Optimal. Leaf size=18 \[ -\left ((x+1) \sqrt {\frac {2}{x+1}-1}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {512, 514, 375, 74} \begin {gather*} -(x+1) \sqrt {\frac {2}{x+1}-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 375
Rule 512
Rule 514
Rubi steps
\begin {align*} \int \frac {x}{(1+x) \sqrt {-1+\frac {2}{1+x}}} \, dx &=\operatorname {Subst}\left (\int \frac {-1+x}{\sqrt {-1+\frac {2}{x}} x} \, dx,x,1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {1-\frac {1}{x}}{\sqrt {-1+\frac {2}{x}}} \, dx,x,1+x\right )\\ &=-\operatorname {Subst}\left (\int \frac {1-x}{x^2 \sqrt {-1+2 x}} \, dx,x,\frac {1}{1+x}\right )\\ &=-\left ((1+x) \sqrt {-1+\frac {2}{1+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} \frac {x-1}{\sqrt {\frac {2}{x+1}-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 21, normalized size = 1.17 \begin {gather*} (-x-1) \sqrt {\frac {1-x}{x+1}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 17, normalized size = 0.94 \begin {gather*} -{\left (x + 1\right )} \sqrt {-\frac {x - 1}{x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 29, normalized size = 1.61 \begin {gather*} -\frac {2}{\sqrt {-\frac {x - 1}{x + 1}} + \frac {1}{\sqrt {-\frac {x - 1}{x + 1}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.94 \begin {gather*} \frac {x -1}{\sqrt {-\frac {x -1}{x +1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 16, normalized size = 0.89 \begin {gather*} \frac {\sqrt {x + 1} {\left (x - 1\right )}}{\sqrt {-x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {x}{\left (x+1\right )\,\sqrt {\frac {2}{x+1}-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- \frac {x - 1}{x + 1}} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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