Optimal. Leaf size=47 \[ 2 \sqrt {x+\sqrt {1+x}}-\tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 47, 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 = {654, 635, 212}
\begin {gather*} 2 \sqrt {x+\sqrt {x+1}}-\tanh ^{-1}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 654
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x+\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {x}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-\text {Subst}\left (\int \frac {1}{\sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-2 \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {1+2 \sqrt {1+x}}{\sqrt {x+\sqrt {1+x}}}\right )\\ &=2 \sqrt {x+\sqrt {1+x}}-\tanh ^{-1}\left (\frac {1+2 \sqrt {1+x}}{2 \sqrt {x+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 43, normalized size = 0.91 \begin {gather*} 2 \sqrt {x+\sqrt {1+x}}+\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 32, normalized size = 0.68
method | result | size |
derivativedivides | \(2 \sqrt {x +\sqrt {1+x}}-\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )\) | \(32\) |
default | \(2 \sqrt {x +\sqrt {1+x}}-\ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.53, size = 47, normalized size = 1.00 \begin {gather*} 2 \, \sqrt {x + \sqrt {x + 1}} + \frac {1}{2} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt {x + 1} - 5\right ) \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 {1}{\sqrt {x + \sqrt {x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.76, size = 33, normalized size = 0.70 \begin {gather*} 2 \, \sqrt {x + \sqrt {x + 1}} + \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x+\sqrt {x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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