Integrand size = 23, antiderivative size = 31 \[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=\frac {1}{2} \log \left (-2 x+x^2+\sqrt {-5+4 x^2-4 x^3+x^4}\right ) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1694, 1121, 635, 212} \[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=-\frac {1}{2} \text {arctanh}\left (\frac {1-(x-1)^2}{\sqrt {(x-1)^4-2 (x-1)^2-4}}\right ) \]
[In]
[Out]
Rule 212
Rule 635
Rule 1121
Rule 1694
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\sqrt {-4-2 x^2+x^4}} \, dx,x,-1+x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-4-2 x+x^2}} \, dx,x,(-1+x)^2\right ) \\ & = \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2 (-2+x) x}{\sqrt {-4-2 (-1+x)^2+(-1+x)^4}}\right ) \\ & = \frac {1}{2} \text {arctanh}\left (\frac {(-2+x) x}{\sqrt {-4-2 (-1+x)^2+(-1+x)^4}}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=\frac {1}{2} \log \left (-2 x+x^2+\sqrt {-5+4 x^2-4 x^3+x^4}\right ) \]
[In]
[Out]
Time = 3.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\ln \left (-2 x +x^{2}+\sqrt {x^{4}-4 x^{3}+4 x^{2}-5}\right )}{2}\) | \(28\) |
pseudoelliptic | \(\frac {\ln \left (-2 x +x^{2}+\sqrt {x^{4}-4 x^{3}+4 x^{2}-5}\right )}{2}\) | \(28\) |
trager | \(-\frac {\ln \left (-x^{2}+\sqrt {x^{4}-4 x^{3}+4 x^{2}-5}+2 x \right )}{2}\) | \(30\) |
elliptic | \(\text {Expression too large to display}\) | \(1110\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=\frac {1}{2} \, \log \left (x^{2} - 2 \, x + \sqrt {x^{4} - 4 \, x^{3} + 4 \, x^{2} - 5}\right ) \]
[In]
[Out]
\[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=\int \frac {x - 1}{\sqrt {x^{4} - 4 x^{3} + 4 x^{2} - 5}}\, dx \]
[In]
[Out]
\[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=\int { \frac {x - 1}{\sqrt {x^{4} - 4 \, x^{3} + 4 \, x^{2} - 5}} \,d x } \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=\frac {1}{4} \, \sqrt {{\left (x^{2} - 2 \, x\right )}^{2} - 5} {\left (x^{2} - 2 \, x\right )} + \frac {5}{4} \, \log \left ({\left | -x^{2} + 2 \, x + \sqrt {{\left (x^{2} - 2 \, x\right )}^{2} - 5} \right |}\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx=\int \frac {x-1}{\sqrt {x^4-4\,x^3+4\,x^2-5}} \,d x \]
[In]
[Out]