Optimal. Leaf size=18 \[ \frac {x}{\sqrt {1-x} \sqrt {x+1}} \]
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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 = {39} \begin {gather*} \frac {x}{\sqrt {1-x} \sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx &=\frac {x}{\sqrt {1-x} \sqrt {1+x}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 0.72 \begin {gather*} \frac {x}{\sqrt {1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 34, normalized size = 1.89 \begin {gather*} \frac {\sqrt {x+1} \left (1-\frac {1-x}{x+1}\right )}{2 \sqrt {1-x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 22, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {x + 1} x \sqrt {-x + 1}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.72, size = 62, normalized size = 3.44 \begin {gather*} \frac {\sqrt {2} - \sqrt {-x + 1}}{4 \, \sqrt {x + 1}} - \frac {\sqrt {x + 1} \sqrt {-x + 1}}{2 \, {\left (x - 1\right )}} - \frac {\sqrt {x + 1}}{4 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {x}{\sqrt {-x +1}\, \sqrt {x +1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 11, normalized size = 0.61 \begin {gather*} \frac {x}{\sqrt {-x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 14, normalized size = 0.78 \begin {gather*} \frac {x}{\sqrt {1-x}\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.86, size = 65, normalized size = 3.61 \begin {gather*} \begin {cases} \frac {1}{\sqrt {-1 + \frac {2}{x + 1}}} - \frac {1}{\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{x - 1} + \frac {i \sqrt {1 - \frac {2}{x + 1}}}{x - 1} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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