Integrand size = 18, antiderivative size = 35 \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=-\frac {1}{\sqrt {-1+x} \sqrt {1+x}}-\arctan \left (\sqrt {-1+x} \sqrt {1+x}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {106, 94, 209} \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=-\arctan \left (\sqrt {x-1} \sqrt {x+1}\right )-\frac {1}{\sqrt {x-1} \sqrt {x+1}} \]
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Rule 94
Rule 106
Rule 209
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{\sqrt {-1+x} \sqrt {1+x}}-\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx \\ & = -\frac {1}{\sqrt {-1+x} \sqrt {1+x}}-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x} \sqrt {1+x}\right ) \\ & = -\frac {1}{\sqrt {-1+x} \sqrt {1+x}}-\tan ^{-1}\left (\sqrt {-1+x} \sqrt {1+x}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=-\frac {1}{\sqrt {-1+x} \sqrt {1+x}}-2 \arctan \left (\sqrt {\frac {-1+x}{1+x}}\right ) \]
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Time = 1.79 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20
method | result | size |
risch | \(-\frac {1}{\sqrt {-1+x}\, \sqrt {1+x}}+\frac {\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right ) \sqrt {\left (-1+x \right ) \left (1+x \right )}}{\sqrt {-1+x}\, \sqrt {1+x}}\) | \(42\) |
default | \(\frac {\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right ) x^{2}-\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )-\sqrt {x^{2}-1}}{\sqrt {x^{2}-1}\, \sqrt {1+x}\, \sqrt {-1+x}}\) | \(51\) |
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none
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=-\frac {2 \, {\left (x^{2} - 1\right )} \arctan \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) + \sqrt {x + 1} \sqrt {x - 1}}{x^{2} - 1} \]
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Result contains complex when optimal does not.
Time = 73.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & 1, 2, \frac {5}{2} \\\frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, 1 & \\\frac {3}{4}, \frac {5}{4} & 0, \frac {1}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}}} \]
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none
Time = 0.32 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=-\frac {1}{\sqrt {x^{2} - 1}} + \arcsin \left (\frac {1}{{\left | x \right |}}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=-\frac {\sqrt {x + 1}}{2 \, \sqrt {x - 1}} + \frac {2}{{\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2} + 2} + 2 \, \arctan \left (\frac {1}{2} \, {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2}\right ) \]
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Timed out. \[ \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx=\int \frac {1}{x\,{\left (x-1\right )}^{3/2}\,{\left (x+1\right )}^{3/2}} \,d x \]
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