Optimal. Leaf size=18 \[ -\frac {\sqrt {1-x}}{\sqrt {1+x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 18, 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}
\begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx &=-\frac {\sqrt {1-x}}{\sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {1+x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 15, normalized size = 0.83
method | result | size |
gosper | \(-\frac {\sqrt {1-x}}{\sqrt {1+x}}\) | \(15\) |
default | \(-\frac {\sqrt {1-x}}{\sqrt {1+x}}\) | \(15\) |
risch | \(\frac {\left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}\, \sqrt {1+x}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 23, normalized size = 1.28 \begin {gather*} -\frac {x + \sqrt {x + 1} \sqrt {-x + 1} + 1}{x + 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 = 0.46, size = 31, normalized size = 1.72 \begin {gather*} \begin {cases} - \sqrt {-1 + \frac {2}{x + 1}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- i \sqrt {1 - \frac {2}{x + 1}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (14) = 28\).
time = 1.63, size = 43, normalized size = 2.39 \begin {gather*} \frac {\sqrt {2} - \sqrt {-x + 1}}{2 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1}}{2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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