Integrand size = 30, antiderivative size = 167 \[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=\frac {1}{8} \sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}}-\sqrt {1+\sqrt {2}} x\right )+\frac {1}{8} \sqrt {-1+\sqrt {2}} \arctan \left (\sqrt {1+\sqrt {2}}+\sqrt {1+\sqrt {2}} x\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}}-\sqrt {-1+\sqrt {2}} x\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\sqrt {-1+\sqrt {2}}+\sqrt {-1+\sqrt {2}} x\right ) \] Output:
-1/8*(2^(1/2)-1)^(1/2)*arctan(-(1+2^(1/2))^(1/2)+(1+2^(1/2))^(1/2)*x)+1/8* (2^(1/2)-1)^(1/2)*arctan((1+2^(1/2))^(1/2)+(1+2^(1/2))^(1/2)*x)-1/8*(1+2^( 1/2))^(1/2)*arctanh(-(2^(1/2)-1)^(1/2)+(2^(1/2)-1)^(1/2)*x)+1/8*(1+2^(1/2) )^(1/2)*arctanh((2^(1/2)-1)^(1/2)+(2^(1/2)-1)^(1/2)*x)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=\frac {1}{16} \text {RootSum}\left [-2+4 \text {$\#$1}^2-4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log (x-\text {$\#$1})-2 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{2 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]-\frac {1}{16} \text {RootSum}\left [-2+4 \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log (x-\text {$\#$1})+2 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{2 \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \] Input:
Integrate[(2*x + x^5)/(4 - 16*x^2 + 12*x^4 - 8*x^6 + x^8),x]
Output:
RootSum[-2 + 4*#1^2 - 4*#1^3 + #1^4 & , (2*Log[x - #1] - 2*Log[x - #1]*#1 + Log[x - #1]*#1^2)/(2*#1 - 3*#1^2 + #1^3) & ]/16 - RootSum[-2 + 4*#1^2 + 4*#1^3 + #1^4 & , (2*Log[x - #1] + 2*Log[x - #1]*#1 + Log[x - #1]*#1^2)/(2 *#1 + 3*#1^2 + #1^3) & ]/16
Time = 0.79 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2027, 2462, 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.
\(\displaystyle \int \frac {x^5+2 x}{x^8-8 x^6+12 x^4-16 x^2+4} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x \left (x^4+2\right )}{x^8-8 x^6+12 x^4-16 x^2+4}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {-x^2-2 x-2}{4 \left (x^4+4 x^3+4 x^2-2\right )}+\frac {x^2-2 x+2}{4 \left (x^4-4 x^3+4 x^2-2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} \sqrt {\sqrt {2}-1} \arctan \left (\frac {1-x}{\sqrt {\sqrt {2}-1}}\right )+\frac {1}{8} \sqrt {\sqrt {2}-1} \arctan \left (\frac {x+1}{\sqrt {\sqrt {2}-1}}\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {1-x}{\sqrt {1+\sqrt {2}}}\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {x+1}{\sqrt {1+\sqrt {2}}}\right )\) |
Input:
Int[(2*x + x^5)/(4 - 16*x^2 + 12*x^4 - 8*x^6 + x^8),x]
Output:
(Sqrt[-1 + Sqrt[2]]*ArcTan[(1 - x)/Sqrt[-1 + Sqrt[2]]])/8 + (Sqrt[-1 + Sqr t[2]]*ArcTan[(1 + x)/Sqrt[-1 + Sqrt[2]]])/8 + (Sqrt[1 + Sqrt[2]]*ArcTanh[( 1 - x)/Sqrt[1 + Sqrt[2]]])/8 + (Sqrt[1 + Sqrt[2]]*ArcTanh[(1 + x)/Sqrt[1 + Sqrt[2]]])/8
