Optimal. Leaf size=63 \[ -\frac {2 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{\sqrt {\sqrt {x^2+1}+x}}-2 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]
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Rubi [F] time = 0.45, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}} \, dx &=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2} \sqrt {x+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.11, size = 117, normalized size = 1.86 \begin {gather*} -\frac {2 \left (\sqrt {x^2+1}+\sqrt {\sqrt {x^2+1}+x}+\left (\sqrt {x^2+1}+x\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+x\right )}{\left (\sqrt {x^2+1}+x\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 63, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}}-2 \tanh ^{-1}\left (\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 78, normalized size = 1.24 \begin {gather*} 2 \, \sqrt {x + \sqrt {x^{2} + 1}} {\left (x - \sqrt {x^{2} + 1}\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x^{2}+1}\, \sqrt {x +\sqrt {x^{2}+1}}}\, dx\]
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 {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\sqrt {x^{2} + 1} \sqrt {x + \sqrt {x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 121, normalized size = 1.92 \begin {gather*} -\frac {2\,\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\sqrt {x+\sqrt {x^2+1}}}-\frac {\ln \left (\sqrt {\frac {1}{x+\sqrt {x^2+1}}+\frac {1}{\sqrt {x+\sqrt {x^2+1}}}}+\frac {1}{\sqrt {x+\sqrt {x^2+1}}}+\frac {1}{2}\right )\,\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\sqrt {x+\sqrt {x^2+1}}\,\sqrt {\frac {1}{x+\sqrt {x^2+1}}+\frac {1}{\sqrt {x+\sqrt {x^2+1}}}}} \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 {\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1}}{\sqrt {x + \sqrt {x^{2} + 1}} \sqrt {x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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