Optimal. Leaf size=54 \[ \frac {2}{\sqrt {\sqrt {x^2+1}+x}}+2 \tan ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right )-2 \tanh ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2119, 453, 329, 298, 203, 206} \begin {gather*} \frac {2}{\sqrt {\sqrt {x^2+1}+x}}+2 \tan ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right )-2 \tanh ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 453
Rule 2119
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {x+\sqrt {1+x^2}}} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2} \left (-1+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {2}{\sqrt {x+\sqrt {1+x^2}}}+2 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {2}{\sqrt {x+\sqrt {1+x^2}}}+4 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=\frac {2}{\sqrt {x+\sqrt {1+x^2}}}-2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=\frac {2}{\sqrt {x+\sqrt {1+x^2}}}+2 \tan ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right )-2 \tanh ^{-1}\left (\sqrt {x+\sqrt {1+x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.92, size = 175, normalized size = 3.24 \begin {gather*} \frac {2}{3} \sqrt {\sqrt {x^2+1}+x} \left (-\frac {\sqrt {x^2+1} \left (\sqrt {x^2+1}+x\right )^4 \left (\left (4 x^2+4 \sqrt {x^2+1} x+2\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\left (x+\sqrt {x^2+1}\right )^2\right )-x^2-\sqrt {x^2+1} x-2\right )}{16 x^6+28 x^4+13 x^2+5 \sqrt {x^2+1} x+16 \sqrt {x^2+1} x^5+20 \sqrt {x^2+1} x^3+1}+\sqrt {x^2+1}-2 x\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 54, normalized size = 1.00 \begin {gather*} \frac {2}{\sqrt {\sqrt {x^2+1}+x}}+2 \tan ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right )-2 \tanh ^{-1}\left (\sqrt {\sqrt {x^2+1}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 69, normalized size = 1.28 \begin {gather*} -2 \, \sqrt {x + \sqrt {x^{2} + 1}} {\left (x - \sqrt {x^{2} + 1}\right )} + 2 \, \arctan \left (\sqrt {x + \sqrt {x^{2} + 1}}\right ) - \log \left (\sqrt {x + \sqrt {x^{2} + 1}} + 1\right ) + \log \left (\sqrt {x + \sqrt {x^{2} + 1}} - 1\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 {x + \sqrt {x^{2} + 1}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 22, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {2}\, \hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}, \frac {3}{4}\right ], \left [\frac {5}{4}, \frac {3}{2}\right ], -\frac {1}{x^{2}}\right )}{\sqrt {x}} \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 {x + \sqrt {x^{2} + 1}} x}\,{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 {1}{x\,\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 [C] time = 1.34, size = 44, normalized size = 0.81 \begin {gather*} - \frac {\Gamma ^{2}\left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4}, \frac {3}{4} \\ \frac {5}{4}, \frac {3}{2} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{2}}} \right )}}{4 \pi \sqrt {x} \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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