Optimal. Leaf size=24 \[ -2 \sqrt {x+1}-2 \log \left (1-\sqrt {x+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 190, 43} \[ -2 \sqrt {x+1}-2 \log \left (1-\sqrt {x+1}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{1-\sqrt {1+x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {x}} \, dx,x,1+x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x}{1-x} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-2 \sqrt {1+x}-2 \log \left (1-\sqrt {1+x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ -2 \sqrt {x+1}-2 \log \left (1-\sqrt {x+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 18, normalized size = 0.75 \[ -2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 19, normalized size = 0.79 \[ -2 \, \sqrt {x + 1} - 2 \, \log \left ({\left | \sqrt {x + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 1.29 \[ -\ln \relax (x )-\ln \left (-1+\sqrt {x +1}\right )+\ln \left (1+\sqrt {x +1}\right )-2 \sqrt {x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 18, normalized size = 0.75 \[ -2 \, \sqrt {x + 1} - 2 \, \log \left (\sqrt {x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 18, normalized size = 0.75 \[ -2\,\ln \left (\sqrt {x+1}-1\right )-2\,\sqrt {x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.83 \[ - 2 \sqrt {x + 1} - 2 \log {\left (\sqrt {x + 1} - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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