Optimal. Leaf size=35 \[ -\frac {1}{\sqrt {-1+x} \sqrt {1+x}}-\tan ^{-1}\left (\sqrt {-1+x} \sqrt {1+x}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {106, 94, 209}
\begin {gather*} -\text {ArcTan}\left (\sqrt {x-1} \sqrt {x+1}\right )-\frac {1}{\sqrt {x-1} \sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 94
Rule 106
Rule 209
Rubi steps
\begin {align*} \int \frac {1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx &=-\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*}
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Mathematica [A]
time = 0.05, size = 33, normalized size = 0.94 \begin {gather*} -\frac {1}{\sqrt {-1+x} \sqrt {1+x}}-2 \tan ^{-1}\left (\sqrt {\frac {-1+x}{1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 51, normalized size = 1.46
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 | \(\frac {\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right ) x^{2}-\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )-\sqrt {x^{2}-1}}{\sqrt {x^{2}-1}\, \sqrt {1+x}\, \sqrt {-1+x}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 15, normalized size = 0.43 \begin {gather*} -\frac {1}{\sqrt {x^{2} - 1}} + \arcsin \left (\frac {1}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 44, normalized size = 1.26 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 88.15, size = 58, normalized size = 1.66 \begin {gather*} - \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 54, normalized size = 1.54 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{x\,{\left (x-1\right )}^{3/2}\,{\left (x+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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