Integrand size = 22, antiderivative size = 109 \[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=-\frac {\arctan \left (\frac {1+2 x}{\sqrt {7-4 \sqrt {2}}}\right )}{2 \sqrt {2 \left (7-4 \sqrt {2}\right )}}+\frac {\arctan \left (\frac {1+2 x}{\sqrt {7+4 \sqrt {2}}}\right )}{2 \sqrt {2 \left (7+4 \sqrt {2}\right )}}-\frac {\text {arctanh}\left (\frac {7+4 \left (\frac {1}{2}+x\right )^2}{4 \sqrt {2}}\right )}{2 \sqrt {2}} \] Output:
-1/2*arctan((1+2*x)/(7-4*2^(1/2))^(1/2))/(14-8*2^(1/2))^(1/2)+1/2*arctan(( 1+2*x)/(7+4*2^(1/2))^(1/2))/(14+8*2^(1/2))^(1/2)-1/4*arctanh(1/8*(7+4*(1/2 +x)^2)*2^(1/2))*2^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.54 \[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=\frac {1}{2} \text {RootSum}\left [2+4 \text {$\#$1}+5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{2+5 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[x/(2 + 4*x + 5*x^2 + 2*x^3 + x^4),x]
Output:
RootSum[2 + 4*#1 + 5*#1^2 + 2*#1^3 + #1^4 & , (Log[x - #1]*#1)/(2 + 5*#1 + 3*#1^2 + 2*#1^3) & ]/2
Time = 0.65 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.30, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2459, 2202, 27, 1406, 216, 1432, 1081, 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.
| \(\displaystyle \int \frac {x}{x^4+2 x^3+5 x^2+4 x+2} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {x}{\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}}d\left (x+\frac {1}{2}\right )\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int -\frac {1}{2 \left (\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}\right )}d\left (x+\frac {1}{2}\right )+\int \frac {x+\frac {1}{2}}{\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}}d\left (x+\frac {1}{2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x+\frac {1}{2}}{\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}}d\left (x+\frac {1}{2}\right )-\frac {1}{2} \int \frac {1}{\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}}d\left (x+\frac {1}{2}\right )\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{\left (x+\frac {1}{2}\right )^2+\frac {1}{4} \left (7+4 \sqrt {2}\right )}d\left (x+\frac {1}{2}\right )}{2 \sqrt {2}}-\frac {\int \frac {1}{\left (x+\frac {1}{2}\right )^2+\frac {1}{4} \left (7-4 \sqrt {2}\right )}d\left (x+\frac {1}{2}\right )}{2 \sqrt {2}}\right )+\int \frac {x+\frac {1}{2}}{\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}}d\left (x+\frac {1}{2}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \int \frac {x+\frac {1}{2}}{\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}}d\left (x+\frac {1}{2}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7+4 \sqrt {2}}}\right )}{\sqrt {2 \left (7+4 \sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7-4 \sqrt {2}}}\right )}{\sqrt {2 \left (7-4 \sqrt {2}\right )}}\right )\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{\left (x+\frac {1}{2}\right )^4+\frac {7}{2} \left (x+\frac {1}{2}\right )^2+\frac {17}{16}}d\left (x+\frac {1}{2}\right )^2+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7+4 \sqrt {2}}}\right )}{\sqrt {2 \left (7+4 \sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7-4 \sqrt {2}}}\right )}{\sqrt {2 \left (7-4 \sqrt {2}\right )}}\right )\) |
| \(\Big \downarrow \) 1081 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\sqrt {2}}{4 \left (x+\frac {1}{2}\right )^2-4 \sqrt {2}+7}-\frac {\sqrt {2}}{4 \left (x+\frac {1}{2}\right )^2+4 \sqrt {2}+7}\right )d\left (x+\frac {1}{2}\right )^2+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7+4 \sqrt {2}}}\right )}{\sqrt {2 \left (7+4 \sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7-4 \sqrt {2}}}\right )}{\sqrt {2 \left (7-4 \sqrt {2}\right )}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7+4 \sqrt {2}}}\right )}{\sqrt {2 \left (7+4 \sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 \left (x+\frac {1}{2}\right )}{\sqrt {7-4 \sqrt {2}}}\right )}{\sqrt {2 \left (7-4 \sqrt {2}\right )}}\right )+\frac {1}{2} \left (\frac {\log \left (4 \left (x+\frac {1}{2}\right )^2-4 \sqrt {2}+7\right )}{2 \sqrt {2}}-\frac {\log \left (4 \left (x+\frac {1}{2}\right )^2+4 \sqrt {2}+7\right )}{2 \sqrt {2}}\right )\) |
