Integrand size = 58, antiderivative size = 186 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {\left (-1+x+x^2\right ) \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{2 x}-\frac {3 \text {arctanh}\left (\frac {\sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}}{\sqrt {2}}\right )}{2 \sqrt {2}}+\frac {1}{5} \sqrt {2 \left (5-\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2}+\frac {1}{2 \sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right )+\frac {1}{5} \sqrt {2 \left (5+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{5+\sqrt {5}}} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}\right ) \]
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 = 4.57 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.41 \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\frac {-20 \sqrt {-1+x+x^2}+10 x \sqrt {-1+x+x^2}+20 x^2 \sqrt {-1+x+x^2}-15 \sqrt {2} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {-2+x+2 x^2}}{\sqrt {2} \sqrt {-1+x+x^2}}\right )+4 \sqrt {2 \left (5-\sqrt {5}\right )} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \sqrt {-2+x+2 x^2}}{\sqrt {-1+x+x^2}}\right )+4 \sqrt {2 \left (5+\sqrt {5}\right )} x \sqrt {-2+x+2 x^2} \text {arctanh}\left (\frac {\sqrt {-\left (\left (-5+\sqrt {5}\right ) \left (-2+x+2 x^2\right )\right )}}{\sqrt {10} \sqrt {-1+x+x^2}}\right )}{20 x \sqrt {-1+x+x^2} \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}}} \]
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]
(-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 |
| \(\displaystyle \frac {\sqrt {2 x^2+x-2} \left (\int \frac {\sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2}}dx+\int \frac {\sqrt {x^2+x-1}}{x^2 \sqrt {2 x^2+x-2}}dx+\int \frac {\sqrt {x^2+x-1}}{x \sqrt {2 x^2+x-2}}dx+\int \frac {\sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx+3 \int \frac {x \sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx-\int \frac {x^2 \sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx-2 \int \frac {x^3 \sqrt {x^2+x-1}}{\sqrt {2 x^2+x-2} \left (x^4+x^3-3 x^2-x+1\right )}dx\right )}{\sqrt {\frac {-2 x^2-x+2}{-x^2-x+1}} \sqrt {x^2+x-1}}\) |
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]
3.24.48.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Time = 6.06 (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 | \(-\frac {\left (2 x^{2}+x -2\right ) \left (15 \sqrt {2}\, \ln \left (\frac {4 \sqrt {2}\, x^{2}+3 x \sqrt {2}-4 \sqrt {2}+4 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}{x}\right ) x -4 \sqrt {10-2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {5}\, x^{2}+3 x \sqrt {5}+2 x^{2}-4 \sqrt {5}+x -2}{2 \sqrt {10+2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) x -4 \sqrt {10+2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {4 \sqrt {5}\, x^{2}+3 x \sqrt {5}-2 x^{2}-4 \sqrt {5}-x +2}{2 \sqrt {10-2 \sqrt {5}}\, \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}}\right ) x -20 \sqrt {2 x^{4}+3 x^{3}-3 x^{2}-3 x +2}\right )}{40 \sqrt {\frac {2 x^{2}+x -2}{x^{2}+x -1}}\, \sqrt {\left (2 x^{2}+x -2\right ) \left (x^{2}+x -1\right )}\, x}\) | \(278\) |
| trager | \(\text {Expression too large to display}\) | \(1371\) |
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)
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 = 0.50 (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=\frac {4 \, x \sqrt {2 \, \sqrt {5} + 10} \log \left (\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} - \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {2 \, \sqrt {5} + 10} + 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} - \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) - 4 \, x \sqrt {2 \, \sqrt {5} + 10} \log \left (-\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} - \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {2 \, \sqrt {5} + 10} - 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} - \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) + 4 \, x \sqrt {-2 \, \sqrt {5} + 10} \log \left (\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} + \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {-2 \, \sqrt {5} + 10} + 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} + \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) - 4 \, x \sqrt {-2 \, \sqrt {5} + 10} \log \left (-\frac {{\left (20575 \, x^{4} + 50235 \, x^{3} - 15795 \, x^{2} + \sqrt {5} {\left (10237 \, x^{4} + 22677 \, x^{3} - 9661 \, x^{2} - 22677 \, x + 10237\right )} - 50235 \, x + 20575\right )} \sqrt {-2 \, \sqrt {5} + 10} - 20 \, {\left (1627 \, x^{4} + 4593 \, x^{3} - 288 \, x^{2} + \sqrt {5} {\left (861 \, x^{4} + 2105 \, x^{3} - 478 \, x^{2} - 2105 \, x + 861\right )} - 4593 \, x + 1627\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{x^{4} + x^{3} - 3 \, x^{2} - x + 1}\right ) + 15 \, \sqrt {2} x \log \left (-\frac {32 \, x^{4} + 48 \, x^{3} - 47 \, x^{2} - 4 \, \sqrt {2} {\left (4 \, x^{4} + 7 \, x^{3} - 5 \, x^{2} - 7 \, x + 4\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}} - 48 \, x + 32}{x^{2}}\right ) + 40 \, {\left (x^{2} + x - 1\right )} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}}{80 \, x} \]
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")
1/80*(4*x*sqrt(2*sqrt(5) + 10)*log(((20575*x^4 + 50235*x^3 - 15795*x^2 - s qrt(5)*(10237*x^4 + 22677*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20 575)*sqrt(2*sqrt(5) + 10) + 20*(1627*x^4 + 4593*x^3 - 288*x^2 - sqrt(5)*(8 61*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)) - 4*x*sqrt(2*sqrt(5) + 10)*log(-((20575*x^4 + 50235*x^3 - 15795*x^2 - sqrt(5)*(10237*x^4 + 2267 7*x^3 - 9661*x^2 - 22677*x + 10237) - 50235*x + 20575)*sqrt(2*sqrt(5) + 10 ) - 20*(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)) + 4*x*sqrt(-2*sqrt(5) + 10)*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(-2*sqrt(5) + 10) + 20*(1627*x^4 + 4593 *x^3 - 288*x^2 + sqrt(5)*(861*x^4 + 2105*x^3 - 478*x^2 - 2105*x + 861) - 4 593*x + 1627)*sqrt((2*x^2 + x - 2)/(x^2 + x - 1)))/(x^4 + x^3 - 3*x^2 - x + 1)) - 4*x*sqrt(-2*sqrt(5) + 10)*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(-2*sqrt(5) + 10) - 20*(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)) + 15*sqrt(2)*x*l og(-(32*x^4 + 48*x^3 - 47*x^2 - 4*sqrt(2)*(4*x^4 + 7*x^3 - 5*x^2 - 7*x ...
Timed out. \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\text {Timed out} \]
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)
\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}} \,d x } \]
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")
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)
\[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3 \, x^{2} + 1\right )} {\left (x^{2} + 1\right )}}{{\left (x^{4} + x^{3} - 3 \, x^{2} - x + 1\right )} x^{2} \sqrt {\frac {2 \, x^{2} + x - 2}{x^{2} + x - 1}}} \,d x } \]
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")
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. \[ \int \frac {\left (1+x^2\right ) \left (1-3 x^2+x^4\right )}{x^2 \sqrt {\frac {-2+x+2 x^2}{-1+x+x^2}} \left (1-x-3 x^2+x^3+x^4\right )} \, dx=\int \frac {\left (x^2+1\right )\,\left (x^4-3\,x^2+1\right )}{x^2\,\sqrt {\frac {2\,x^2+x-2}{x^2+x-1}}\,\left (x^4+x^3-3\,x^2-x+1\right )} \,d x \]
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)