Optimal. Leaf size=110 \[ \frac {\sqrt {x^2+1} \left (2 \sqrt {\sqrt {x^2+1}+x}+5\right )+(2 x+2) \sqrt {\sqrt {x^2+1}+x}+5 x-1}{2 \sqrt {x^2+1}+2 x}+\frac {1}{2} \log \left (\sqrt {x^2+1}+x\right )-2 \log \left (\sqrt {\sqrt {x^2+1}+x}+1\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 0.76, 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} \begin {gather*} \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 ) \end {gather*}
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 = 0.04, size = 84, normalized size = 0.76 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 110, normalized size = 1.00 \begin {gather*} \frac {-1+5 x+(2+2 x) \sqrt {x+\sqrt {1+x^2}}+\sqrt {1+x^2} \left (5+2 \sqrt {x+\sqrt {1+x^2}}\right )}{2 x+2 \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 {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 66, normalized size = 0.60 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, 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 \begin {gather*} \int \frac {1}{\sqrt {x + \sqrt {x^{2} + 1}} + 1}\,{d x} \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}{\sqrt {x+\sqrt {x^2+1}}+1} \,d x \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^{2} + 1}} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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