Integrand size = 32, antiderivative size = 97 \[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=\frac {1}{8} \sqrt {1+\sqrt {2}} \arctan \left (\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}-\frac {1}{2} \sqrt {-1+\sqrt {2}} x^2\right )-\frac {1}{8} \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\frac {1}{2} \sqrt {1+\sqrt {2}} x^2\right ) \] Output:
-1/8*(1+2^(1/2))^(1/2)*arctan(-1/2*(-2+2*2^(1/2))^(1/2)+1/2*(2^(1/2)-1)^(1 /2)*x^2)-1/8*(2^(1/2)-1)^(1/2)*arctanh(1/2*(2+2*2^(1/2))^(1/2)+1/2*(1+2^(1 /2))^(1/2)*x^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=-\frac {1}{8} \text {RootSum}\left [4+32 \text {$\#$1}+4 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-2 \log \left (x^2-\text {$\#$1}\right )+4 \log \left (x^2-\text {$\#$1}\right ) \text {$\#$1}+\log \left (x^2-\text {$\#$1}\right ) \text {$\#$1}^2}{8+2 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \] Input:
Integrate[(2*x - 4*x^3 - x^5)/(4 + 32*x^2 + 4*x^4 + x^8),x]
Output:
-1/8*RootSum[4 + 32*#1 + 4*#1^2 + #1^4 & , (-2*Log[x^2 - #1] + 4*Log[x^2 - #1]*#1 + Log[x^2 - #1]*#1^2)/(8 + 2*#1 + #1^3) & ]
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.
\(\displaystyle \int \frac {-x^5-4 x^3+2 x}{x^8+4 x^4+32 x^2+4} \, dx\) |
\(\Big \downarrow \) 2028 |
\(\displaystyle \int \frac {x \left (-x^4-4 x^2+2\right )}{x^8+4 x^4+32 x^2+4}dx\) |
\(\Big \downarrow \) 7266 |
\(\displaystyle \frac {1}{2} \int \frac {-x^4-4 x^2+2}{x^8+4 x^4+32 x^2+4}dx^2\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {x^4}{x^8+4 x^4+32 x^2+4}-\frac {4 x^2}{x^8+4 x^4+32 x^2+4}+\frac {2}{x^8+4 x^4+32 x^2+4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{x^8+4 x^4+32 x^2+4}dx^2-4 \int \frac {x^2}{x^8+4 x^4+32 x^2+4}dx^2-\int \frac {x^4}{x^8+4 x^4+32 x^2+4}dx^2\right )\) |
Input:
Int[(2*x - 4*x^3 - x^5)/(4 + 32*x^2 + 4*x^4 + x^8),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{2}+x^{2}-2 \textit {\_R} +1\right )\right )}{16}\) | \(31\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z}^{2}+32 \textit {\_Z} +4\right )}{\sum }\frac {\left (-\textit {\_R}^{2}-4 \textit {\_R} +2\right ) \ln \left (x^{2}-\textit {\_R} \right )}{\textit {\_R}^{3}+2 \textit {\_R} +8}\right )}{8}\) | \(49\) |
Input:
int((-x^5-4*x^3+2*x)/(x^8+4*x^4+32*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 = 87, normalized size of antiderivative = 0.90 \[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=-\frac {1}{8} \, \sqrt {\sqrt {2} + 1} \arctan \left (-\frac {1}{2} \, {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\sqrt {2} + 1}\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} - 1} \log \left (x^{2} + \sqrt {2} + 2 \, \sqrt {\sqrt {2} - 1}\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} - 1} \log \left (x^{2} + \sqrt {2} - 2 \, \sqrt {\sqrt {2} - 1}\right ) \] Input:
integrate((-x^5-4*x^3+2*x)/(x^8+4*x^4+32*x^2+4),x, algorithm="fricas")
Output:
-1/8*sqrt(sqrt(2) + 1)*arctan(-1/2*(x^2 - sqrt(2)*(x^2 + 1) + 2)*sqrt(sqrt (2) + 1)) - 1/16*sqrt(sqrt(2) - 1)*log(x^2 + sqrt(2) + 2*sqrt(sqrt(2) - 1) ) + 1/16*sqrt(sqrt(2) - 1)*log(x^2 + sqrt(2) - 2*sqrt(sqrt(2) - 1))
