Optimal. Leaf size=33 \[ \frac {2 \sqrt {-1+x}}{1-x}+\frac {2 \sqrt {1+x}}{-1-x} \]
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Rubi [A]
time = 0.19, antiderivative size = 19, normalized size of antiderivative = 0.58, number of steps
used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6820}
\begin {gather*} -\frac {2}{\sqrt {x+1}}-\frac {2}{\sqrt {x-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6820
Rubi steps
\begin {align*} \int \frac {(-1+x)^{3/2}+(1+x)^{3/2}}{(-1+x)^{3/2} (1+x)^{3/2}} \, dx &=\int \left (\frac {1}{(-1+x)^{3/2}}+\frac {1}{(1+x)^{3/2}}\right ) \, dx\\ &=-\frac {2}{\sqrt {-1+x}}-\frac {2}{\sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 0.58 \begin {gather*} -\frac {2}{\sqrt {-1+x}}-\frac {2}{\sqrt {1+x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 16, normalized size = 0.48
method | result | size |
default | \(-\frac {2}{\sqrt {1+x}}-\frac {2}{\sqrt {-1+x}}\) | \(16\) |
meijerg | \(\frac {2 \sqrt {\pi }-\frac {2 \sqrt {\pi }}{\sqrt {1+x}}}{\sqrt {\pi }}-\frac {2 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {3}{2}} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-x}}\right )}{\sqrt {\pi }\, \mathrm {signum}\left (-1+x \right )^{\frac {3}{2}}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2}{\sqrt {x + 1}} - \frac {2}{\sqrt {x - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 28, normalized size = 0.85 \begin {gather*} -\frac {2 \, {\left ({\left (x + 1\right )} \sqrt {x - 1} + \sqrt {x + 1} {\left (x - 1\right )}\right )}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (24) = 48\).
time = 1.33, size = 56, normalized size = 1.70 \begin {gather*} - \frac {2 x \sqrt {x - 1}}{x^{2} - 1} - \frac {2 x \sqrt {x + 1}}{x^{2} - 1} - \frac {2 \sqrt {x - 1}}{x^{2} - 1} + \frac {2 \sqrt {x + 1}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2}{\sqrt {x + 1}} - \frac {2}{\sqrt {x - 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 15, normalized size = 0.45 \begin {gather*} -\frac {2}{\sqrt {x-1}}-\frac {2}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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