Integrand size = 58, antiderivative size = 186 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {\left (-1+x+x^2\right ) \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{2 x}-\frac {3 \text {arctanh}\left (\frac {\sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {1}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {1}{2 \sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right )+\frac {1}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5+\sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right ) \] Output:
1/2*(x^2+x-1)*((2*x^2+x-2)/(x^2+x-1))^(1/2)/x-3/4*arctanh(1/2*((2*x^2+x-2) /(x^2+x-1))^(1/2)*2^(1/2))*2^(1/2)+1/5*(10-2*5^(1/2))^(1/2)*arctanh(1/10*( 50+10*5^(1/2))^(1/2)*((2*x^2+x-2)/(x^2+x-1))^(1/2))+1/5*(10+2*5^(1/2))^(1/ 2)*arctanh(2^(1/2)/(5+5^(1/2))^(1/2)*((2*x^2+x-2)/(x^2+x-1))^(1/2))
Time = 4.61 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.41 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {-20 \sqrt {-1+x+x^2}+10 x \sqrt {-1+x+x^2}+20 x^2 \sqrt {-1+x+x^2}-15 \sqrt {2} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {-2+x+2 x^2}}{\sqrt {2} \sqrt {-1+x+x^2}}\right )+4 \sqrt {2 \left (5-\sqrt {5}\right )} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \sqrt {-2+x+2 x^2}}{\sqrt {-1+x+x^2}}\right )+4 \sqrt {2 \left (5+\sqrt {5}\right )} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {-\left (\left (-5+\sqrt {5}\right ) \left (-2+x+2 x^2\right )\right )}}{\sqrt {10} \sqrt {-1+x+x^2}}\right )}{20 x \sqrt {-1+x+x^2} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}} \] Input:
Integrate[((1 + x^2)*(1 - 3*x^2 + x^4))/(x^2*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)]*(1 - x - 3*x^2 + x^3 + x^4)),x]
Output:
(-20*Sqrt[-1 + x + x^2] + 10*x*Sqrt[-1 + x + x^2] + 20*x^2*Sqrt[-1 + x + x ^2] - 15*Sqrt[2]*x*Sqrt[-2 + x + 2*x^2]*ArcTanh[Sqrt[-2 + x + 2*x^2]/(Sqrt [2]*Sqrt[-1 + x + x^2])] + 4*Sqrt[2*(5 - Sqrt[5])]*x*Sqrt[-2 + x + 2*x^2]* ArcTanh[(Sqrt[(5 + Sqrt[5])/10]*Sqrt[-2 + x + 2*x^2])/Sqrt[-1 + x + x^2]] + 4*Sqrt[2*(5 + Sqrt[5])]*x*Sqrt[-2 + x + 2*x^2]*ArcTanh[Sqrt[-((-5 + Sqrt [5])*(-2 + x + 2*x^2))]/(Sqrt[10]*Sqrt[-1 + x + x^2])])/(20*x*Sqrt[-1 + x + x^2]*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2)])
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.
