Optimal. Leaf size=44 \[ 2 \sqrt {x+\sqrt {2 x-1}+1}-\sqrt {2} \sinh ^{-1}\left (\frac {\sqrt {2 x-1}+1}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {640, 619, 215} \begin {gather*} \sqrt {2} \sqrt {2 x+2 \sqrt {2 x-1}+2}-\sqrt {2} \sinh ^{-1}\left (\frac {\sqrt {2 x-1}+1}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x+\sqrt {-1+2 x}}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {\frac {3}{2}+x+\frac {x^2}{2}}} \, dx,x,\sqrt {-1+2 x}\right )\\ &=\sqrt {2} \sqrt {2+2 x+2 \sqrt {-1+2 x}}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {3}{2}+x+\frac {x^2}{2}}} \, dx,x,\sqrt {-1+2 x}\right )\\ &=\sqrt {2} \sqrt {2+2 x+2 \sqrt {-1+2 x}}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2}}} \, dx,x,1+\sqrt {-1+2 x}\right )\\ &=\sqrt {2} \sqrt {2+2 x+2 \sqrt {-1+2 x}}-\sqrt {2} \sinh ^{-1}\left (\frac {1+\sqrt {-1+2 x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.00 \begin {gather*} 2 \sqrt {x+\sqrt {2 x-1}+1}-\sqrt {2} \sinh ^{-1}\left (\frac {\sqrt {2 x-1}+1}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 67, normalized size = 1.52 \begin {gather*} \sqrt {2} \sqrt {2 x+2 \sqrt {2 x-1}+2}+\sqrt {2} \log \left (-\sqrt {2 x-1}+\sqrt {2 x+2 \sqrt {2 x-1}+2}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.79, size = 85, normalized size = 1.93 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-8 \, x^{2} - 8 \, {\left (2 \, x + 1\right )} \sqrt {2 \, x - 1} + 2 \, {\left (\sqrt {2} {\left (2 \, x + 3\right )} \sqrt {2 \, x - 1} + \sqrt {2} {\left (6 \, x - 1\right )}\right )} \sqrt {x + \sqrt {2 \, x - 1} + 1} - 24 \, x + 7\right ) + 2 \, \sqrt {x + \sqrt {2 \, x - 1} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 49, normalized size = 1.11 \begin {gather*} \sqrt {2} {\left (\sqrt {2 \, x + 2 \, \sqrt {2 \, x - 1} + 2} + \log \left (\sqrt {2 \, x + 2 \, \sqrt {2 \, x - 1} + 2} - \sqrt {2 \, x - 1} - 1\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.86 \begin {gather*} -\sqrt {2}\, \arcsinh \left (\frac {\left (1+\sqrt {2 x -1}\right ) \sqrt {2}}{2}\right )+\sqrt {4 x +4+4 \sqrt {2 x -1}} \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 {2 \, x - 1} + 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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x+\sqrt {2\,x-1}+1}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x + \sqrt {2 x - 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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