Integrand size = 43, antiderivative size = 93 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \left (1-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}-\frac {1}{4} \text {arctanh}\left (3-2 x+2 x^2+2 \sqrt {-2+x^2-2 x^3+x^4}\right )-\frac {7}{8} \log \left (-x+x^2+\sqrt {-2+x^2-2 x^3+x^4}\right ) \]
1/4*(x^2-x+1)*(x^4-2*x^3+x^2-2)^(1/2)-1/4*arctanh(3-2*x+2*x^2+2*(x^4-2*x^3 +x^2-2)^(1/2))-7/8*ln(-x+x^2+(x^4-2*x^3+x^2-2)^(1/2))
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \left (1-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}+\frac {1}{4} \text {arctanh}\left (3-2 x+2 x^2-2 \sqrt {-2+x^2-2 x^3+x^4}\right )+\frac {7}{8} \log \left (x-x^2+\sqrt {-2+x^2-2 x^3+x^4}\right ) \]
((1 - x + x^2)*Sqrt[-2 + x^2 - 2*x^3 + x^4])/4 + ArcTanh[3 - 2*x + 2*x^2 - 2*Sqrt[-2 + x^2 - 2*x^3 + x^4]]/4 + (7*Log[x - x^2 + Sqrt[-2 + x^2 - 2*x^ 3 + x^4]])/8
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.
\(\displaystyle \int \frac {(2 x-1) \left (x^2-x+2\right ) \sqrt {x^4-2 x^3+x^2-2}}{2 x^2-2 x+3} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (-\frac {\sqrt {x^4-2 x^3+x^2-2} (1-2 x)}{2 \left (2 x^2-2 x+3\right )}+x \sqrt {x^4-2 x^3+x^2-2}-\frac {1}{2} \sqrt {x^4-2 x^3+x^2-2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {\sqrt {x^4-2 x^3+x^2-2}}{4 x-2 i \sqrt {5}-2}dx+\int \frac {\sqrt {x^4-2 x^3+x^2-2}}{4 x+2 i \sqrt {5}-2}dx+\frac {1}{2} \text {arctanh}\left (\frac {4 (1-x) x}{\sqrt {(2 x-1)^4-2 (1-2 x)^2-31}}\right )-\frac {1}{16} (1-x) \sqrt {(2 x-1)^4-2 (1-2 x)^2-31} x\) |
3.13.86.3.1 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 = 10.67 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {\left (x^{2}-x +1\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}}{4}-\frac {7 \ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}-2}\right )}{8}+\frac {\operatorname {arctanh}\left (\frac {3 x^{2}-3 x +4}{\sqrt {x^{4}-2 x^{3}+x^{2}-2}}\right )}{8}\) | \(81\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {3 x^{2}-3 x +4}{\sqrt {x^{4}-2 x^{3}+x^{2}-2}}\right )}{8}-\frac {7 \ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}-2}\right )}{8}+\frac {\left (2 x^{2}-2 x +2\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}}{8}\) | \(83\) |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {3 x^{2}-3 x +4}{\sqrt {x^{4}-2 x^{3}+x^{2}-2}}\right )}{8}-\frac {7 \ln \left (-x +x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}-2}\right )}{8}+\frac {\left (2 x^{2}-2 x +2\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}}{8}\) | \(83\) |
trager | \(\left (\frac {1}{4} x^{2}-\frac {1}{4} x +\frac {1}{4}\right ) \sqrt {x^{4}-2 x^{3}+x^{2}-2}+\frac {\ln \left (-\frac {1-2 x \sqrt {x^{4}-2 x^{3}+x^{2}-2}+8 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{14}-56 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{13}+184 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{12}-376 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{11}+504 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{10}-408 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{9}+96 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{8}+216 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{7}+26 x^{2} \sqrt {x^{4}-2 x^{3}+x^{2}-2}-14 x -186 x^{4}-624 x^{7}+176 x^{5}+12 x^{2}+60 x^{3}+784 x^{8}+152 x^{6}-368 x^{10}+864 x^{11}-872 x^{12}+560 x^{13}-240 x^{14}+64 x^{15}-8 x^{16}-360 x^{9}-2 \sqrt {x^{4}-2 x^{3}+x^{2}-2}-300 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{6}+164 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{5}-4 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{4}-52 \sqrt {x^{4}-2 x^{3}+x^{2}-2}\, x^{3}}{2 x^{2}-2 x +3}\right )}{8}\) | \(420\) |
elliptic | \(\text {Expression too large to display}\) | \(3475\) |
1/4*(x^2-x+1)*(x^4-2*x^3+x^2-2)^(1/2)-7/8*ln(-x+x^2+(x^4-2*x^3+x^2-2)^(1/2 ))+1/8*arctanh((3*x^2-3*x+4)/(x^4-2*x^3+x^2-2)^(1/2))
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \, \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} {\left (x^{2} - x + 1\right )} + \frac {7}{8} \, \log \left (-x^{2} + x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2}\right ) + \frac {1}{8} \, \log \left (\frac {3 \, x^{2} - 3 \, x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} + 4}{2 \, x^{2} - 2 \, x + 3}\right ) \]
1/4*sqrt(x^4 - 2*x^3 + x^2 - 2)*(x^2 - x + 1) + 7/8*log(-x^2 + x + sqrt(x^ 4 - 2*x^3 + x^2 - 2)) + 1/8*log((3*x^2 - 3*x + sqrt(x^4 - 2*x^3 + x^2 - 2) + 4)/(2*x^2 - 2*x + 3))
\[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\int \frac {\left (2 x - 1\right ) \left (x^{2} - x + 2\right ) \sqrt {x^{4} - 2 x^{3} + x^{2} - 2}}{2 x^{2} - 2 x + 3}\, dx \]
\[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\int { \frac {\sqrt {x^{4} - 2 \, x^{3} + x^{2} - 2} {\left (x^{2} - x + 2\right )} {\left (2 \, x - 1\right )}}{2 \, x^{2} - 2 \, x + 3} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\frac {1}{4} \, \sqrt {{\left (x^{2} - x\right )}^{2} - 2} {\left (x^{2} - x + 1\right )} + \frac {1}{8} \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} - 2} + 2\right ) - \frac {1}{8} \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} - 2} + 1\right ) + \frac {7}{8} \, \log \left ({\left | -x^{2} + x + \sqrt {{\left (x^{2} - x\right )}^{2} - 2} \right |}\right ) \]
1/4*sqrt((x^2 - x)^2 - 2)*(x^2 - x + 1) + 1/8*log(x^2 - x - sqrt((x^2 - x) ^2 - 2) + 2) - 1/8*log(x^2 - x - sqrt((x^2 - x)^2 - 2) + 1) + 7/8*log(abs( -x^2 + x + sqrt((x^2 - x)^2 - 2)))
Timed out. \[ \int \frac {(-1+2 x) \left (2-x+x^2\right ) \sqrt {-2+x^2-2 x^3+x^4}}{3-2 x+2 x^2} \, dx=\int \frac {\left (2\,x-1\right )\,\left (x^2-x+2\right )\,\sqrt {x^4-2\,x^3+x^2-2}}{2\,x^2-2\,x+3} \,d x \]