Optimal. Leaf size=35 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\sqrt {x-1}}{x+1} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {47, 63, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\sqrt {x-1}}{x+1} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 203
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+x}}{(1+x)^2} \, dx &=-\frac {\sqrt {-1+x}}{1+x}+\frac {1}{2} \int \frac {1}{\sqrt {-1+x} (1+x)} \, dx\\ &=-\frac {\sqrt {-1+x}}{1+x}+\operatorname {Subst}\left (\int \frac {1}{2+x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=-\frac {\sqrt {-1+x}}{1+x}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 1.46 \[ \frac {-2 x-\sqrt {2-2 x} (x+1) \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )+2}{2 \sqrt {x-1} (x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 33, normalized size = 0.94 \[ \frac {\sqrt {2} {\left (x + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) - 2 \, \sqrt {x - 1}}{2 \, {\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 29, normalized size = 0.83 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) - \frac {\sqrt {x - 1}}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.86 \[ \frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x -1}\, \sqrt {2}}{2}\right )}{2}-\frac {\sqrt {x -1}}{x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 29, normalized size = 0.83 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) - \frac {\sqrt {x - 1}}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 29, normalized size = 0.83 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-1}}{2}\right )}{2}-\frac {\sqrt {x-1}}{x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.50, size = 104, normalized size = 2.97 \[ \begin {cases} \frac {\sqrt {2} i \operatorname {acosh}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{2} + \frac {i}{\sqrt {-1 + \frac {2}{x + 1}} \sqrt {x + 1}} - \frac {2 i}{\sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{\frac {3}{2}}} & \text {for}\: \frac {2}{\left |{x + 1}\right |} > 1 \\- \frac {\sqrt {1 - \frac {2}{x + 1}}}{\sqrt {x + 1}} - \frac {\sqrt {2} \operatorname {asin}{\left (\frac {\sqrt {2}}{\sqrt {x + 1}} \right )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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