Integrand size = 17, antiderivative size = 23 \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=\frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\arcsin (x) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {49, 41, 222} \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=\frac {2 \sqrt {x+1}}{\sqrt {1-x}}-\arcsin (x) \]
[In]
[Out]
Rule 41
Rule 49
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = \frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {2 \sqrt {1+x}}{\sqrt {1-x}}-\sin ^{-1}(x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=-\frac {2 \sqrt {1-x^2}}{-1+x}+2 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(19)=38\).
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78
method | result | size |
risch | \(\frac {2 \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(64\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=\frac {2 \, {\left ({\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + x - \sqrt {x + 1} \sqrt {-x + 1} - 1\right )}}{x - 1} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=\begin {cases} 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=-\frac {2 \, \sqrt {x + 1} \sqrt {-x + 1}}{x - 1} - 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {1+x}}{(1-x)^{3/2}} \, dx=\int \frac {\sqrt {x+1}}{{\left (1-x\right )}^{3/2}} \,d x \]
[In]
[Out]