Integrand size = 60, antiderivative size = 22 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=-\frac {1}{2} \text {arctanh}\left (\frac {2 x}{3-2 \sqrt {2}+x^2}\right ) \] Output:
-1/2*arctanh(2*x/(3-2*2^(1/2)+x^2))
Leaf count is larger than twice the leaf count of optimal. \(45\) vs. \(2(22)=44\).
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=-\frac {1}{4} \log \left (-3+2 \sqrt {2}-2 x-x^2\right )+\frac {1}{4} \log \left (-3+2 \sqrt {2}+2 x-x^2\right ) \] Input:
Integrate[((-3 + 2*Sqrt[2])^2 - x^4)/(-99 + 70*Sqrt[2] + (-39 + 28*Sqrt[2] )*x^2 + (-5 + 6*Sqrt[2])*x^4 - x^6),x]
Output:
-1/4*Log[-3 + 2*Sqrt[2] - 2*x - x^2] + Log[-3 + 2*Sqrt[2] + 2*x - x^2]/4
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.
\(\displaystyle \int \frac {\left (2 \sqrt {2}-3\right )^2-x^4}{-x^6+\left (6 \sqrt {2}-5\right ) x^4+\left (28 \sqrt {2}-39\right ) x^2+70 \sqrt {2}-99} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^4-\left (2 \sqrt {2}-3\right )^2}{x^6-\left (6 \sqrt {2}-5\right ) x^4-\left (28 \sqrt {2}-39\right ) x^2+99 \left (1-\frac {70 \sqrt {2}}{99}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^4}{x^6+5 \left (1-\frac {6 \sqrt {2}}{5}\right ) x^4+39 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^2+99 \left (1-\frac {70 \sqrt {2}}{99}\right )}+\frac {12 \sqrt {2}-17}{x^6+5 \left (1-\frac {6 \sqrt {2}}{5}\right ) x^4+39 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^2+99 \left (1-\frac {70 \sqrt {2}}{99}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {x^4}{x^6+5 \left (1-\frac {6 \sqrt {2}}{5}\right ) x^4+39 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^2+99 \left (1-\frac {70 \sqrt {2}}{99}\right )}dx-\left (17-12 \sqrt {2}\right ) \int \frac {1}{x^6+5 \left (1-\frac {6 \sqrt {2}}{5}\right ) x^4+39 \left (1-\frac {28 \sqrt {2}}{39}\right ) x^2+99 \left (1-\frac {70 \sqrt {2}}{99}\right )}dx\) |
Input:
Int[((-3 + 2*Sqrt[2])^2 - x^4)/(-99 + 70*Sqrt[2] + (-39 + 28*Sqrt[2])*x^2 + (-5 + 6*Sqrt[2])*x^4 - x^6),x]
Output:
$Aborted
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55
method | result | size |
default | \(\frac {\ln \left (x^{2}-2 \sqrt {2}-2 x +3\right )}{4}-\frac {\ln \left (x^{2}-2 \sqrt {2}+2 x +3\right )}{4}\) | \(34\) |
risch | \(\frac {\ln \left (x^{2}-2 \sqrt {2}-2 x +3\right )}{4}-\frac {\ln \left (x^{2}-2 \sqrt {2}+2 x +3\right )}{4}\) | \(34\) |
parallelrisch | \(\frac {\ln \left (x^{2}-2 \sqrt {2}-2 x +3\right )}{4}-\frac {\ln \left (x^{2}-2 \sqrt {2}+2 x +3\right )}{4}\) | \(34\) |
Input:
int(((-3+2*2^(1/2))^2-x^4)/(-99+70*2^(1/2)+(-39+28*2^(1/2))*x^2+(-5+6*2^(1 /2))*x^4-x^6),x,method=_RETURNVERBOSE)
Output:
1/4*ln(x^2-2*2^(1/2)-2*x+3)-1/4*ln(x^2-2*2^(1/2)+2*x+3)
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=-\frac {1}{4} \, \log \left (x^{2} + 2 \, x - 2 \, \sqrt {2} + 3\right ) + \frac {1}{4} \, \log \left (x^{2} - 2 \, x - 2 \, \sqrt {2} + 3\right ) \] Input:
integrate(((-3+2*2^(1/2))^2-x^4)/(-99+70*2^(1/2)+(-39+28*2^(1/2))*x^2+(-5+ 6*2^(1/2))*x^4-x^6),x, algorithm="fricas")
Output:
-1/4*log(x^2 + 2*x - 2*sqrt(2) + 3) + 1/4*log(x^2 - 2*x - 2*sqrt(2) + 3)
