Optimal. Leaf size=17 \[ \frac {\sqrt {x+1}}{\sqrt {1-x}} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \[ \frac {\sqrt {x+1}}{\sqrt {1-x}} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{\sqrt {1-x}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ \frac {\sqrt {x+1}}{\sqrt {1-x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 23, normalized size = 1.35 \[ \frac {x - \sqrt {x + 1} \sqrt {-x + 1} - 1}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 19, normalized size = 1.12 \[ -\frac {\sqrt {x + 1} \sqrt {-x + 1}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.82 \[ \frac {\sqrt {x +1}}{\sqrt {-x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 16, normalized size = 0.94 \[ -\frac {\sqrt {-x^{2} + 1}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 13, normalized size = 0.76 \[ \frac {\sqrt {x+1}}{\sqrt {1-x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 29, normalized size = 1.71 \[ \begin {cases} \frac {1}{\sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {i}{\sqrt {1 - \frac {2}{x + 1}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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