Integrand size = 36, antiderivative size = 98 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {\left (1+2 x+x^2\right ) \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )+2 \log (x)-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}\right ) \] Output:
(x^2+2*x+1)*(x^4+3*x^2+1)^(1/2)/x/(x^2+x+1)-3*2^(1/2)*arctanh(2^(1/2)*x/(1 +x+x^2+(x^4+3*x^2+1)^(1/2)))+2*ln(x)-2*ln(1+x^2+(x^4+3*x^2+1)^(1/2))
Time = 0.87 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {(1+x)^2 \sqrt {1+3 x^2+x^4}}{x \left (1+x+x^2\right )}-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )+2 \log (x)-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}\right ) \] Input:
Integrate[((-1 + x^2)*(1 + x^2)*Sqrt[1 + 3*x^2 + x^4])/(x^2*(1 + x + x^2)^ 2),x]
Output:
((1 + x)^2*Sqrt[1 + 3*x^2 + x^4])/(x*(1 + x + x^2)) - 3*Sqrt[2]*ArcTanh[(S qrt[2]*x)/(1 + x + x^2 + Sqrt[1 + 3*x^2 + x^4])] + 2*Log[x] - 2*Log[1 + x^ 2 + Sqrt[1 + 3*x^2 + x^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.
| \(\displaystyle \int \frac {\left (x^2-1\right ) \left (x^2+1\right ) \sqrt {x^4+3 x^2+1}}{x^2 \left (x^2+x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4+3 x^2+1} (-2 x-1)}{\left (x^2+x+1\right )^2}+\frac {2 \sqrt {x^4+3 x^2+1}}{x}-\frac {2 x \sqrt {x^4+3 x^2+1}}{x^2+x+1}-\frac {\sqrt {x^4+3 x^2+1}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (3+i \sqrt {3}\right ) x \left (2 x^2+\sqrt {5}+3\right )}{6 \sqrt {x^4+3 x^2+1}}+\frac {\left (3-i \sqrt {3}\right ) x \left (2 x^2+\sqrt {5}+3\right )}{6 \sqrt {x^4+3 x^2+1}}-\frac {x \left (2 x^2+\sqrt {5}+3\right )}{\sqrt {x^4+3 x^2+1}}+\frac {1}{2} \sqrt {\frac {1}{3} \left (1+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \arctan \left (\frac {2 \left (2-i \sqrt {3}\right ) x^2-3 i \sqrt {3}+1}{4 \sqrt {1+i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )-\frac {1}{2} \left (i-\sqrt {3}\right ) \sqrt {\frac {1}{3} \left (1-i \sqrt {3}\right )} \arctan \left (\frac {2 \left (2+i \sqrt {3}\right ) x^2+3 i \sqrt {3}+1}{4 \sqrt {1-i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )-\frac {1}{12} \left (9+i \sqrt {3}\right ) \text {arctanh}\left (\frac {2 x^2+3}{2 \sqrt {x^4+3 x^2+1}}\right )-\frac {1}{12} \left (9-i \sqrt {3}\right ) \text {arctanh}\left (\frac {2 x^2+3}{2 \sqrt {x^4+3 x^2+1}}\right )+\frac {3}{2} \text {arctanh}\left (\frac {2 x^2+3}{2 \sqrt {x^4+3 x^2+1}}\right )-\text {arctanh}\left (\frac {3 x^2+2}{2 \sqrt {x^4+3 x^2+1}}\right )+\frac {\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\frac {\left (3-\sqrt {5}\right ) x^2+2}{\left (3+\sqrt {5}\right ) x^2+2}} \left (\left (3+\sqrt {5}\right ) x^2+2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {x^4+3 x^2+1}}-\frac {\left (i+\sqrt {3}\right ) \sqrt {\frac {1}{6} \left (3+\sqrt {5}\right )} \sqrt {\frac {\left (3-\sqrt {5}\right ) x^2+2}{\left (3+\sqrt {5}\right ) x^2+2}} \left (\left (3+\sqrt {5}\right ) x^2+2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {x^4+3 x^2+1}}+\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {1}{6} \left (3+\sqrt {5}\right )} \sqrt {\frac {\left (3-\sqrt {5}\right ) x^2+2}{\left (3+\sqrt {5}\right ) x^2+2}} \left (\left (3+\sqrt {5}\right ) x^2+2\right ) E\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )|\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {x^4+3 x^2+1}}-\frac {3 \sqrt {\frac {\left (3-\sqrt {5}\right ) x^2+2}{\left (3+\sqrt {5}\right ) x^2+2}} \left (\left (3+\sqrt {5}\right ) x^2+2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {x^4+3 x^2+1}}+\frac {\left (i+\sqrt {3}\right ) \sqrt {\frac {1}{3} \left (9-4 \sqrt {5}\right )} \sqrt {\frac {\left (3-\sqrt {5}\right ) x^2+2}{\left (3+\sqrt {5}\right ) x^2+2}} \left (\left (3+\sqrt {5}\right ) x^2+2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {x^4+3 x^2+1}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {\frac {1}{3} \left (9-4 \sqrt {5}\right )} \sqrt {\frac {\left (3-\sqrt {5}\right ) x^2+2}{\left (3+\sqrt {5}\right ) x^2+2}} \left (\left (3+\sqrt {5}\right ) x^2+2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x\right ),\frac {1}{2} \left (-5+3 \sqrt {5}\right )\right )}{2 \sqrt {x^4+3 x^2+1}}-\frac {i \left (i-3 \sqrt {3}+i \sqrt {5}+\sqrt {15}\right )^{5/2} \left (\frac {x^2}{2+i \sqrt {3}+\sqrt {5}}+\frac {2}{1+i \left (3 \sqrt {3}-i \sqrt {5}-\sqrt {15}\right )}\right ) \operatorname {EllipticPi}\left (\frac {4}{2+i \sqrt {3}+\sqrt {5}},\arctan \left (\sqrt {\frac {2 \left (2 i-\sqrt {3}+i \sqrt {5}\right )}{i-3 \sqrt {3}+i \sqrt {5}+\sqrt {15}}} x\right ),\frac {15+5 i \sqrt {3}-\sqrt {5}+i \sqrt {15}}{\left (2+i \sqrt {3}+\sqrt {5}\right )^2}\right )}{4 \sqrt {3 \left (2 i-\sqrt {3}+i \sqrt {5}\right )} \sqrt {-\frac {\left (i-3 \sqrt {3}+i \sqrt {5}+\sqrt {15}\right )^2 \left (\frac {x^2}{2+i \sqrt {3}+\sqrt {5}}+\frac {2}{1+i \left (3 \sqrt {3}-i \sqrt {5}-\sqrt {15}\right )}\right )}{2 \left (2+i \sqrt {3}+\sqrt {5}\right ) x^2-i \sqrt {15}+\sqrt {5}+3 i \sqrt {3}+1}} \sqrt {x^4+3 x^2+1}}+\frac {\left (9 i+\sqrt {3}-5 i \sqrt {5}-\sqrt {15}\right ) \left (\frac {x^2}{2-i \sqrt {3}+\sqrt {5}}+\frac {2}{1-i \left (3 \sqrt {3}+i \sqrt {5}-\sqrt {15}\right )}\right ) \operatorname {EllipticPi}\left (\frac {4}{2-i \left (\sqrt {3}+i \sqrt {5}\right )},\arctan \left (\sqrt {\frac {2 \left (2 i+\sqrt {3}+i \sqrt {5}\right )}{i+3 \sqrt {3}+i \sqrt {5}-\sqrt {15}}} x\right ),-\frac {15-5 i \sqrt {3}-\sqrt {5}-i \sqrt {15}}{\left (2 i+\sqrt {3}+i \sqrt {5}\right )^2}\right )}{2 \sqrt {\frac {3 \left (2 i+\sqrt {3}+i \sqrt {5}\right )}{i+3 \sqrt {3}+i \sqrt {5}-\sqrt {15}}} \sqrt {-\frac {\left (9-i \sqrt {3}-5 \sqrt {5}+i \sqrt {15}\right ) \left (\frac {x^2}{2-i \sqrt {3}+\sqrt {5}}+\frac {2}{1-i \left (3 \sqrt {3}+i \sqrt {5}-\sqrt {15}\right )}\right )}{2 \left (2-i \sqrt {3}+\sqrt {5}\right ) x^2+i \sqrt {15}+\sqrt {5}-3 i \sqrt {3}+1}} \sqrt {x^4+3 x^2+1}}-\frac {4}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {x^4+3 x^2+1}}{\left (-2 x+i \sqrt {3}-1\right )^2}dx+\frac {4}{3} \int \frac {\sqrt {x^4+3 x^2+1}}{\left (-2 x+i \sqrt {3}-1\right )^2}dx-\frac {4}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {x^4+3 x^2+1}}{\left (2 x+i \sqrt {3}+1\right )^2}dx+\frac {4}{3} \int \frac {\sqrt {x^4+3 x^2+1}}{\left (2 x+i \sqrt {3}+1\right )^2}dx+\frac {\sqrt {x^4+3 x^2+1}}{x}-\frac {1}{6} \left (3+i \sqrt {3}\right ) \sqrt {x^4+3 x^2+1}-\frac {1}{6} \left (3-i \sqrt {3}\right ) \sqrt {x^4+3 x^2+1}+\sqrt {x^4+3 x^2+1}\) |
Input:
Int[((-1 + x^2)*(1 + x^2)*Sqrt[1 + 3*x^2 + x^4])/(x^2*(1 + x + x^2)^2),x]
Output:
$Aborted
Time = 3.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79
| method | result | size |
| risch | \(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}-2 \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right )+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )}{2}\) | \(77\) |
| default | \(\frac {3 \sqrt {2}\, x \left (x^{2}+x +1\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+2 \left (1+x \right )^{2} \sqrt {x^{4}+3 x^{2}+1}-4 \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right ) x \left (x^{2}+x +1\right )}{2 x \left (x^{2}+x +1\right )}\) | \(91\) |
| pseudoelliptic | \(\frac {3 \sqrt {2}\, x \left (x^{2}+x +1\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+2 \left (1+x \right )^{2} \sqrt {x^{4}+3 x^{2}+1}-4 \,\operatorname {arcsinh}\left (\frac {x^{2}+1}{x}\right ) x \left (x^{2}+x +1\right )}{2 x \left (x^{2}+x +1\right )}\) | \(91\) |
| trager | \(\frac {\left (x^{2}+2 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{x \left (x^{2}+x +1\right )}-2 \ln \left (\frac {1+x^{2}+\sqrt {x^{4}+3 x^{2}+1}}{x}\right )-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +2 \sqrt {x^{4}+3 x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}+x +1}\right )}{2}\) | \(118\) |
| elliptic | \(-\ln \left (\frac {3}{2}+x^{2}+\sqrt {x^{4}+3 x^{2}+1}\right )-\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-\frac {2 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )-6 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )-12 \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\right )}{6 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (1+\frac {x^{2}-1}{-x^{2}-1}\right )^{2}}}\, \left (1+\frac {x^{2}-1}{-x^{2}-1}\right ) \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+3\right )}+\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )-\sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )\right )}{3 \left (1+\frac {x^{2}-1}{-x^{2}-1}\right ) \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (1+\frac {x^{2}-1}{-x^{2}-1}\right )^{2}}}}-\operatorname {arctanh}\left (\frac {3 x^{2}+2}{2 \sqrt {x^{4}+3 x^{2}+1}}\right )+\frac {\left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}-\frac {1}{2 \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}-\frac {3 \ln \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{2}-\frac {1}{2 \left (-1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}+\frac {3 \ln \left (-1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(725\) |
Input:
int((x^2-1)*(x^2+1)*(x^4+3*x^2+1)^(1/2)/x^2/(x^2+x+1)^2,x,method=_RETURNVE RBOSE)
Output:
(x^2+2*x+1)*(x^4+3*x^2+1)^(1/2)/x/(x^2+x+1)-2*arcsinh((x^2+1)/x)+3/2*2^(1/ 2)*arctanh(1/2*(x^2-x+1)*2^(1/2)/(x^4+3*x^2+1)^(1/2))
Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.53 \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {3 \, \sqrt {2} {\left (x^{3} + x^{2} + x\right )} \log \left (\frac {3 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} - x + 1\right )} + 9 \, x^{2} - 2 \, x + 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + 8 \, {\left (x^{3} + x^{2} + x\right )} \log \left (-\frac {x^{2} - \sqrt {x^{4} + 3 \, x^{2} + 1} + 1}{x}\right ) + 4 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 2 \, x + 1\right )}}{4 \, {\left (x^{3} + x^{2} + x\right )}} \] Input:
integrate((x^2-1)*(x^2+1)*(x^4+3*x^2+1)^(1/2)/x^2/(x^2+x+1)^2,x, algorithm ="fricas")
Output:
1/4*(3*sqrt(2)*(x^3 + x^2 + x)*log((3*x^4 - 2*x^3 + 2*sqrt(2)*sqrt(x^4 + 3 *x^2 + 1)*(x^2 - x + 1) + 9*x^2 - 2*x + 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) ) + 8*(x^3 + x^2 + x)*log(-(x^2 - sqrt(x^4 + 3*x^2 + 1) + 1)/x) + 4*sqrt(x ^4 + 3*x^2 + 1)*(x^2 + 2*x + 1))/(x^3 + x^2 + x)
\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + 3 x^{2} + 1}}{x^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \] Input:
integrate((x**2-1)*(x**2+1)*(x**4+3*x**2+1)**(1/2)/x**2/(x**2+x+1)**2,x)
Output:
Integral((x - 1)*(x + 1)*(x**2 + 1)*sqrt(x**4 + 3*x**2 + 1)/(x**2*(x**2 + x + 1)**2), x)
\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 1\right )} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} x^{2}} \,d x } \] Input:
integrate((x^2-1)*(x^2+1)*(x^4+3*x^2+1)^(1/2)/x^2/(x^2+x+1)^2,x, algorithm ="maxima")
Output:
integrate(sqrt(x^4 + 3*x^2 + 1)*(x^2 + 1)*(x^2 - 1)/((x^2 + x + 1)^2*x^2), x)
\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 1\right )} {\left (x^{2} - 1\right )}}{{\left (x^{2} + x + 1\right )}^{2} x^{2}} \,d x } \] Input:
integrate((x^2-1)*(x^2+1)*(x^4+3*x^2+1)^(1/2)/x^2/(x^2+x+1)^2,x, algorithm ="giac")
Output:
integrate(sqrt(x^4 + 3*x^2 + 1)*(x^2 + 1)*(x^2 - 1)/((x^2 + x + 1)^2*x^2), x)
Timed out. \[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\int \frac {\left (x^2-1\right )\,\left (x^2+1\right )\,\sqrt {x^4+3\,x^2+1}}{x^2\,{\left (x^2+x+1\right )}^2} \,d x \] Input:
int(((x^2 - 1)*(x^2 + 1)*(3*x^2 + x^4 + 1)^(1/2))/(x^2*(x + x^2 + 1)^2),x)
Output:
int(((x^2 - 1)*(x^2 + 1)*(3*x^2 + x^4 + 1)^(1/2))/(x^2*(x + x^2 + 1)^2), x )
\[ \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \sqrt {1+3 x^2+x^4}}{x^2 \left (1+x+x^2\right )^2} \, dx=\frac {\sqrt {x^{4}+3 x^{2}+1}\, x^{2}+5 \sqrt {x^{4}+3 x^{2}+1}\, x +\sqrt {x^{4}+3 x^{2}+1}+6 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+1}\, x^{3}}{x^{8}+2 x^{7}+6 x^{6}+8 x^{5}+11 x^{4}+8 x^{3}+6 x^{2}+2 x +1}d x \right ) x^{3}+6 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+1}\, x^{3}}{x^{8}+2 x^{7}+6 x^{6}+8 x^{5}+11 x^{4}+8 x^{3}+6 x^{2}+2 x +1}d x \right ) x^{2}+6 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+1}\, x^{3}}{x^{8}+2 x^{7}+6 x^{6}+8 x^{5}+11 x^{4}+8 x^{3}+6 x^{2}+2 x +1}d x \right ) x -6 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+1}\, x}{x^{8}+2 x^{7}+6 x^{6}+8 x^{5}+11 x^{4}+8 x^{3}+6 x^{2}+2 x +1}d x \right ) x^{3}-6 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+1}\, x}{x^{8}+2 x^{7}+6 x^{6}+8 x^{5}+11 x^{4}+8 x^{3}+6 x^{2}+2 x +1}d x \right ) x^{2}-6 \left (\int \frac {\sqrt {x^{4}+3 x^{2}+1}\, x}{x^{8}+2 x^{7}+6 x^{6}+8 x^{5}+11 x^{4}+8 x^{3}+6 x^{2}+2 x +1}d x \right ) x +2 \,\mathrm {log}\left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}-1\right ) x^{3}+2 \,\mathrm {log}\left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}-1\right ) x^{2}+2 \,\mathrm {log}\left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}-1\right ) x -2 \,\mathrm {log}\left (x \right ) x^{3}-2 \,\mathrm {log}\left (x \right ) x^{2}-2 \,\mathrm {log}\left (x \right ) x}{x \left (x^{2}+x +1\right )} \] Input:
int((x^2-1)*(x^2+1)*(x^4+3*x^2+1)^(1/2)/x^2/(x^2+x+1)^2,x)
Output:
(sqrt(x**4 + 3*x**2 + 1)*x**2 + 5*sqrt(x**4 + 3*x**2 + 1)*x + sqrt(x**4 + 3*x**2 + 1) + 6*int((sqrt(x**4 + 3*x**2 + 1)*x**3)/(x**8 + 2*x**7 + 6*x**6 + 8*x**5 + 11*x**4 + 8*x**3 + 6*x**2 + 2*x + 1),x)*x**3 + 6*int((sqrt(x** 4 + 3*x**2 + 1)*x**3)/(x**8 + 2*x**7 + 6*x**6 + 8*x**5 + 11*x**4 + 8*x**3 + 6*x**2 + 2*x + 1),x)*x**2 + 6*int((sqrt(x**4 + 3*x**2 + 1)*x**3)/(x**8 + 2*x**7 + 6*x**6 + 8*x**5 + 11*x**4 + 8*x**3 + 6*x**2 + 2*x + 1),x)*x - 6* int((sqrt(x**4 + 3*x**2 + 1)*x)/(x**8 + 2*x**7 + 6*x**6 + 8*x**5 + 11*x**4 + 8*x**3 + 6*x**2 + 2*x + 1),x)*x**3 - 6*int((sqrt(x**4 + 3*x**2 + 1)*x)/ (x**8 + 2*x**7 + 6*x**6 + 8*x**5 + 11*x**4 + 8*x**3 + 6*x**2 + 2*x + 1),x) *x**2 - 6*int((sqrt(x**4 + 3*x**2 + 1)*x)/(x**8 + 2*x**7 + 6*x**6 + 8*x**5 + 11*x**4 + 8*x**3 + 6*x**2 + 2*x + 1),x)*x + 2*log(sqrt(x**4 + 3*x**2 + 1) - x**2 - 1)*x**3 + 2*log(sqrt(x**4 + 3*x**2 + 1) - x**2 - 1)*x**2 + 2*l og(sqrt(x**4 + 3*x**2 + 1) - x**2 - 1)*x - 2*log(x)*x**3 - 2*log(x)*x**2 - 2*log(x)*x)/(x*(x**2 + x + 1))