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.20
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{2}+x^{2}-2 \textit {\_R} -1\right )\right )}{16}\) | \(33\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-8 \textit {\_Z}^{3}+12 \textit {\_Z}^{2}-16 \textit {\_Z} +4\right )}{\sum }\frac {\left (\textit {\_R}^{2}+2\right ) \ln \left (x^{2}-\textit {\_R} \right )}{\textit {\_R}^{3}-6 \textit {\_R}^{2}+6 \textit {\_R} -4}\right )}{8}\) | \(54\) |
Input:
int((x^5+2*x)/(x^8-8*x^6+12*x^4-16*x^2+4),x,method=_RETURNVERBOSE)
Output:
1/16*sum(_R*ln(-_R^2+x^2-2*_R-1),_R=RootOf(_Z^4-2*_Z^2-1))
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.54 \[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=-\frac {1}{8} \, \sqrt {\sqrt {2} - 1} \arctan \left (\frac {1}{2} \, {\left (x^{2} + \sqrt {2} {\left (x^{2} - 1\right )}\right )} \sqrt {\sqrt {2} - 1}\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} + 1} \log \left (x^{2} - \sqrt {2} + 2 \, \sqrt {\sqrt {2} + 1} - 2\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} + 1} \log \left (x^{2} - \sqrt {2} - 2 \, \sqrt {\sqrt {2} + 1} - 2\right ) \] Input:
integrate((x^5+2*x)/(x^8-8*x^6+12*x^4-16*x^2+4),x, algorithm="fricas")
Output:
-1/8*sqrt(sqrt(2) - 1)*arctan(1/2*(x^2 + sqrt(2)*(x^2 - 1))*sqrt(sqrt(2) - 1)) - 1/16*sqrt(sqrt(2) + 1)*log(x^2 - sqrt(2) + 2*sqrt(sqrt(2) + 1) - 2) + 1/16*sqrt(sqrt(2) + 1)*log(x^2 - sqrt(2) - 2*sqrt(sqrt(2) + 1) - 2)
Time = 0.21 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.84 \[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=- \sqrt {\frac {1}{256} + \frac {\sqrt {2}}{256}} \log {\left (x^{2} - 2 - \sqrt {2} + 32 \sqrt {\frac {1}{256} + \frac {\sqrt {2}}{256}} \right )} + \sqrt {\frac {1}{256} + \frac {\sqrt {2}}{256}} \log {\left (x^{2} - 32 \sqrt {\frac {1}{256} + \frac {\sqrt {2}}{256}} - 2 - \sqrt {2} \right )} - 2 \sqrt {- \frac {1}{256} + \frac {\sqrt {2}}{256}} \operatorname {atan}{\left (\frac {x^{2}}{2 \sqrt {-1 + \sqrt {2}}} - \frac {1}{\sqrt {-1 + \sqrt {2}}} + \frac {\sqrt {2}}{2 \sqrt {-1 + \sqrt {2}}} \right )} \] Input:
integrate((x**5+2*x)/(x**8-8*x**6+12*x**4-16*x**2+4),x)
Output:
-sqrt(1/256 + sqrt(2)/256)*log(x**2 - 2 - sqrt(2) + 32*sqrt(1/256 + sqrt(2 )/256)) + sqrt(1/256 + sqrt(2)/256)*log(x**2 - 32*sqrt(1/256 + sqrt(2)/256 ) - 2 - sqrt(2)) - 2*sqrt(-1/256 + sqrt(2)/256)*atan(x**2/(2*sqrt(-1 + sqr t(2))) - 1/sqrt(-1 + sqrt(2)) + sqrt(2)/(2*sqrt(-1 + sqrt(2))))
\[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=\int { \frac {x^{5} + 2 \, x}{x^{8} - 8 \, x^{6} + 12 \, x^{4} - 16 \, x^{2} + 4} \,d x } \] Input:
