Integrand size = 18, antiderivative size = 27 \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=-2 \sqrt {2+x-x^2}-2 \arcsin \left (\frac {1}{3} (1-2 x)\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {654, 633, 222} \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=-2 \arcsin \left (\frac {1}{3} (1-2 x)\right )-2 \sqrt {-x^2+x+2} \]
[In]
[Out]
Rule 222
Rule 633
Rule 654
Rubi steps \begin{align*} \text {integral}& = -2 \sqrt {2+x-x^2}+2 \int \frac {1}{\sqrt {2+x-x^2}} \, dx \\ & = -2 \sqrt {2+x-x^2}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,1-2 x\right ) \\ & = -2 \sqrt {2+x-x^2}-2 \arcsin \left (\frac {1}{3} (1-2 x)\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=-2 \sqrt {2+x-x^2}-4 \arctan \left (\frac {\sqrt {2+x-x^2}}{1+x}\right ) \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
default | \(2 \arcsin \left (-\frac {1}{3}+\frac {2 x}{3}\right )-2 \sqrt {-x^{2}+x +2}\) | \(22\) |
risch | \(\frac {2 x^{2}-2 x -4}{\sqrt {-x^{2}+x +2}}+2 \arcsin \left (-\frac {1}{3}+\frac {2 x}{3}\right )\) | \(30\) |
trager | \(-2 \sqrt {-x^{2}+x +2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {-x^{2}+x +2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\) | \(51\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=-2 \, \sqrt {-x^{2} + x + 2} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + x + 2} {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - x - 2\right )}}\right ) \]
[In]
[Out]
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=- 2 \sqrt {- x^{2} + x + 2} + 2 \operatorname {asin}{\left (\frac {2 x}{3} - \frac {1}{3} \right )} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=-2 \, \sqrt {-x^{2} + x + 2} - 2 \, \arcsin \left (-\frac {2}{3} \, x + \frac {1}{3}\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=-2 \, \sqrt {-x^{2} + x + 2} + 2 \, \arcsin \left (\frac {2}{3} \, x - \frac {1}{3}\right ) \]
[In]
[Out]
Time = 0.42 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx=\mathrm {asin}\left (\frac {2\,x}{3}-\frac {1}{3}\right )-2\,\sqrt {-x^2+x+2}-\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2+x+2}-\frac {1}{2}{}\mathrm {i}\right )\,1{}\mathrm {i} \]
[In]
[Out]