Optimal. Leaf size=96 \[ \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x+\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{x}+1}+3}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right )-\frac {3}{4} \tanh ^{-1}\left (\frac {1-3 \sqrt {\frac {1}{x}+1}}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1014, 1033, 724, 206, 204} \[ \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x+\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{x}+1}+3}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right )-\frac {3}{4} \tanh ^{-1}\left (\frac {1-3 \sqrt {\frac {1}{x}+1}}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 724
Rule 1014
Rule 1033
Rubi steps
\begin {align*} \int \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x \sqrt {-1+x+x^2}}{\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+\frac {1}{x}}\right )\right )\\ &=\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} x-\operatorname {Subst}\left (\int \frac {\frac {1}{2}+x}{\left (-1+x^2\right ) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+\frac {1}{x}}\right )\\ &=\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} x-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+\frac {1}{x}}\right )-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {-1+x+x^2}} \, dx,x,\sqrt {1+\frac {1}{x}}\right )\\ &=\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} x+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3-\sqrt {1+\frac {1}{x}}}{\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+3 \sqrt {1+\frac {1}{x}}}{\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}\right )\\ &=\sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}} x+\frac {1}{4} \tan ^{-1}\left (\frac {3+\sqrt {1+\frac {1}{x}}}{2 \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}\right )-\frac {3}{4} \tanh ^{-1}\left (\frac {1-3 \sqrt {1+\frac {1}{x}}}{2 \sqrt {\sqrt {1+\frac {1}{x}}+\frac {1}{x}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.20, size = 98, normalized size = 1.02 \[ \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} x-\frac {1}{4} \tan ^{-1}\left (\frac {-\sqrt {\frac {1}{x}+1}-3}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right )+\frac {3}{4} \tanh ^{-1}\left (\frac {3 \sqrt {\frac {1}{x}+1}-1}{2 \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 2.86, size = 122, normalized size = 1.27 \[ x \sqrt {\frac {x \sqrt {\frac {x + 1}{x}} + 1}{x}} + \frac {1}{4} \, \arctan \left (\frac {2 \, {\left (x \sqrt {\frac {x + 1}{x}} - 3 \, x\right )} \sqrt {\frac {x \sqrt {\frac {x + 1}{x}} + 1}{x}}}{8 \, x - 1}\right ) + \frac {3}{4} \, \log \left (2 \, {\left (x \sqrt {\frac {x + 1}{x}} + x\right )} \sqrt {\frac {x \sqrt {\frac {x + 1}{x}} + 1}{x}} + 2 \, x \sqrt {\frac {x + 1}{x}} + 2 \, x + 3\right ) \]
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 \[ \int \sqrt {\sqrt {\frac {1}{x} + 1} + \frac {1}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {1}{x}+\sqrt {\frac {1}{x}+1}}\, dx \]
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 \[ \int \sqrt {\sqrt {\frac {1}{x} + 1} + \frac {1}{x}}\,{d x} \]
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 \[ \int \sqrt {\sqrt {\frac {1}{x}+1}+\frac {1}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {1 + \frac {1}{x}} + \frac {1}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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