Optimal. Leaf size=23 \[ -\frac {1}{3} (-1+x)^{3/2}+\frac {1}{3} (1+x)^{3/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6821}
\begin {gather*} \frac {1}{3} (x+1)^{3/2}-\frac {1}{3} (x-1)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 6821
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x}+\sqrt {1+x}} \, dx &=-\left (\frac {1}{2} \int \left (\sqrt {-1+x}-\sqrt {1+x}\right ) \, dx\right )\\ &=-\frac {1}{3} (-1+x)^{3/2}+\frac {1}{3} (1+x)^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 23, normalized size = 1.00 \begin {gather*} -\frac {1}{3} (-1+x)^{3/2}+\frac {1}{3} (1+x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 16, normalized size = 0.70
method | result | size |
default | \(-\frac {\left (-1+x \right )^{\frac {3}{2}}}{3}+\frac {\left (1+x \right )^{\frac {3}{2}}}{3}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 15, normalized size = 0.65 \begin {gather*} \frac {1}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {1}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} \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. 51 vs.
\(2 (15) = 30\).
time = 0.19, size = 51, normalized size = 2.22 \begin {gather*} \frac {4 x}{3 \sqrt {x - 1} + 3 \sqrt {x + 1}} + \frac {2 \sqrt {x - 1} \sqrt {x + 1}}{3 \sqrt {x - 1} + 3 \sqrt {x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.28, size = 15, normalized size = 0.65 \begin {gather*} \frac {1}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {1}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.84, size = 15, normalized size = 0.65 \begin {gather*} \frac {{\left (x+1\right )}^{3/2}}{3}-\frac {{\left (x-1\right )}^{3/2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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