| \(\displaystyle \int \frac {\left (x^2+1\right ) \left (x^4-3 x^2+1\right )}{x^2 \sqrt {\frac {2 x^2+x-2}{x^2+x-1}} \left (x^4+x^3-3 x^2-x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt {2 x^2+x-2} \int \frac {\left (x^2+1\right ) \sqrt {x^2+x-1} \left (x^4-3 x^2+1\right )}{x^2 \sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx}{\sqrt {\frac {-2 x^2-x+2}{-x^2-x+1}} \sqrt {x^2+x-1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {2 x^2+x-2} \int \left (\frac {\sqrt {x^2+x-1} \left (-2 x^3-x^2+3 x+1\right )}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}+\frac {\sqrt {x^2+x-1}}{x \sqrt {2 x^2+x-2}}+\frac {\sqrt {x^2+x-1}}{x^2 \sqrt {2 x^2+x-2}}+\frac {\sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2}}\right )dx}{\sqrt {\frac {-2 x^2-x+2}{-x^2-x+1}} \sqrt {x^2+x-1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {2 x^2+x-2} \left (\int \frac {\sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2}}dx+\int \frac {\sqrt {x^2+x-1}}{x^2 \sqrt {2 x^2+x-2}}dx+\int \frac {\sqrt {x^2+x-1}}{x \sqrt {2 x^2+x-2}}dx+\int \frac {\sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx+3 \int \frac {x \sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx-\int \frac {x^2 \sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx-2 \int \frac {x^3 \sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx\right )}{\sqrt {\frac {-2 x^2-x+2}{-x^2-x+1}} \sqrt {x^2+x-1}}\) |
Input:
Int[((1 + x^2)*(1 - 3*x^2 + x^4))/(x^2*Sqrt[(-2 + x + 2*x^2)/(-1 + x + x^2 )]*(1 - x - 3*x^2 + x^3 + x^4)),x]
Output:
$Aborted
Time = 6.95 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(\frac {2 x^{2}+x -2}{2 x \sqrt {\frac {2 x^{2}+x -2}{x^{2}+x -1}}}+\frac {\left (-\frac {3 \sqrt {2}\, \ln \left (\frac {4 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}+\sqrt {2}\, \left (4 x^{2}+3 x -4\right )}{x}\right )}{8}+\frac {\operatorname {arctanh}\left (\frac {\left (4 x^{2}+3 x -4\right ) \sqrt {5}+2 x^{2}+x -2}{2 \sqrt {10+2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) \sqrt {10-2 \sqrt {5}}}{10}+\frac {\operatorname {arctanh}\left (\frac {\left (4 x^{2}+3 x -4\right ) \sqrt {5}-2 x^{2}-x +2}{2 \sqrt {10-2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) \sqrt {10+2 \sqrt {5}}}{10}\right ) \sqrt {\left (2 x^{2}+x -2\right ) \left (x^{2}+x -1\right )}}{\sqrt {\frac {2 x^{2}+x -2}{x^{2}+x -1}}\, \left (x^{2}+x -1\right )}\) | \(265\) |
| default | \(-\frac {\left (2 x^{2}+x -2\right ) \left (15 \sqrt {2}\, \ln \left (\frac {4 \sqrt {2}\, x^{2}+3 x \sqrt {2}-4 \sqrt {2}+4 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}{x}\right ) x -4 \,\operatorname {arctanh}\left (\frac {4 \sqrt {5}\, x^{2}+3 x \sqrt {5}+2 x^{2}-4 \sqrt {5}+x -2}{2 \sqrt {10+2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) \sqrt {10-2 \sqrt {5}}\, x -4 \,\operatorname {arctanh}\left (\frac {4 \sqrt {5}\, x^{2}+3 x \sqrt {5}-2 x^{2}-4 \sqrt {5}-x +2}{2 \sqrt {10-2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) \sqrt {10+2 \sqrt {5}}\, x -20 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}\right )}{40 \sqrt {\frac {2 x^{2}+x -2}{x^{2}+x -1}}\, \sqrt {\left (2 x^{2}+x -2\right ) \left (x^{2}+x -1\right )}\, x}\) | \(278\) |
| trager | \(\text {Expression too large to display}\) | \(1371\) |
Input:
int((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2 -x+1),x,method=_RETURNVERBOSE)
Output:
1/2*(2*x^2+x-2)/x/((2*x^2+x-2)/(x^2+x-1))^(1/2)+(-3/8*2^(1/2)*ln((4*(2*x^4 +3*x^3-3*x^2-3*x+2)^(1/2)+2^(1/2)*(4*x^2+3*x-4))/x)+1/10*arctanh(1/2*((4*x ^2+3*x-4)*5^(1/2)+2*x^2+x-2)/(10+2*5^(1/2))^(1/2)/(2*x^4+3*x^3-3*x^2-3*x+2 )^(1/2))*(10-2*5^(1/2))^(1/2)+1/10*arctanh(1/2*((4*x^2+3*x-4)*5^(1/2)-2*x^ 2-x+2)/(10-2*5^(1/2))^(1/2)/(2*x^4+3*x^3-3*x^2-3*x+2)^(1/2))*(10+2*5^(1/2) )^(1/2))/((2*x^2+x-2)/(x^2+x-1))^(1/2)*((2*x^2+x-2)*(x^2+x-1))^(1/2)/(x^2+ x-1)
Leaf count of result is larger than twice the leaf count of optimal. 723 vs. \(2 (145) = 290\).
