Integrand size = 53, antiderivative size = 84 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx=\frac {\sqrt {1+x^2+x^4} \left (9 x+2 x^3+9 x^5\right )}{8 \left (1+x^4\right )^2}+\frac {31}{8} \text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right ) \]
1/8*(x^4+x^2+1)^(1/2)*(9*x^5+2*x^3+9*x)/(x^4+1)^2+31/8*arctanh(x/(x^4+x^2+ 1)^(1/2))-3*2^(1/2)*arctanh(2^(1/2)*x/(x^4+x^2+1)^(1/2))
Time = 0.99 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx=\frac {x \sqrt {1+x^2+x^4} \left (9+2 x^2+9 x^4\right )}{8 \left (1+x^4\right )^2}+\frac {31}{8} \text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right ) \]
Integrate[((-1 + x^4)*Sqrt[1 + x^2 + x^4]*(1 + x^2 + 3*x^4 + x^6 + x^8))/( (1 + x^4)^3*(1 - x^2 + x^4)),x]
(x*Sqrt[1 + x^2 + x^4]*(9 + 2*x^2 + 9*x^4))/(8*(1 + x^4)^2) + (31*ArcTanh[ x/Sqrt[1 + x^2 + x^4]])/8 - 3*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + 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.
| \(\displaystyle \int \frac {\left (x^4-1\right ) \sqrt {x^4+x^2+1} \left (x^8+x^6+3 x^4+x^2+1\right )}{\left (x^4+1\right )^3 \left (x^4-x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 \sqrt {x^4+x^2+1} x^2}{\left (x^4+1\right )^3}-\frac {2 \left (3 x^2+1\right ) \sqrt {x^4+x^2+1}}{x^4+1}+\frac {3 \left (2 x^2-1\right ) \sqrt {x^4+x^2+1}}{x^4-x^2+1}+\frac {\left (4-x^2\right ) \sqrt {x^4+x^2+1}}{\left (x^4+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {x^2 \sqrt {x^4+x^2+1}}{\left (x^4+1\right )^3}dx-\int \frac {x^2 \sqrt {x^4+x^2+1}}{\left (x^4+1\right )^2}dx+\frac {3 \left (\sqrt {3}+2 i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\left (\sqrt {3}+3 i\right ) \sqrt {x^4+x^2+1}}+\frac {3 \left (-\sqrt {3}+2 i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{\left (-\sqrt {3}+3 i\right ) \sqrt {x^4+x^2+1}}-\frac {9 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{4}\right )}{2 \sqrt {x^4+x^2+1}}-\frac {3 \left (-\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan (x),\frac {1}{4}\right )}{2 \left (\sqrt {3}+3 i\right ) \sqrt {x^4+x^2+1}}-\frac {3 \left (\sqrt {3}+i\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan (x),\frac {1}{4}\right )}{2 \left (-\sqrt {3}+3 i\right ) \sqrt {x^4+x^2+1}}+4 \text {arctanh}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right )-\frac {3 \sqrt {\frac {3}{2}} \left (1+i \sqrt {3}\right ) \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+x^2+1}}\right )}{\sqrt {3}+3 i}-\frac {3 \sqrt {\frac {3}{2}} \left (\sqrt {3}+i\right ) \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+x^2+1}}\right )}{3+i \sqrt {3}}+\frac {i \sqrt {x^4+x^2+1} x}{2 \left (-x^2+i\right )}+\frac {i \sqrt {x^4+x^2+1} x}{2 \left (x^2+i\right )}\) |
Int[((-1 + x^4)*Sqrt[1 + x^2 + x^4]*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x ^4)^3*(1 - x^2 + x^4)),x]
3.12.28.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 5.87 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.52
| method | result | size |
