Integrand size = 17, antiderivative size = 21 \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=-\sqrt {1-x} \sqrt {1+x}+\arcsin (x) \]
[Out]
Time = 0.00 (sec) , antiderivative size = 21, 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 = {52, 41, 222} \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=\arcsin (x)-\sqrt {1-x} \sqrt {x+1} \]
[In]
[Out]
Rule 41
Rule 52
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\sqrt {1-x} \sqrt {1+x}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -\sqrt {1-x} \sqrt {1+x}+\int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\sqrt {1-x} \sqrt {1+x}+\sin ^{-1}(x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=-\sqrt {1-x^2}-2 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(17)=34\).
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00
method | result | size |
default | \(-\sqrt {1-x}\, \sqrt {1+x}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(42\) |
risch | \(\frac {\left (-1+x \right ) \sqrt {1+x}\, \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(65\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=-\sqrt {x + 1} \sqrt {-x + 1} - 2 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.88 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.71 \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=\begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} + \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} - \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=-\sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=-\sqrt {x + 1} \sqrt {-x + 1} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx=\mathrm {asin}\left (x\right )-\sqrt {1-x^2} \]
[In]
[Out]