Optimal. Leaf size=51 \[ -\frac {\sqrt {\sqrt {x^2+1}+1}}{x}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right )}{\sqrt {2}} \]
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Rubi [F] time = 0.33, 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 {1+x^2}}}{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 {1+x^2}}}{x^2 \sqrt {1+x^2}} \, dx &=\int \frac {\sqrt {1+\sqrt {1+x^2}}}{x^2 \sqrt {1+x^2}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.11, size = 45, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {\sqrt {x^2+1}+1} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {\sqrt {x^2+1}}{2}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 51, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {\sqrt {x^2+1}+1}}{x}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {\sqrt {x^2+1}+1}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 43, normalized size = 0.84 \begin {gather*} \frac {\sqrt {2} x \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^{2} + 1} + 1}}{x}\right ) - 2 \, \sqrt {\sqrt {x^{2} + 1} + 1}}{2 \, x} \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 {\sqrt {\sqrt {x^{2} + 1} + 1}}{\sqrt {x^{2} + 1} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+\sqrt {x^{2}+1}}}{x^{2} \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^{2} + 1} + 1}}{\sqrt {x^{2} + 1} x^{2}}\,{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^2+1}+1}}{x^2\,\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.19, size = 34, normalized size = 0.67 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{\pi x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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