Optimal. Leaf size=51 \[ -\frac {\sqrt {1+\sqrt {1+x^2}}}{x}-\frac {\text {ArcTan}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )}{\sqrt {2}} \]
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Rubi [F]
time = 0.20, 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 [A]
time = 0.07, size = 51, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1+\sqrt {1+x^2}}}{x}-\frac {\text {ArcTan}\left (\frac {x}{\sqrt {2} \sqrt {1+\sqrt {1+x^2}}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {1+\sqrt {x^{2}+1}}}{x^{2} \sqrt {x^{2}+1}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, 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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Sympy [C] Result contains complex when optimal does not.
time = 0.73, 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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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