| elliptic | \(\frac {\left (3 \ln \left (\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{2 x}-1\right )-\frac {8 \left (-\frac {9 \left (x^{4}+x^{2}+1\right )^{\frac {3}{2}} \sqrt {2}}{64 x^{3}}+\frac {7 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{64 x}\right )}{\left (\frac {x^{4}+x^{2}+1}{x^{2}}-1\right )^{2}}+\frac {31 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )}{8}-3 \ln \left (1+\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{2 x}\right )\right ) \sqrt {2}}{2}\) | \(128\) |
| trager | \(\frac {\left (9 x^{4}+2 x^{2}+9\right ) x \sqrt {x^{4}+x^{2}+1}}{8 \left (x^{4}+1\right )^{2}}+\frac {31 \ln \left (-\frac {x^{4}+2 x \sqrt {x^{4}+x^{2}+1}+2 x^{2}+1}{x^{4}+1}\right )}{16}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+4 x \sqrt {x^{4}+x^{2}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{4}-x^{2}+1}\right )}{2}\) | \(133\) |
| risch | \(\frac {\left (9 x^{4}+2 x^{2}+9\right ) x \sqrt {x^{4}+x^{2}+1}}{8 \left (x^{4}+1\right )^{2}}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x +\sqrt {3}\, x^{2}+\sqrt {3}\right ) \sqrt {2}}{2 \sqrt {x^{4}+x^{2}+1}}\right )}{2}-\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\sqrt {3}\, x^{2}+\sqrt {3}-x \right ) \sqrt {2}}{2 \sqrt {x^{4}+x^{2}+1}}\right )}{2}+\frac {31 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, x^{2}+\sqrt {2}-x}{\sqrt {x^{4}+x^{2}+1}}\right )}{16}-\frac {31 \,\operatorname {arctanh}\left (\frac {x +\sqrt {2}\, x^{2}+\sqrt {2}}{\sqrt {x^{4}+x^{2}+1}}\right )}{16}\) | \(156\) |
| default | \(\frac {-24 \sqrt {2}\, \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {\left (\sqrt {3}\, x^{2}+\sqrt {3}-x \right ) \sqrt {2}}{2 \sqrt {x^{4}+x^{2}+1}}\right )+24 \sqrt {2}\, \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {\left (x +\sqrt {3}\, x^{2}+\sqrt {3}\right ) \sqrt {2}}{2 \sqrt {x^{4}+x^{2}+1}}\right )+31 \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {2}\, x^{2}+\sqrt {2}-x}{\sqrt {x^{4}+x^{2}+1}}\right )-31 \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {x +\sqrt {2}\, x^{2}+\sqrt {2}}{\sqrt {x^{4}+x^{2}+1}}\right )+\left (18 x^{5}+4 x^{3}+18 x \right ) \sqrt {x^{4}+x^{2}+1}}{16 \left (x^{2}-x \sqrt {2}+1\right )^{2} \left (x^{2}+x \sqrt {2}+1\right )^{2}}\) | \(204\) |
| pseudoelliptic | \(\frac {-24 \sqrt {2}\, \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {\left (\sqrt {3}\, x^{2}+\sqrt {3}-x \right ) \sqrt {2}}{2 \sqrt {x^{4}+x^{2}+1}}\right )+24 \sqrt {2}\, \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {\left (x +\sqrt {3}\, x^{2}+\sqrt {3}\right ) \sqrt {2}}{2 \sqrt {x^{4}+x^{2}+1}}\right )+31 \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {2}\, x^{2}+\sqrt {2}-x}{\sqrt {x^{4}+x^{2}+1}}\right )-31 \left (x^{4}+1\right )^{2} \operatorname {arctanh}\left (\frac {x +\sqrt {2}\, x^{2}+\sqrt {2}}{\sqrt {x^{4}+x^{2}+1}}\right )+\left (18 x^{5}+4 x^{3}+18 x \right ) \sqrt {x^{4}+x^{2}+1}}{16 \left (x^{2}-x \sqrt {2}+1\right )^{2} \left (x^{2}+x \sqrt {2}+1\right )^{2}}\) | \(204\) |
int((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x^2+1), x,method=_RETURNVERBOSE)
1/2*(3*ln(1/2*(x^4+x^2+1)^(1/2)*2^(1/2)/x-1)-8*(-9/64*(x^4+x^2+1)^(3/2)*2^ (1/2)/x^3+7/64*(x^4+x^2+1)^(1/2)*2^(1/2)/x)/((x^4+x^2+1)/x^2-1)^2+31/8*2^( 1/2)*arctanh(1/x*(x^4+x^2+1)^(1/2))-3*ln(1+1/2*(x^4+x^2+1)^(1/2)*2^(1/2)/x ))*2^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (70) = 140\).