Time = 0.36 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.89 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx =\text {Too large to display} \] Input:
integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3 -3*x^2-x+1),x, algorithm="fricas")
Output:
1/80*(8*x*sqrt(1/2*sqrt(5) + 5/2)*log(((20575*x^4 + 50235*x^3 - 15795*x^2 - sqrt(5)*(10237*x^4 + 22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(1/2*sqrt(5) + 5/2) + 10*(1627*x^4 + 4593*x^3 - 288*x^2 - sqrt (5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2 *x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) - 8*x*sqrt(1/2* sqrt(5) + 5/2)*log(-((20575*x^4 + 50235*x^3 - 15795*x^2 - sqrt(5)*(10237*x ^4 + 22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(1/2*s qrt(5) + 5/2) - 10*(1627*x^4 + 4593*x^3 - 288*x^2 - sqrt(5)*(861*x^4 + 210 5*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) + 8*x*sqrt(-1/2*sqrt(5) + 5/2)*lo g(((20575*x^4 + 50235*x^3 - 15795*x^2 + sqrt(5)*(10237*x^4 + 22677*x^3 - 9 661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(-1/2*sqrt(5) + 5/2) + 1 0*(1627*x^4 + 4593*x^3 - 288*x^2 + sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) - 8*x*sqrt(-1/2*sqrt(5) + 5/2)*log(-((20575*x^4 + 50235*x^3 - 15795*x^2 + sqrt(5)*(10237*x^4 + 22677*x^3 - 9661*x^2 - 22677* x + 10237) - 50235*x + 20575)*sqrt(-1/2*sqrt(5) + 5/2) - 10*(1627*x^4 + 45 93*x^3 - 288*x^2 + sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) + 15*sqrt(2)*x*log(-(32*x^4 + 48*x^3 - 47*x^2 - 4*sqrt(2)*(4*x^...
Timed out. \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((x**2+1)*(x**4-3*x**2+1)/x**2/((2*x**2+x-2)/(x**2+x-1))**(1/2)/( x**4+x**3-3*x**2-x+1),x)
Output:
Timed out
\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}} \,d x } \] Input:
integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3 -3*x^2-x+1),x, algorithm="maxima")
Output:
integrate((x^4 - 3*x^2 + 1)*(x^2 + 1)/((x^4 + x^3 - 3*x^2 - x + 1)*x^2*sqr t((2*x^2 + x - 2)/(x^2 + x - 1))), x)
\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}} \,d x } \] Input:
integrate((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3 -3*x^2-x+1),x, algorithm="giac")
Output:
integrate((x^4 - 3*x^2 + 1)*(x^2 + 1)/((x^4 + x^3 - 3*x^2 - x + 1)*x^2*sqr t((2*x^2 + x - 2)/(x^2 + x - 1))), x)
Timed out. \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int \frac {\left (x^2+1\right )\,\left (x^4-3\,x^2+1\right )}{x^2\,\sqrt {\frac {2\,x^2+x-2}{x^2+x-1}}\,\left (x^4+x^3-3\,x^2-x+1\right )} \,d x \] Input:
int(((x^2 + 1)*(x^4 - 3*x^2 + 1))/(x^2*((x + 2*x^2 - 2)/(x + x^2 - 1))^(1/ 2)*(x^3 - 3*x^2 - x + x^4 + 1)),x)
Output:
