Optimal. Leaf size=35 \[ \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2117, 14} \begin {gather*} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x+\sqrt {1+x^2}}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{x^{5/2}} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} \sqrt {\sqrt {x^2+1}+x}-\frac {1}{3 \left (\sqrt {x^2+1}+x\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 35, normalized size = 1.00 \begin {gather*} -\frac {1}{3 \left (x+\sqrt {1+x^2}\right )^{3/2}}+\sqrt {x+\sqrt {1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 28, normalized size = 0.80 \begin {gather*} -\frac {2}{3} \, {\left (x^{2} - \sqrt {x^{2} + 1} x - 1\right )} \sqrt {x + \sqrt {x^{2} + 1}} \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}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 62, normalized size = 1.77
method | result | size |
meijerg | \(-\frac {-\frac {32 \sqrt {\pi }\, \sqrt {2}\, \cosh \left (\frac {3 \arcsinh \left (\frac {1}{x}\right )}{2}\right )}{3 x^{\frac {3}{2}}}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, x^{\frac {3}{2}} \left (-\frac {4}{3 x^{4}}-\frac {2}{3 x^{2}}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {1}{x}\right )}{2}\right )}{\sqrt {1+\frac {1}{x^{2}}}}}{8 \sqrt {\pi }}\) | \(62\) |
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}}}\,{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 = -0.03 \begin {gather*} \int \frac {1}{\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 [A] time = 0.34, size = 42, normalized size = 1.20 \begin {gather*} \frac {4 x}{3 \sqrt {x + \sqrt {x^{2} + 1}}} + \frac {2 \sqrt {x^{2} + 1}}{3 \sqrt {x + \sqrt {x^{2} + 1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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