Integrand size = 16, antiderivative size = 45 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=\frac {67+47 x}{18 \left (13+4 x+x^2\right )}-\frac {61}{54} \arctan \left (\frac {2+x}{3}\right )+\frac {1}{2} \log \left (13+4 x+x^2\right ) \] Output:
(67+47*x)/(18*x^2+72*x+234)-61/54*arctan(2/3+1/3*x)+1/2*ln(x^2+4*x+13)
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=\frac {67+47 x}{18 \left (13+4 x+x^2\right )}-\frac {61}{54} \arctan \left (\frac {2+x}{3}\right )+\frac {1}{2} \log \left (13+4 x+x^2\right ) \] Input:
Integrate[(1 + x^3)/(13 + 4*x + x^2)^2,x]
Output:
(67 + 47*x)/(18*(13 + 4*x + x^2)) - (61*ArcTan[(2 + x)/3])/54 + Log[13 + 4 *x + x^2]/2
Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2191, 27, 1142, 27, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {x^3+1}{\left (x^2+4 x+13\right )^2} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{36} \int -\frac {2 (25-18 x)}{x^2+4 x+13}dx+\frac {47 x+67}{18 \left (x^2+4 x+13\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {47 x+67}{18 \left (x^2+4 x+13\right )}-\frac {1}{18} \int \frac {25-18 x}{x^2+4 x+13}dx\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{18} \left (9 \int \frac {2 (x+2)}{x^2+4 x+13}dx-61 \int \frac {1}{x^2+4 x+13}dx\right )+\frac {47 x+67}{18 \left (x^2+4 x+13\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{18} \left (18 \int \frac {x+2}{x^2+4 x+13}dx-61 \int \frac {1}{x^2+4 x+13}dx\right )+\frac {47 x+67}{18 \left (x^2+4 x+13\right )}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{18} \left (18 \int \frac {x+2}{x^2+4 x+13}dx+122 \int \frac {1}{-(2 x+4)^2-36}d(2 x+4)\right )+\frac {47 x+67}{18 \left (x^2+4 x+13\right )}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{18} \left (18 \int \frac {x+2}{x^2+4 x+13}dx-\frac {61}{3} \arctan \left (\frac {1}{6} (2 x+4)\right )\right )+\frac {47 x+67}{18 \left (x^2+4 x+13\right )}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{18} \left (9 \log \left (x^2+4 x+13\right )-\frac {61}{3} \arctan \left (\frac {1}{6} (2 x+4)\right )\right )+\frac {47 x+67}{18 \left (x^2+4 x+13\right )}\) |
Input:
Int[(1 + x^3)/(13 + 4*x + x^2)^2,x]
Output:
(67 + 47*x)/(18*(13 + 4*x + x^2)) + ((-61*ArcTan[(4 + 2*x)/6])/3 + 9*Log[1 3 + 4*x + x^2])/18
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*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 [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 2.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {\frac {47 x}{18}+\frac {67}{18}}{x^{2}+4 x +13}+\frac {\ln \left (x^{2}+4 x +13\right )}{2}-\frac {61 \arctan \left (\frac {2}{3}+\frac {x}{3}\right )}{54}\) | \(37\) |
risch | \(\frac {\frac {47 x}{18}+\frac {67}{18}}{x^{2}+4 x +13}+\frac {\ln \left (x^{2}+4 x +13\right )}{2}-\frac {61 \arctan \left (\frac {2}{3}+\frac {x}{3}\right )}{54}\) | \(37\) |
parallelrisch | \(\frac {10309 i \ln \left (x +2-3 i\right )-793 i \ln \left (x +2+3 i\right ) x^{2}-3172 i \ln \left (x +2+3 i\right ) x +3172 i \ln \left (x +2-3 i\right ) x +702 \ln \left (x +2-3 i\right ) x^{2}+702 \ln \left (x +2+3 i\right ) x^{2}-10309 i \ln \left (x +2+3 i\right )+793 i \ln \left (x +2-3 i\right ) x^{2}+2808 \ln \left (x +2-3 i\right ) x +2808 \ln \left (x +2+3 i\right ) x -402 x^{2}+9126 \ln \left (x +2-3 i\right )+9126 \ln \left (x +2+3 i\right )+2058 x}{1404 x^{2}+5616 x +18252}\) | \(140\) |
