Optimal. Leaf size=92 \[ 4 \sqrt {\sqrt {x+1}+1}-\frac {2}{5} \left (3 \sqrt {5}-5\right ) \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}}{\sqrt {5}-1}\right )-\frac {2}{5} \left (5+3 \sqrt {5}\right ) \tanh ^{-1}\left (\frac {2 \sqrt {\sqrt {x+1}+1}}{1+\sqrt {5}}\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 114, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {824, 826, 1166, 207} \begin {gather*} 4 \sqrt {\sqrt {x+1}+1}-2 \sqrt {\frac {2}{5} \left (7+3 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {\sqrt {x+1}+1}\right )-2 \sqrt {\frac {2}{5} \left (7-3 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {\sqrt {x+1}+1}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 207
Rule 824
Rule 826
Rule 1166
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\sqrt {1+x}}}{x-\sqrt {1+x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {1+x}}{-1-x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}+2 \operatorname {Subst}\left (\int \frac {1+2 x}{\sqrt {1+x} \left (-1-x+x^2\right )} \, dx,x,\sqrt {1+x}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}+4 \operatorname {Subst}\left (\int \frac {-1+2 x^2}{1-3 x^2+x^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}+\frac {1}{5} \left (4 \left (5-2 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{5} \left (4 \left (5+2 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}-2 \sqrt {\frac {2}{5} \left (7+3 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{5} \sqrt {70-30 \sqrt {5}} \tanh ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 125, normalized size = 1.36 \begin {gather*} \frac {1}{5} \left (20 \sqrt {\sqrt {x+1}+1}+\sqrt {6-2 \sqrt {5}} \left (\sqrt {5}-5\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{3-\sqrt {5}}} \sqrt {\sqrt {x+1}+1}\right )-\sqrt {2 \left (3+\sqrt {5}\right )} \left (5+\sqrt {5}\right ) \tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} \sqrt {\sqrt {x+1}+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 92, normalized size = 1.00 \begin {gather*} 4 \sqrt {1+\sqrt {1+x}}-\frac {2}{5} \left (5+3 \sqrt {5}\right ) \tanh ^{-1}\left (\frac {1}{2} \left (-1+\sqrt {5}\right ) \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{5} \left (-5+3 \sqrt {5}\right ) \tanh ^{-1}\left (\frac {1}{2} \left (1+\sqrt {5}\right ) \sqrt {1+\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 234, normalized size = 2.54 \begin {gather*} \frac {3}{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 ) + \frac {3}{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 ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + \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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giac [A] time = 0.56, size = 136, normalized size = 1.48 \begin {gather*} \frac {3}{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 ) + \frac {3}{5} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}{{\left | \sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} - 1 \right |}}\right ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + \log \left ({\left | \sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 97, normalized size = 1.05
method | result | size |
derivativedivides | \(4 \sqrt {1+\sqrt {1+x}}-\ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}+\ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\) | \(97\) |
default | \(4 \sqrt {1+\sqrt {1+x}}-\ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}+\ln \left (\sqrt {1+x}-\sqrt {1+\sqrt {1+x}}\right )-\frac {6 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}-1\right ) \sqrt {5}}{5}\right )}{5}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 132, normalized size = 1.43 \begin {gather*} \frac {3}{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 ) + \frac {3}{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 ) + 4 \, \sqrt {\sqrt {x + 1} + 1} - \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + \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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\sqrt {x+1}+1}}{x-\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.35, size = 260, normalized size = 2.83 \begin {gather*} 4 \sqrt {\sqrt {x + 1} + 1} + 12 \left (\begin {cases} - \frac {\sqrt {5} \operatorname {acoth}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} - \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} - \frac {1}{2}\right )^{2} > \frac {5}{4} \\- \frac {\sqrt {5} \operatorname {atanh}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} - \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} - \frac {1}{2}\right )^{2} < \frac {5}{4} \end {cases}\right ) + 12 \left (\begin {cases} - \frac {\sqrt {5} \operatorname {acoth}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} > \frac {5}{4} \\- \frac {\sqrt {5} \operatorname {atanh}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} < \frac {5}{4} \end {cases}\right ) + \log {\left (\sqrt {x + 1} - \sqrt {\sqrt {x + 1} + 1} \right )} - \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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