Optimal. Leaf size=63 \[ 4 \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1397, 843, 620, 206, 724} \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {1-3 \sqrt {x}}{2 \sqrt {2} \sqrt {x-\sqrt {x}}}\right )+4 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 724
Rule 843
Rule 1397
Rubi steps
\begin {align*} \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {-\sqrt {x}+x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{(1+x) \sqrt {-x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-x+x^2}} \, dx,x,\sqrt {x}\right )-2 \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {-x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-\sqrt {x}+x}}\right )+4 \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {1-3 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}\right )\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {1-3 \sqrt {x}}{2 \sqrt {2} \sqrt {-\sqrt {x}+x}}\right )+4 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-\sqrt {x}+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 1.25 \begin {gather*} -\sqrt {2} \log \left (\sqrt {x}+1\right )+2 \log \left (-2 \sqrt {x}-2 \sqrt {x-\sqrt {x}}+1\right )+\sqrt {2} \log \left (-3 \sqrt {x}+2 \sqrt {2} \sqrt {x-\sqrt {x}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 63, normalized size = 1.00 \begin {gather*} 4 \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.53, size = 102, normalized size = 1.62 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {17 \, x^{2} - 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) + \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 76, normalized size = 1.21 \begin {gather*} -\sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} - \sqrt {x - \sqrt {x}} + \sqrt {x} + 1\right )}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} - 2 \right |}}\right ) - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 67, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {-\sqrt {x}+x}\, \left (\sqrt {2}\, \arctanh \left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-2 \ln \left (\sqrt {x}-\frac {1}{2}+\sqrt {-\sqrt {x}+x}\right )\right )}{\sqrt {\sqrt {x}\, \left (-1+\sqrt {x}\right )}} \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}} {\left (\sqrt {x} + 1\right )}}\,{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 {x}}\,\left (\sqrt {x}+1\right )} \,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 {- \sqrt {x} + x} \left (\sqrt {x} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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