Optimal. Leaf size=73 \[ \frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {646, 31}
\begin {gather*} \frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{x-\sqrt {1+\sqrt {1+x}}} \, dx &=2 \text {Subst}\left (\int \frac {x}{-1+x^2-\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \text {Subst}\left (\int \frac {-1+x}{-1-x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1-\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 69, normalized size = 0.95 \begin {gather*} -\frac {2}{5} \left (-5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(174\) vs.
\(2(53)=106\).
time = 0.10, size = 175, normalized size = 2.40
method | result | size |
derivativedivides | \(2 \ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\) | \(46\) |
default | \(\frac {\sqrt {5}\, \arctanh \left (\frac {\left (1+2 \sqrt {1+x}\right ) \sqrt {5}}{5}\right )}{5}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+x}-1\right ) \sqrt {5}}{5}\right )}{5}-\ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {\ln \left (x -\sqrt {1+x}\right )}{2}+\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x -1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (x^{2}-x -1\right )}{2}+\frac {2 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}+\ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )-\frac {\ln \left (x +\sqrt {1+x}\right )}{2}+\frac {2 \arctanh \left (\frac {\left (1+2 \sqrt {1+\sqrt {1+x}}\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{5}\) | \(175\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 63, normalized size = 0.86 \begin {gather*} -\frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (51) = 102\).
time = 0.35, size = 112, normalized size = 1.53 \begin {gather*} \frac {2}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} + \sqrt {5} {\left (3 \, x + 1\right )} + {\left (\sqrt {5} {\left (x + 2\right )} + 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} + 5\right )} \sqrt {x + 1} + 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1}\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}{x - \sqrt {\sqrt {x + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.02, size = 92, normalized size = 1.26 \begin {gather*} 4 \left (\frac {\ln \left |\sqrt {x+1}+1-\sqrt {\sqrt {x+1}+1}-1\right |}{2}-\frac {\ln \left (\frac {\left |2 \sqrt {\sqrt {x+1}+1}-1-\sqrt {5}\right |}{\left |2 \sqrt {\sqrt {x+1}+1}-1+\sqrt {5}\right |}\right )}{2 \sqrt {5}}\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 = -0.01 \begin {gather*} \int \frac {1}{x-\sqrt {\sqrt {x+1}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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