Integrand size = 17, antiderivative size = 41 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=-\frac {2 (1-x)^{3/2}}{\sqrt {1+x}}-3 \sqrt {1-x} \sqrt {1+x}-3 \arcsin (x) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {49, 52, 41, 222} \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=-3 \arcsin (x)-\frac {2 (1-x)^{3/2}}{\sqrt {x+1}}-3 \sqrt {x+1} \sqrt {1-x} \]
[In]
[Out]
Rule 41
Rule 49
Rule 52
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-x)^{3/2}}{\sqrt {1+x}}-3 \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx \\ & = -\frac {2 (1-x)^{3/2}}{\sqrt {1+x}}-3 \sqrt {1-x} \sqrt {1+x}-3 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = -\frac {2 (1-x)^{3/2}}{\sqrt {1+x}}-3 \sqrt {1-x} \sqrt {1+x}-3 \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {2 (1-x)^{3/2}}{\sqrt {1+x}}-3 \sqrt {1-x} \sqrt {1+x}-3 \sin ^{-1}(x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=-\frac {\sqrt {1-x} (5+x)}{\sqrt {1+x}}+6 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.73
method | result | size |
risch | \(\frac {\left (x^{2}+4 x -5\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(71\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=-\frac {{\left (x + 5\right )} \sqrt {x + 1} \sqrt {-x + 1} - 6 \, {\left (x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 5 \, x + 5}{x + 1} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 3.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.20 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=\begin {cases} 6 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} - \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} + \frac {8 i}{\sqrt {x - 1} \sqrt {x + 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 6 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} - \frac {8}{\sqrt {1 - x} \sqrt {x + 1}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2} + 2 \, x + 1} - \frac {6 \, \sqrt {-x^{2} + 1}}{x + 1} - 3 \, \arcsin \left (x\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (33) = 66\).
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.71 \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=-\sqrt {x + 1} \sqrt {-x + 1} + \frac {2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} - \frac {2 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 6 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {(1-x)^{3/2}}{(1+x)^{3/2}} \, dx=\int \frac {{\left (1-x\right )}^{3/2}}{{\left (x+1\right )}^{3/2}} \,d x \]
[In]
[Out]