2 + x^3 + 1) - log(x + 1) - log(x + (3^(1/2)*1i)/2 + 1 /2)*((3^(1/2)*25i)/18 - 1/2)
Time = 0.00 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.68 \[ \int \frac {-2-3 x+x^2}{(1+x)^2 \left (1+x+x^2\right )^2} \, dx=\frac {-50 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{3}-100 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x^{2}-100 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right ) x -50 \sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right )+9 \,\mathrm {log}\left (x^{2}+x +1\right ) x^{3}+18 \,\mathrm {log}\left (x^{2}+x +1\right ) x^{2}+18 \,\mathrm {log}\left (x^{2}+x +1\right ) x +9 \,\mathrm {log}\left (x^{2}+x +1\right )-18 \,\mathrm {log}\left (x +1\right ) x^{3}-36 \,\mathrm {log}\left (x +1\right ) x^{2}-36 \,\mathrm {log}\left (x +1\right ) x -18 \,\mathrm {log}\left (x +1\right )+33 x^{3}-42 x -45}{18 x^{3}+36 x^{2}+36 x +18} \]
( - 50*sqrt(3)*atan((2*x + 1)/sqrt(3))*x**3 - 100*sqrt(3)*atan((2*x + 1)/s qrt(3))*x**2 - 100*sqrt(3)*atan((2*x + 1)/sqrt(3))*x - 50*sqrt(3)*atan((2* x + 1)/sqrt(3)) + 9*log(x**2 + x + 1)*x**3 + 18*log(x**2 + x + 1)*x**2 + 1 8*log(x**2 + x + 1)*x + 9*log(x**2 + x + 1) - 18*log(x + 1)*x**3 - 36*log( x + 1)*x**2 - 36*log(x + 1)*x - 18*log(x + 1) + 33*x**3 - 42*x - 45)/(18*( x**3 + 2*x**2 + 2*x + 1))