Optimal. Leaf size=16 \[ \sqrt {x-1} x \sqrt {x+1} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {384} \begin {gather*} \sqrt {x-1} x \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 384
Rubi steps
\begin {align*} \int \frac {-1+2 x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx &=\sqrt {-1+x} x \sqrt {1+x}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 66, normalized size = 4.12 \begin {gather*} \frac {\sqrt {x-1} \left (x \sqrt {1-x^2}-2 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right )}{\sqrt {1-x}}+2 \tanh ^{-1}\left (\sqrt {\frac {x-1}{x+1}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [B] time = 0.05, size = 46, normalized size = 2.88 \begin {gather*} \frac {2 \left (\frac {(x-1)^{3/2}}{(x+1)^{3/2}}+\frac {\sqrt {x-1}}{\sqrt {x+1}}\right )}{\left (\frac {x-1}{x+1}-1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 12, normalized size = 0.75 \begin {gather*} \sqrt {x + 1} \sqrt {x - 1} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 12, normalized size = 0.75 \begin {gather*} \sqrt {x + 1} \sqrt {x - 1} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} \sqrt {x -1}\, \sqrt {x +1}\, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.43, size = 9, normalized size = 0.56 \begin {gather*} \sqrt {x^{2} - 1} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.80, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (x^2+x\right )\,\sqrt {x-1}}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 43.48, size = 129, normalized size = 8.06 \begin {gather*} - \begin {cases} 2 \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 2 i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} & \text {otherwise} \end {cases} + \frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & - \frac {1}{2}, - \frac {1}{2}, 0, 1 \\-1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 0 & \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & - \frac {3}{2}, -1, -1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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