Exception generated. \[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate(((-3+2*2**(1/2))**2-x**4)/(-99+70*2**(1/2)+(-39+28*2**(1/2))*x** 2+(-5+6*2**(1/2))*x**4-x**6),x)
Output:
Exception raised: PolynomialError >> 1/(-160817378869521623700434126076296 32108508413135104*_t**4 + 113715059131284938253449200806460980121432594954 24*sqrt(2)*_t**4 - 525076531527889516631004780624192141786868300832*_t**2 + 3712851760852
\[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=\int { \frac {x^{4} - {\left (2 \, \sqrt {2} - 3\right )}^{2}}{x^{6} - x^{4} {\left (6 \, \sqrt {2} - 5\right )} - x^{2} {\left (28 \, \sqrt {2} - 39\right )} - 70 \, \sqrt {2} + 99} \,d x } \] Input:
integrate(((-3+2*2^(1/2))^2-x^4)/(-99+70*2^(1/2)+(-39+28*2^(1/2))*x^2+(-5+ 6*2^(1/2))*x^4-x^6),x, algorithm="maxima")
Output:
integrate((x^4 - (2*sqrt(2) - 3)^2)/(x^6 - x^4*(6*sqrt(2) - 5) - x^2*(28*s qrt(2) - 39) - 70*sqrt(2) + 99), x)
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=-\frac {1}{4} \, \log \left ({\left | x^{2} + 2 \, x - 2 \, \sqrt {2} + 3 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x^{2} - 2 \, x - 2 \, \sqrt {2} + 3 \right |}\right ) \] Input:
integrate(((-3+2*2^(1/2))^2-x^4)/(-99+70*2^(1/2)+(-39+28*2^(1/2))*x^2+(-5+ 6*2^(1/2))*x^4-x^6),x, algorithm="giac")
Output:
-1/4*log(abs(x^2 + 2*x - 2*sqrt(2) + 3)) + 1/4*log(abs(x^2 - 2*x - 2*sqrt( 2) + 3))
Time = 22.75 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=-\frac {\mathrm {atanh}\left (\frac {x\,\left (16\,\sqrt {2}-16\right )}{2\,\left (20\,\sqrt {2}+4\,\sqrt {2}\,x^2-4\,x^2-28\right )}\right )}{2} \] Input:
int(((2*2^(1/2) - 3)^2 - x^4)/(x^4*(6*2^(1/2) - 5) + x^2*(28*2^(1/2) - 39) + 70*2^(1/2) - x^6 - 99),x)
Output:
-atanh((x*(16*2^(1/2) - 16))/(2*(20*2^(1/2) + 4*2^(1/2)*x^2 - 4*x^2 - 28)) )/2
\[ \int \frac {\left (-3+2 \sqrt {2}\right )^2-x^4}{-99+70 \sqrt {2}+\left (-39+28 \sqrt {2}\right ) x^2+\left (-5+6 \sqrt {2}\right ) x^4-x^6} \, dx=6 \sqrt {2}\, \left (\int \frac {x^{4}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )+4 \sqrt {2}\, \left (\int \frac {x^{2}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )-2 \sqrt {2}\, \left (\int \frac {1}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )+\int \frac {x^{6}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x -\left (\int \frac {x^{4}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )+27 \left (\int \frac {x^{2}}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right )-3 \left (\int \frac {1}{x^{8}+4 x^{6}+6 x^{4}-124 x^{2}+1}d x \right ) \] Input:
int(((-3+2*2^(1/2))^2-x^4)/(-99+70*2^(1/2)+(-39+28*2^(1/2))*x^2+(-5+6*2^(1 /2))*x^4-x^6),x)
Output:
6*sqrt(2)*int(x**4/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) + 4*sqrt(2)* int(x**2/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) - 2*sqrt(2)*int(1/(x** 8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) + int(x**6/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) - int(x**4/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) + 27*int(x**2/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x) - 3*int(1/(x**8 + 4*x**6 + 6*x**4 - 124*x**2 + 1),x)