Input:
Int[x/(2 + 4*x + 5*x^2 + 2*x^3 + x^4),x]
Output:
(-(ArcTan[(2*(1/2 + x))/Sqrt[7 - 4*Sqrt[2]]]/Sqrt[2*(7 - 4*Sqrt[2])]) + Ar cTan[(2*(1/2 + x))/Sqrt[7 + 4*Sqrt[2]]]/Sqrt[2*(7 + 4*Sqrt[2])])/2 + (Log[ 7 - 4*Sqrt[2] + 4*(1/2 + x)^2]/(2*Sqrt[2]) - Log[7 + 4*Sqrt[2] + 4*(1/2 + x)^2]/(2*Sqrt[2]))/2
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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.46
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}+5 \textit {\_Z}^{2}+4 \textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}+5 \textit {\_R} +2}\right )}{2}\) | \(50\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}+5 \textit {\_Z}^{2}+4 \textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}^{2}+5 \textit {\_R} +2}\right )}{2}\) | \(50\) |
Input:
int(x/(x^4+2*x^3+5*x^2+4*x+2),x,method=_RETURNVERBOSE)
Output:
1/2*sum(_R/(2*_R^3+3*_R^2+5*_R+2)*ln(x-_R),_R=RootOf(_Z^4+2*_Z^3+5*_Z^2+4* _Z+2))
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85 \[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=-\frac {1}{2} \, \sqrt {\frac {2}{17} \, \sqrt {2} + \frac {7}{34}} \arctan \left (\sqrt {2} {\left (2 \, x + 1\right )} \sqrt {\frac {2}{17} \, \sqrt {2} + \frac {7}{34}}\right ) + \frac {1}{2} \, \sqrt {-\frac {2}{17} \, \sqrt {2} + \frac {7}{34}} \arctan \left (\sqrt {2} {\left (2 \, x + 1\right )} \sqrt {-\frac {2}{17} \, \sqrt {2} + \frac {7}{34}}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (x^{2} + x + \sqrt {2} + 2\right ) + \frac {1}{8} \, \sqrt {2} \log \left (x^{2} + x - \sqrt {2} + 2\right ) \] Input:
integrate(x/(x^4+2*x^3+5*x^2+4*x+2),x, algorithm="fricas")
Output:
-1/2*sqrt(2/17*sqrt(2) + 7/34)*arctan(sqrt(2)*(2*x + 1)*sqrt(2/17*sqrt(2) + 7/34)) + 1/2*sqrt(-2/17*sqrt(2) + 7/34)*arctan(sqrt(2)*(2*x + 1)*sqrt(-2 /17*sqrt(2) + 7/34)) - 1/8*sqrt(2)*log(x^2 + x + sqrt(2) + 2) + 1/8*sqrt(2 )*log(x^2 + x - sqrt(2) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (88) = 176\).
Time = 0.44 (sec) , antiderivative size = 1216, normalized size of antiderivative = 11.16 \[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=\text {Too large to display} \] Input:
integrate(x/(x**4+2*x**3+5*x**2+4*x+2),x)
Output:
sqrt(2)*log(x**2 + x*(-2243*sqrt(2)/208 - 19*sqrt(297 - 68*sqrt(2))/52 + 2 49/26 + 123*sqrt(2)*sqrt(297 - 68*sqrt(2))/208) - 104443*sqrt(297 - 68*sqr t(2))/10816 - 451653*sqrt(2)/5408 + 10053*sqrt(2)*sqrt(297 - 68*sqrt(2))/2 704 + 1958011/10816)/8 - sqrt(2)*log(x**2 + x*(-123*sqrt(2)*sqrt(68*sqrt(2 ) + 297)/208 - 19*sqrt(68*sqrt(2) + 297)/52 + 249/26 + 2243*sqrt(2)/208) - 104443*sqrt(68*sqrt(2) + 297)/10816 - 10053*sqrt(2)*sqrt(68*sqrt(2) + 297 )/2704 + 451653*sqrt(2)/5408 + 1958011/10816)/8 + 2*sqrt(41/544 - sqrt(68* sqrt(2) + 297)/272)*atan(416*sqrt(17)*x/(135*sqrt(2)*sqrt(41 - 2*sqrt(68*s qrt(2) + 297)) + 41*sqrt(2)*sqrt(41 - 2*sqrt(68*sqrt(2) + 297))*sqrt(68*sq rt(2) + 297) + 1292*sqrt(41 - 2*sqrt(68*sqrt(2) + 297))) - 123*sqrt(34)*sq rt(68*sqrt(2) + 297)/(135*sqrt(2)*sqrt(41 - 2*sqrt(68*sqrt(2) + 297)) + 41 *sqrt(2)*sqrt(41 - 2*sqrt(68*sqrt(2) + 297))*sqrt(68*sqrt(2) + 297) + 1292 *sqrt(41 - 2*sqrt(68*sqrt(2) + 297))) - 76*sqrt(17)*sqrt(68*sqrt(2) + 297) /(135*sqrt(2)*sqrt(41 - 2*sqrt(68*sqrt(2) + 297)) + 41*sqrt(2)*sqrt(41 - 2 *sqrt(68*sqrt(2) + 297))*sqrt(68*sqrt(2) + 297) + 1292*sqrt(41 - 2*sqrt(68 *sqrt(2) + 297))) + 1992*sqrt(17)/(135*sqrt(2)*sqrt(41 - 2*sqrt(68*sqrt(2) + 297)) + 41*sqrt(2)*sqrt(41 - 2*sqrt(68*sqrt(2) + 297))*sqrt(68*sqrt(2) + 297) + 1292*sqrt(41 - 2*sqrt(68*sqrt(2) + 297))) + 2243*sqrt(34)/(135*sq rt(2)*sqrt(41 - 2*sqrt(68*sqrt(2) + 297)) + 41*sqrt(2)*sqrt(41 - 2*sqrt(68 *sqrt(2) + 297))*sqrt(68*sqrt(2) + 297) + 1292*sqrt(41 - 2*sqrt(68*sqrt...
