Optimal. Leaf size=47 \[ 2 \sqrt {\sqrt {x^2+1}+1}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+1}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {371, 1398, 783, 80, 63, 207} \begin {gather*} 2 \sqrt {\sqrt {x^2+1}+1}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+1}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 207
Rule 371
Rule 783
Rule 1398
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\sqrt {1+x^2}}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {1+x}}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {x}}}{-1+x} \, dx,x,1+x^2\right )\\ &=\operatorname {Subst}\left (\int \frac {x \sqrt {1+x}}{-1+x^2} \, dx,x,\sqrt {1+x^2}\right )\\ &=\operatorname {Subst}\left (\int \frac {x}{(-1+x) \sqrt {1+x}} \, dx,x,\sqrt {1+x^2}\right )\\ &=2 \sqrt {1+\sqrt {1+x^2}}+\operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {1+x}} \, dx,x,\sqrt {1+x^2}\right )\\ &=2 \sqrt {1+\sqrt {1+x^2}}+2 \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {1+\sqrt {1+x^2}}\right )\\ &=2 \sqrt {1+\sqrt {1+x^2}}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 1.00 \begin {gather*} 2 \sqrt {\sqrt {x^2+1}+1}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+1}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 47, normalized size = 1.00 \begin {gather*} 2 \sqrt {1+\sqrt {1+x^2}}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+x^2}}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 67, normalized size = 1.43 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {x^{2} - 2 \, {\left (\sqrt {2} \sqrt {x^{2} + 1} + \sqrt {2}\right )} \sqrt {\sqrt {x^{2} + 1} + 1} + 4 \, \sqrt {x^{2} + 1} + 4}{x^{2}}\right ) + 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 56, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x^{2} + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {x^{2} + 1} + 1}}\right ) + 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 51, normalized size = 1.09
method | result | size |
meijerg | \(-\frac {-4 \left (-4 \ln \relax (2)+4+2 \ln \relax (x )\right ) \sqrt {\pi }\, \sqrt {2}-\frac {\hypergeom \left (\left [\frac {3}{4}, 1, 1, \frac {5}{4}\right ], \left [\frac {3}{2}, 2, 2\right ], -x^{2}\right ) \sqrt {\pi }\, \sqrt {2}\, x^{2}}{2}}{8 \sqrt {\pi }}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 56, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x^{2} + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {x^{2} + 1} + 1}}\right ) + 2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.14, size = 49, normalized size = 1.04 \begin {gather*} \frac {x^{2} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{4}F_{3}\left (\begin {matrix} \frac {3}{4}, 1, 1, \frac {5}{4} \\ \frac {3}{2}, 2, 2 \end {matrix}\middle | {x^{2} e^{i \pi }} \right )}}{4 \pi } + \frac {\log {\left (x^{2} \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{2 \pi } \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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