integrate((x^5+2*x)/(x^8-8*x^6+12*x^4-16*x^2+4),x, algorithm="maxima")
Output:
integrate((x^5 + 2*x)/(x^8 - 8*x^6 + 12*x^4 - 16*x^2 + 4), x)
Time = 0.16 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=0 \] Input:
integrate((x^5+2*x)/(x^8-8*x^6+12*x^4-16*x^2+4),x, algorithm="giac")
Output:
0
Time = 10.10 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.83 \[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=\frac {\mathrm {atanh}\left (\frac {32768\,\sqrt {1-\sqrt {2}}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2+155648\,x^2-48128}-\frac {20480\,\sqrt {2}\,\sqrt {1-\sqrt {2}}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2+155648\,x^2-48128}-\frac {101376\,x^2\,\sqrt {1-\sqrt {2}}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2+155648\,x^2-48128}+\frac {70656\,\sqrt {2}\,x^2\,\sqrt {1-\sqrt {2}}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2+155648\,x^2-48128}\right )\,\sqrt {1-\sqrt {2}}}{8}-\frac {\mathrm {atanh}\left (\frac {32768\,\sqrt {\sqrt {2}+1}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2-155648\,x^2+48128}+\frac {20480\,\sqrt {2}\,\sqrt {\sqrt {2}+1}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2-155648\,x^2+48128}-\frac {101376\,x^2\,\sqrt {\sqrt {2}+1}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2-155648\,x^2+48128}-\frac {70656\,\sqrt {2}\,x^2\,\sqrt {\sqrt {2}+1}}{33792\,\sqrt {2}-111104\,\sqrt {2}\,x^2-155648\,x^2+48128}\right )\,\sqrt {\sqrt {2}+1}}{8} \] Input:
int((2*x + x^5)/(12*x^4 - 16*x^2 - 8*x^6 + x^8 + 4),x)
Output:
(atanh((32768*(1 - 2^(1/2))^(1/2))/(33792*2^(1/2) - 111104*2^(1/2)*x^2 + 1 55648*x^2 - 48128) - (20480*2^(1/2)*(1 - 2^(1/2))^(1/2))/(33792*2^(1/2) - 111104*2^(1/2)*x^2 + 155648*x^2 - 48128) - (101376*x^2*(1 - 2^(1/2))^(1/2) )/(33792*2^(1/2) - 111104*2^(1/2)*x^2 + 155648*x^2 - 48128) + (70656*2^(1/ 2)*x^2*(1 - 2^(1/2))^(1/2))/(33792*2^(1/2) - 111104*2^(1/2)*x^2 + 155648*x ^2 - 48128))*(1 - 2^(1/2))^(1/2))/8 - (atanh((32768*(2^(1/2) + 1)^(1/2))/( 33792*2^(1/2) - 111104*2^(1/2)*x^2 - 155648*x^2 + 48128) + (20480*2^(1/2)* (2^(1/2) + 1)^(1/2))/(33792*2^(1/2) - 111104*2^(1/2)*x^2 - 155648*x^2 + 48 128) - (101376*x^2*(2^(1/2) + 1)^(1/2))/(33792*2^(1/2) - 111104*2^(1/2)*x^ 2 - 155648*x^2 + 48128) - (70656*2^(1/2)*x^2*(2^(1/2) + 1)^(1/2))/(33792*2 ^(1/2) - 111104*2^(1/2)*x^2 - 155648*x^2 + 48128))*(2^(1/2) + 1)^(1/2))/8
\[ \int \frac {2 x+x^5}{4-16 x^2+12 x^4-8 x^6+x^8} \, dx=\int \frac {x^{5}}{x^{8}-8 x^{6}+12 x^{4}-16 x^{2}+4}d x +2 \left (\int \frac {x}{x^{8}-8 x^{6}+12 x^{4}-16 x^{2}+4}d x \right ) \] Input:
int((x^5+2*x)/(x^8-8*x^6+12*x^4-16*x^2+4),x)
Output:
int(x**5/(x**8 - 8*x**6 + 12*x**4 - 16*x**2 + 4),x) + 2*int(x/(x**8 - 8*x* *6 + 12*x**4 - 16*x**2 + 4),x)