Optimal. Leaf size=35 \[ -\frac {\sqrt {-1+x}}{1+x}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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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 = {43, 65, 209}
\begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {\sqrt {x-1}}{x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 209
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}+\text {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 = 0.04, size = 35, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-1+x}}{1+x}+\frac {\tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.83, size = 87, normalized size = 2.49 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-\sqrt {\frac {1-x}{1+x}}+\frac {\text {ArcCosh}\left [\frac {\sqrt {2}}{\sqrt {1+x}}\right ] \sqrt {2+2 x}}{2}\right )}{\sqrt {1+x}},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},-\frac {\sqrt {1-\frac {2}{1+x}}}{\sqrt {1+x}}-\frac {\sqrt {2} \text {ArcSin}\left [\frac {\sqrt {2}}{\sqrt {1+x}}\right ]}{2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.17, size = 30, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {\sqrt {-1+x}}{1+x}\) | \(30\) |
default | \(\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {\sqrt {-1+x}}{1+x}\) | \(30\) |
risch | \(\frac {\arctan \left (\frac {\sqrt {-1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {\sqrt {-1+x}}{1+x}\) | \(30\) |
trager | \(-\frac {\sqrt {-1+x}}{1+x}+\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {-1+x}+3 \RootOf \left (\textit {\_Z}^{2}+2\right )}{1+x}\right )}{4}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 29, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x - 1}\right ) - \frac {\sqrt {x - 1}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 33, normalized size = 0.94 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.75, size = 105, normalized size = 3.00 \begin {gather*} \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 {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 36, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {x-1}}{x+1}+\frac {\arctan \left (\frac {\sqrt {x-1}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
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 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-1}}{2}\right )}{2}-\frac {\sqrt {x-1}}{x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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