Time = 0.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx=\frac {12 \, \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )} \log \left (-\frac {x^{8} + 14 \, x^{6} + 19 \, x^{4} - 4 \, \sqrt {2} {\left (x^{5} + 3 \, x^{3} + x\right )} \sqrt {x^{4} + x^{2} + 1} + 14 \, x^{2} + 1}{x^{8} - 2 \, x^{6} + 3 \, x^{4} - 2 \, x^{2} + 1}\right ) + 31 \, {\left (x^{8} + 2 \, x^{4} + 1\right )} \log \left (-\frac {x^{4} + 2 \, x^{2} + 2 \, \sqrt {x^{4} + x^{2} + 1} x + 1}{x^{4} + 1}\right ) + 2 \, {\left (9 \, x^{5} + 2 \, x^{3} + 9 \, x\right )} \sqrt {x^{4} + x^{2} + 1}}{16 \, {\left (x^{8} + 2 \, x^{4} + 1\right )}} \]
integrate((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x ^2+1),x, algorithm="fricas")
1/16*(12*sqrt(2)*(x^8 + 2*x^4 + 1)*log(-(x^8 + 14*x^6 + 19*x^4 - 4*sqrt(2) *(x^5 + 3*x^3 + x)*sqrt(x^4 + x^2 + 1) + 14*x^2 + 1)/(x^8 - 2*x^6 + 3*x^4 - 2*x^2 + 1)) + 31*(x^8 + 2*x^4 + 1)*log(-(x^4 + 2*x^2 + 2*sqrt(x^4 + x^2 + 1)*x + 1)/(x^4 + 1)) + 2*(9*x^5 + 2*x^3 + 9*x)*sqrt(x^4 + x^2 + 1))/(x^8 + 2*x^4 + 1)
\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx=\int \frac {\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{8} + x^{6} + 3 x^{4} + x^{2} + 1\right )}{\left (x^{4} + 1\right )^{3} \left (x^{4} - x^{2} + 1\right )}\, dx \]
integrate((x**4-1)*(x**4+x**2+1)**(1/2)*(x**8+x**6+3*x**4+x**2+1)/(x**4+1) **3/(x**4-x**2+1),x)
Integral(sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x - 1)*(x + 1)*(x**2 + 1)*(x **8 + x**6 + 3*x**4 + x**2 + 1)/((x**4 + 1)**3*(x**4 - x**2 + 1)), x)
\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1\right )} \sqrt {x^{4} + x^{2} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} - x^{2} + 1\right )} {\left (x^{4} + 1\right )}^{3}} \,d x } \]
integrate((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x ^2+1),x, algorithm="maxima")
integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*sqrt(x^4 + x^2 + 1)*(x^4 - 1)/((x^ 4 - x^2 + 1)*(x^4 + 1)^3), x)
\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1\right )} \sqrt {x^{4} + x^{2} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} - x^{2} + 1\right )} {\left (x^{4} + 1\right )}^{3}} \,d x } \]
integrate((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x ^2+1),x, algorithm="giac")
integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*sqrt(x^4 + x^2 + 1)*(x^4 - 1)/((x^ 4 - x^2 + 1)*(x^4 + 1)^3), x)
Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {x^4+x^2+1}\,\left (x^8+x^6+3\,x^4+x^2+1\right )}{{\left (x^4+1\right )}^3\,\left (x^4-x^2+1\right )} \,d x \]
int(((x^4 - 1)*(x^2 + x^4 + 1)^(1/2)*(x^2 + 3*x^4 + x^6 + x^8 + 1))/((x^4 + 1)^3*(x^4 - x^2 + 1)),x)