int(((x^2 + 1)*(x^4 - 3*x^2 + 1))/(x^2*((x + 2*x^2 - 2)/(x + x^2 - 1))^(1/ 2)*(x^3 - 3*x^2 - x + x^4 + 1)), x)
\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {2 \sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}\, x +14 \sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}-8 \left (\int \frac {\sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}\, x^{7}}{2 x^{8}+5 x^{7}-6 x^{6}-17 x^{5}+7 x^{4}+17 x^{3}-6 x^{2}-5 x +2}d x \right ) x -17 \left (\int \frac {\sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}\, x^{6}}{2 x^{8}+5 x^{7}-6 x^{6}-17 x^{5}+7 x^{4}+17 x^{3}-6 x^{2}-5 x +2}d x \right ) x +23 \left (\int \frac {\sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}\, x^{4}}{2 x^{8}+5 x^{7}-6 x^{6}-17 x^{5}+7 x^{4}+17 x^{3}-6 x^{2}-5 x +2}d x \right ) x -24 \left (\int \frac {\sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}\, x^{2}}{2 x^{8}+5 x^{7}-6 x^{6}-17 x^{5}+7 x^{4}+17 x^{3}-6 x^{2}-5 x +2}d x \right ) x +17 \left (\int \frac {\sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}\, x}{2 x^{8}+5 x^{7}-6 x^{6}-17 x^{5}+7 x^{4}+17 x^{3}-6 x^{2}-5 x +2}d x \right ) x -21 \left (\int \frac {\sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}}{2 x^{9}+5 x^{8}-6 x^{7}-17 x^{6}+7 x^{5}+17 x^{4}-6 x^{3}-5 x^{2}+2 x}d x \right ) x +24 \left (\int \frac {\sqrt {2 x^{2}+x -2}\, \sqrt {x^{2}+x -1}}{2 x^{8}+5 x^{7}-6 x^{6}-17 x^{5}+7 x^{4}+17 x^{3}-6 x^{2}-5 x +2}d x \right ) x}{28 x} \] Input:
int((x^2+1)*(x^4-3*x^2+1)/x^2/((2*x^2+x-2)/(x^2+x-1))^(1/2)/(x^4+x^3-3*x^2 -x+1),x)
Output:
(2*sqrt(2*x**2 + x - 2)*sqrt(x**2 + x - 1)*x + 14*sqrt(2*x**2 + x - 2)*sqr t(x**2 + x - 1) - 8*int((sqrt(2*x**2 + x - 2)*sqrt(x**2 + x - 1)*x**7)/(2* x**8 + 5*x**7 - 6*x**6 - 17*x**5 + 7*x**4 + 17*x**3 - 6*x**2 - 5*x + 2),x) *x - 17*int((sqrt(2*x**2 + x - 2)*sqrt(x**2 + x - 1)*x**6)/(2*x**8 + 5*x** 7 - 6*x**6 - 17*x**5 + 7*x**4 + 17*x**3 - 6*x**2 - 5*x + 2),x)*x + 23*int( (sqrt(2*x**2 + x - 2)*sqrt(x**2 + x - 1)*x**4)/(2*x**8 + 5*x**7 - 6*x**6 - 17*x**5 + 7*x**4 + 17*x**3 - 6*x**2 - 5*x + 2),x)*x - 24*int((sqrt(2*x**2 + x - 2)*sqrt(x**2 + x - 1)*x**2)/(2*x**8 + 5*x**7 - 6*x**6 - 17*x**5 + 7 *x**4 + 17*x**3 - 6*x**2 - 5*x + 2),x)*x + 17*int((sqrt(2*x**2 + x - 2)*sq rt(x**2 + x - 1)*x)/(2*x**8 + 5*x**7 - 6*x**6 - 17*x**5 + 7*x**4 + 17*x**3 - 6*x**2 - 5*x + 2),x)*x - 21*int((sqrt(2*x**2 + x - 2)*sqrt(x**2 + x - 1 ))/(2*x**9 + 5*x**8 - 6*x**7 - 17*x**6 + 7*x**5 + 17*x**4 - 6*x**3 - 5*x** 2 + 2*x),x)*x + 24*int((sqrt(2*x**2 + x - 2)*sqrt(x**2 + x - 1))/(2*x**8 + 5*x**7 - 6*x**6 - 17*x**5 + 7*x**4 + 17*x**3 - 6*x**2 - 5*x + 2),x)*x)/(2 8*x)