Optimal. Leaf size=48 \[ 4 \sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-4 \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right ) \]
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Rubi [F] time = 0.14, 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}} \, 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}} \, dx &=\int \frac {\sqrt {1+\sqrt {x+\sqrt {1+x^2}}}}{\sqrt {1+x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 1.00 \begin {gather*} 4 \left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}-\tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {x^2+1}+x}+1}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 48, normalized size = 1.00 \begin {gather*} 4 \sqrt {1+\sqrt {x+\sqrt {1+x^2}}}-4 \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 = 0.44, size = 58, normalized size = 1.21 \begin {gather*} 4 \, \sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} - 2 \, \log \left (\sqrt {\sqrt {x + \sqrt {x^{2} + 1}} + 1} + 1\right ) + 2 \, \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 \[\int \frac {\sqrt {1+\sqrt {x +\sqrt {x^{2}+1}}}}{\sqrt {x^{2}+1}}\, dx\]
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}}\,{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 = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {x+\sqrt {x^2+1}}+1}}{\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^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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