Optimal. Leaf size=197 \[ \frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (16 x-2)+\sqrt {x^2+1} \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (32 x+3)+16 \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (32 x^2+3 x+8\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{24 \left (\sqrt {x^2+1}+x\right )^{3/2}}-\frac {1}{8} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]
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Rubi [F] time = 0.13, 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 {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 {x+\sqrt {1+x^2}}} \, dx &=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx\\ \end {align*}
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Mathematica [F] time = 2.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {x+\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.25, size = 197, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (16 x-2)+\sqrt {x^2+1} \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1} (32 x+3)+16 \sqrt {\sqrt {x^2+1}+x} \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )+\left (32 x^2+3 x+8\right ) \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}}{24 \left (\sqrt {x^2+1}+x\right )^{3/2}}-\frac {1}{8} \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 108, normalized size = 0.55 \begin {gather*} -\frac {1}{24} \, {\left ({\left (16 \, x^{2} - \sqrt {x^{2} + 1} {\left (16 \, x + 3\right )} + 3 \, x - 8\right )} \sqrt {x + \sqrt {x^{2} + 1}} - 2 \, x + 2 \, \sqrt {x^{2} + 1} - 16\right )} \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - \frac {1}{16} \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + \frac {1}{16} \, \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 \begin {gather*} \int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x +\sqrt {x^{2}+1}}}\, dx \end {gather*}
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 + \sqrt {x^{2} + 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 {\sqrt {\sqrt {x+\sqrt {x^2+1}}+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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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}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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