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.30 \[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=- \operatorname {RootSum} {\left (65536 t^{4} + 512 t^{2} - 1, \left ( t \mapsto t \log {\left (256 t^{2} + 32 t + x^{2} + 1 \right )} \right )\right )} \] Input:
integrate((-x**5-4*x**3+2*x)/(x**8+4*x**4+32*x**2+4),x)
Output:
-RootSum(65536*_t**4 + 512*_t**2 - 1, Lambda(_t, _t*log(256*_t**2 + 32*_t + x**2 + 1)))
\[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=\int { -\frac {x^{5} + 4 \, x^{3} - 2 \, x}{x^{8} + 4 \, x^{4} + 32 \, x^{2} + 4} \,d x } \] Input:
integrate((-x^5-4*x^3+2*x)/(x^8+4*x^4+32*x^2+4),x, algorithm="maxima")
Output:
-integrate((x^5 + 4*x^3 - 2*x)/(x^8 + 4*x^4 + 32*x^2 + 4), x)
Time = 0.14 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=0 \] Input:
integrate((-x^5-4*x^3+2*x)/(x^8+4*x^4+32*x^2+4),x, algorithm="giac")
Output:
0
Time = 0.15 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.14 \[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=\frac {\mathrm {atanh}\left (\frac {114688\,\sqrt {\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2+594944\,x^2+74752}+\frac {77824\,\sqrt {2}\,\sqrt {\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2+594944\,x^2+74752}+\frac {920576\,x^2\,\sqrt {\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2+594944\,x^2+74752}+\frac {652288\,\sqrt {2}\,x^2\,\sqrt {\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2+594944\,x^2+74752}\right )\,\sqrt {\sqrt {2}-1}}{8}-\frac {\mathrm {atanh}\left (\frac {114688\,\sqrt {-\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2-594944\,x^2-74752}-\frac {77824\,\sqrt {2}\,\sqrt {-\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2-594944\,x^2-74752}+\frac {920576\,x^2\,\sqrt {-\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2-594944\,x^2-74752}-\frac {652288\,\sqrt {2}\,x^2\,\sqrt {-\sqrt {2}-1}}{58368\,\sqrt {2}+421376\,\sqrt {2}\,x^2-594944\,x^2-74752}\right )\,\sqrt {-\sqrt {2}-1}}{8} \] Input:
int(-(4*x^3 - 2*x + x^5)/(32*x^2 + 4*x^4 + x^8 + 4),x)
Output:
(atanh((114688*(2^(1/2) - 1)^(1/2))/(58368*2^(1/2) + 421376*2^(1/2)*x^2 + 594944*x^2 + 74752) + (77824*2^(1/2)*(2^(1/2) - 1)^(1/2))/(58368*2^(1/2) + 421376*2^(1/2)*x^2 + 594944*x^2 + 74752) + (920576*x^2*(2^(1/2) - 1)^(1/2 ))/(58368*2^(1/2) + 421376*2^(1/2)*x^2 + 594944*x^2 + 74752) + (652288*2^( 1/2)*x^2*(2^(1/2) - 1)^(1/2))/(58368*2^(1/2) + 421376*2^(1/2)*x^2 + 594944 *x^2 + 74752))*(2^(1/2) - 1)^(1/2))/8 - (atanh((114688*(- 2^(1/2) - 1)^(1/ 2))/(58368*2^(1/2) + 421376*2^(1/2)*x^2 - 594944*x^2 - 74752) - (77824*2^( 1/2)*(- 2^(1/2) - 1)^(1/2))/(58368*2^(1/2) + 421376*2^(1/2)*x^2 - 594944*x ^2 - 74752) + (920576*x^2*(- 2^(1/2) - 1)^(1/2))/(58368*2^(1/2) + 421376*2 ^(1/2)*x^2 - 594944*x^2 - 74752) - (652288*2^(1/2)*x^2*(- 2^(1/2) - 1)^(1/ 2))/(58368*2^(1/2) + 421376*2^(1/2)*x^2 - 594944*x^2 - 74752))*(- 2^(1/2) - 1)^(1/2))/8
\[ \int \frac {2 x-4 x^3-x^5}{4+32 x^2+4 x^4+x^8} \, dx=-\left (\int \frac {x^{5}}{x^{8}+4 x^{4}+32 x^{2}+4}d x \right )-4 \left (\int \frac {x^{3}}{x^{8}+4 x^{4}+32 x^{2}+4}d x \right )+2 \left (\int \frac {x}{x^{8}+4 x^{4}+32 x^{2}+4}d x \right ) \] Input:
int((-x^5-4*x^3+2*x)/(x^8+4*x^4+32*x^2+4),x)
Output:
- int(x**5/(x**8 + 4*x**4 + 32*x**2 + 4),x) - 4*int(x**3/(x**8 + 4*x**4 + 32*x**2 + 4),x) + 2*int(x/(x**8 + 4*x**4 + 32*x**2 + 4),x)