Optimal. Leaf size=84 \[ \sqrt {\sqrt {x^2+1}+x}+\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\frac {1}{2 \left (\sqrt {x^2+1}+x\right )}+\frac {1}{2} \log \left (\sqrt {x^2+1}+x\right )-2 \log \left (\sqrt {\sqrt {x^2+1}+x}+1\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2117, 1821, 1620} \[ \sqrt {\sqrt {x^2+1}+x}+\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\frac {1}{2 \left (\sqrt {x^2+1}+x\right )}+\frac {1}{2} \log \left (\sqrt {x^2+1}+x\right )-2 \log \left (\sqrt {\sqrt {x^2+1}+x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 1620
Rule 1821
Rule 2117
Rubi steps
\begin {align*} \int \frac {1}{1+\sqrt {x+\sqrt {1+x^2}}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{\left (1+\sqrt {x}\right ) x^2} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\operatorname {Subst}\left (\int \frac {1+x^4}{x^3 (1+x)} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=\operatorname {Subst}\left (\int \left (1+\frac {1}{x^3}-\frac {1}{x^2}+\frac {1}{x}-\frac {2}{1+x}\right ) \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{2 \left (x+\sqrt {1+x^2}\right )}+\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\sqrt {x+\sqrt {1+x^2}}+\frac {1}{2} \log \left (x+\sqrt {1+x^2}\right )-2 \log \left (1+\sqrt {x+\sqrt {1+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 84, normalized size = 1.00 \[ \sqrt {\sqrt {x^2+1}+x}+\frac {1}{\sqrt {\sqrt {x^2+1}+x}}-\frac {1}{2 \left (\sqrt {x^2+1}+x\right )}+\frac {1}{2} \log \left (\sqrt {x^2+1}+x\right )-2 \log \left (\sqrt {\sqrt {x^2+1}+x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 66, normalized size = 0.79 \[ -\sqrt {x + \sqrt {x^{2} + 1}} {\left (x - \sqrt {x^{2} + 1} - 1\right )} + \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {x^{2} + 1} - 2 \, \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) + \log \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{1+\sqrt {x +\sqrt {x^{2}+1}}}\, dx \]
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 \[ \int \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {x+\sqrt {x^2+1}}+1} \,d x \]
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 \[ \int \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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