Integrand size = 13, antiderivative size = 56 \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=-\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {\arctan \left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 44, 65, 209} \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=\frac {\arctan \left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {\sqrt {x-1}}{8 (x+1)}-\frac {\sqrt {x-1}}{2 (x+1)^2} \]
[In]
[Out]
Rule 43
Rule 44
Rule 65
Rule 209
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {1}{4} \int \frac {1}{\sqrt {-1+x} (1+x)^2} \, dx \\ & = -\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {1}{16} \int \frac {1}{\sqrt {-1+x} (1+x)} \, dx \\ & = -\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-1+x}\right ) \\ & = -\frac {\sqrt {-1+x}}{2 (1+x)^2}+\frac {\sqrt {-1+x}}{8 (1+x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{8 \sqrt {2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=\frac {(-3+x) \sqrt {-1+x}}{8 (1+x)^2}+\frac {\arctan \left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {x^{2}-4 x +3}{8 \left (1+x \right )^{2} \sqrt {-1+x}}+\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{16}\) | \(38\) |
derivativedivides | \(\frac {\frac {\left (-1+x \right )^{\frac {3}{2}}}{8}-\frac {\sqrt {-1+x}}{4}}{\left (1+x \right )^{2}}+\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{16}\) | \(40\) |
default | \(\frac {\frac {\left (-1+x \right )^{\frac {3}{2}}}{8}-\frac {\sqrt {-1+x}}{4}}{\left (1+x \right )^{2}}+\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{16}\) | \(40\) |
trager | \(\frac {\left (-3+x \right ) \sqrt {-1+x}}{8 \left (1+x \right )^{2}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {-1+x}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{1+x}\right )}{32}\) | \(57\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=\frac {\sqrt {2} {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + 2 \, \sqrt {x - 1} {\left (x - 3\right )}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 2.32 (sec) , antiderivative size = 168, normalized size of antiderivative = 3.00 \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=\begin {cases} \frac {\sqrt {2} i \operatorname {acosh}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{16} - \frac {i}{8 \sqrt {-1 + \frac {2}{x + 1}} \sqrt {x + 1}} + \frac {3 i}{4 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}} - \frac {i}{\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{\frac {5}{2}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{16} + \frac {1}{8 \sqrt {1 - \frac {2}{x + 1}} \sqrt {x + 1}} - \frac {3}{4 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}} + \frac {1}{\sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + \frac {{\left (x - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x - 1}}{8 \, {\left ({\left (x - 1\right )}^{2} + 4 \, x\right )}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) + \frac {{\left (x - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x - 1}}{8 \, {\left (x + 1\right )}^{2}} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {-1+x}}{(1+x)^3} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-1}}{2}\right )}{16}-\frac {\frac {\sqrt {x-1}}{4}-\frac {{\left (x-1\right )}^{3/2}}{8}}{4\,x+{\left (x-1\right )}^2} \]
[In]
[Out]