Input:
int((x^3+1)/(x^2+4*x+13)^2,x,method=_RETURNVERBOSE)
Output:
(47/18*x+67/18)/(x^2+4*x+13)+1/2*ln(x^2+4*x+13)-61/54*arctan(2/3+1/3*x)
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=-\frac {61 \, {\left (x^{2} + 4 \, x + 13\right )} \arctan \left (\frac {1}{3} \, x + \frac {2}{3}\right ) - 27 \, {\left (x^{2} + 4 \, x + 13\right )} \log \left (x^{2} + 4 \, x + 13\right ) - 141 \, x - 201}{54 \, {\left (x^{2} + 4 \, x + 13\right )}} \] Input:
integrate((x^3+1)/(x^2+4*x+13)^2,x, algorithm="fricas")
Output:
-1/54*(61*(x^2 + 4*x + 13)*arctan(1/3*x + 2/3) - 27*(x^2 + 4*x + 13)*log(x ^2 + 4*x + 13) - 141*x - 201)/(x^2 + 4*x + 13)
Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=\frac {47 x + 67}{18 x^{2} + 72 x + 234} + \frac {\log {\left (x^{2} + 4 x + 13 \right )}}{2} - \frac {61 \operatorname {atan}{\left (\frac {x}{3} + \frac {2}{3} \right )}}{54} \] Input:
integrate((x**3+1)/(x**2+4*x+13)**2,x)
Output:
(47*x + 67)/(18*x**2 + 72*x + 234) + log(x**2 + 4*x + 13)/2 - 61*atan(x/3 + 2/3)/54
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=\frac {47 \, x + 67}{18 \, {\left (x^{2} + 4 \, x + 13\right )}} - \frac {61}{54} \, \arctan \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{2} \, \log \left (x^{2} + 4 \, x + 13\right ) \] Input:
integrate((x^3+1)/(x^2+4*x+13)^2,x, algorithm="maxima")
Output:
1/18*(47*x + 67)/(x^2 + 4*x + 13) - 61/54*arctan(1/3*x + 2/3) + 1/2*log(x^ 2 + 4*x + 13)
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=\frac {47 \, x + 67}{18 \, {\left (x^{2} + 4 \, x + 13\right )}} - \frac {61}{54} \, \arctan \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{2} \, \log \left (x^{2} + 4 \, x + 13\right ) \] Input:
integrate((x^3+1)/(x^2+4*x+13)^2,x, algorithm="giac")
Output:
1/18*(47*x + 67)/(x^2 + 4*x + 13) - 61/54*arctan(1/3*x + 2/3) + 1/2*log(x^ 2 + 4*x + 13)
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=\frac {\ln \left (x^2+4\,x+13\right )}{2}-\frac {61\,\mathrm {atan}\left (\frac {x}{3}+\frac {2}{3}\right )}{54}+\frac {47\,x}{18\,\left (x^2+4\,x+13\right )}+\frac {67}{18\,\left (x^2+4\,x+13\right )} \] Input:
int((x^3 + 1)/(4*x + x^2 + 13)^2,x)
Output:
log(4*x + x^2 + 13)/2 - (61*atan(x/3 + 2/3))/54 + (47*x)/(18*(4*x + x^2 + 13)) + 67/(18*(4*x + x^2 + 13))
Time = 0.16 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.89 \[ \int \frac {1+x^3}{\left (13+4 x+x^2\right )^2} \, dx=\frac {-244 \mathit {atan} \left (\frac {x}{3}+\frac {2}{3}\right ) x^{2}-976 \mathit {atan} \left (\frac {x}{3}+\frac {2}{3}\right ) x -3172 \mathit {atan} \left (\frac {x}{3}+\frac {2}{3}\right )+108 \,\mathrm {log}\left (x^{2}+4 x +13\right ) x^{2}+432 \,\mathrm {log}\left (x^{2}+4 x +13\right ) x +1404 \,\mathrm {log}\left (x^{2}+4 x +13\right )-141 x^{2}-1029}{216 x^{2}+864 x +2808} \] Input:
int((x^3+1)/(x^2+4*x+13)^2,x)
Output:
( - 244*atan((x + 2)/3)*x**2 - 976*atan((x + 2)/3)*x - 3172*atan((x + 2)/3 ) + 108*log(x**2 + 4*x + 13)*x**2 + 432*log(x**2 + 4*x + 13)*x + 1404*log( x**2 + 4*x + 13) - 141*x**2 - 1029)/(216*(x**2 + 4*x + 13))