\[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=\int { \frac {x}{x^{4} + 2 \, x^{3} + 5 \, x^{2} + 4 \, x + 2} \,d x } \] Input:
integrate(x/(x^4+2*x^3+5*x^2+4*x+2),x, algorithm="maxima")
Output:
integrate(x/(x^4 + 2*x^3 + 5*x^2 + 4*x + 2), x)
\[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=\int { \frac {x}{x^{4} + 2 \, x^{3} + 5 \, x^{2} + 4 \, x + 2} \,d x } \] Input:
integrate(x/(x^4+2*x^3+5*x^2+4*x+2),x, algorithm="giac")
Output:
integrate(x/(x^4 + 2*x^3 + 5*x^2 + 4*x + 2), x)
Time = 10.44 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=\sum _{k=1}^4\ln \left (x-\mathrm {root}\left (z^4-\frac {5\,z^2}{136}+\frac {z}{136}+\frac {1}{544},z,k\right )\,\left (-6\,x+\mathrm {root}\left (z^4-\frac {5\,z^2}{136}+\frac {z}{136}+\frac {1}{544},z,k\right )\,\left (32\,x+\mathrm {root}\left (z^4-\frac {5\,z^2}{136}+\frac {z}{136}+\frac {1}{544},z,k\right )\,\left (224\,x+112\right )+32\right )+4\right )\right )\,\mathrm {root}\left (z^4-\frac {5\,z^2}{136}+\frac {z}{136}+\frac {1}{544},z,k\right ) \] Input:
int(x/(4*x + 5*x^2 + 2*x^3 + x^4 + 2),x)
Output:
symsum(log(x - root(z^4 - (5*z^2)/136 + z/136 + 1/544, z, k)*(root(z^4 - ( 5*z^2)/136 + z/136 + 1/544, z, k)*(32*x + root(z^4 - (5*z^2)/136 + z/136 + 1/544, z, k)*(224*x + 112) + 32) - 6*x + 4))*root(z^4 - (5*z^2)/136 + z/1 36 + 1/544, z, k), k, 1, 4)
Time = 0.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.82 \[ \int \frac {x}{2+4 x+5 x^2+2 x^3+x^4} \, dx=\frac {7 \sqrt {4 \sqrt {2}+7}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {4 \sqrt {2}+7}}\right )}{68}-\frac {2 \sqrt {4 \sqrt {2}+7}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {4 \sqrt {2}+7}}\right )}{17}+\frac {7 \sqrt {4 \sqrt {2}-7}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {4 \sqrt {2}-7}+2 x +1\right )}{136}-\frac {7 \sqrt {4 \sqrt {2}-7}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {4 \sqrt {2}-7}+2 x +1\right )}{136}+\frac {\sqrt {4 \sqrt {2}-7}\, \mathrm {log}\left (-\sqrt {4 \sqrt {2}-7}+2 x +1\right )}{17}-\frac {\sqrt {4 \sqrt {2}-7}\, \mathrm {log}\left (\sqrt {4 \sqrt {2}-7}+2 x +1\right )}{17}+\frac {\sqrt {2}\, \mathrm {log}\left (-\sqrt {4 \sqrt {2}-7}+2 x +1\right )}{8}+\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {4 \sqrt {2}-7}+2 x +1\right )}{8}-\frac {\sqrt {2}\, \mathrm {log}\left (\sqrt {2}+x^{2}+x +2\right )}{8} \] Input:
int(x/(x^4+2*x^3+5*x^2+4*x+2),x)
Output:
(14*sqrt(4*sqrt(2) + 7)*sqrt(2)*atan((2*x + 1)/sqrt(4*sqrt(2) + 7)) - 16*s qrt(4*sqrt(2) + 7)*atan((2*x + 1)/sqrt(4*sqrt(2) + 7)) + 7*sqrt(4*sqrt(2) - 7)*sqrt(2)*log( - sqrt(4*sqrt(2) - 7) + 2*x + 1) - 7*sqrt(4*sqrt(2) - 7) *sqrt(2)*log(sqrt(4*sqrt(2) - 7) + 2*x + 1) + 8*sqrt(4*sqrt(2) - 7)*log( - sqrt(4*sqrt(2) - 7) + 2*x + 1) - 8*sqrt(4*sqrt(2) - 7)*log(sqrt(4*sqrt(2) - 7) + 2*x + 1) + 17*sqrt(2)*log( - sqrt(4*sqrt(2) - 7) + 2*x + 1) + 17*s qrt(2)*log(sqrt(4*sqrt(2) - 7) + 2*x + 1) - 17*sqrt(2)*log(sqrt(2) + x**2 + x + 2))/136