Integrand size = 17, antiderivative size = 41 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \arcsin (x) \]
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Time = 0.00 (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} \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=-3 \arcsin (x)+\frac {2 (x+1)^{3/2}}{\sqrt {1-x}}+3 \sqrt {1-x} \sqrt {x+1} \]
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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 \\ & = 3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx \\ & = 3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = 3 \sqrt {1-x} \sqrt {1+x}+\frac {2 (1+x)^{3/2}}{\sqrt {1-x}}-3 \sin ^{-1}(x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=\frac {(-5+x) \sqrt {1-x^2}}{-1+x}+6 \arctan \left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(33)=66\).
Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.76
method | result | size |
risch | \(-\frac {\left (x^{2}-4 x -5\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (-1+x \right ) \left (1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}-\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(72\) |
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none
Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=\frac {\sqrt {x + 1} {\left (x - 5\right )} \sqrt {-x + 1} + 6 \, {\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 5 \, x - 5}{x - 1} \]
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Result contains complex when optimal does not.
Time = 1.97 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.41 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=\begin {cases} 6 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} - \frac {6 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- 6 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} + \frac {6 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
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none
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=-\frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2} - 2 \, x + 1} - \frac {6 \, \sqrt {-x^{2} + 1}}{x - 1} - 3 \, \arcsin \left (x\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=\frac {\sqrt {x + 1} {\left (x - 5\right )} \sqrt {-x + 1}}{x - 1} - 6 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
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Timed out. \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2}} \, dx=\int \frac {{\left (x+1\right )}^{3/2}}{{\left (1-x\right )}^{3/2}} \,d x \]
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