Optimal. Leaf size=109 \[ \frac {2 x}{\sqrt {\sqrt {a x^2+b^2}+b}}-\frac {2 \sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]
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Rubi [F] time = 0.05, 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 {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx &=\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ \end {align*}
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Mathematica [C] time = 0.39, size = 149, normalized size = 1.37 \begin {gather*} \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (-2 b \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )+4 \sqrt {a x^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}-b} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a x^2+b^2}-b}}{\sqrt {2} \sqrt {b}}\right )-2 b\right )}{2 a x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 77, normalized size = 0.71 \begin {gather*} \frac {2 x}{\sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 46.12, size = 239, normalized size = 2.19 \begin {gather*} \left [\frac {\sqrt {2} a x \sqrt {-\frac {b}{a}} \log \left (-\frac {a x^{3} + 4 \, b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} b x + 2 \, {\left (2 \, \sqrt {2} \sqrt {a x^{2} + b^{2}} b \sqrt {-\frac {b}{a}} - \sqrt {2} {\left (a x^{2} + 2 \, b^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) - 4 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}} {\left (b - \sqrt {a x^{2} + b^{2}}\right )}}{2 \, a x}, \frac {\sqrt {2} a x \sqrt {\frac {b}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {b + \sqrt {a x^{2} + b^{2}}} \sqrt {\frac {b}{a}}}{x}\right ) - 2 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}} {\left (b - \sqrt {a x^{2} + b^{2}}\right )}}{a x}\right ] \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 {1}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 29, normalized size = 0.27 \begin {gather*} \frac {\sqrt {2}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}, \frac {3}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{2 \left (b^{2}\right )^{\frac {1}{4}}} \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 {1}{\sqrt {b + \sqrt {a x^{2} + b^{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 = -0.01 \begin {gather*} \int \frac {1}{\sqrt {b+\sqrt {b^2+a\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.81, size = 42, normalized size = 0.39 \begin {gather*} \frac {x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{2 \pi \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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