Optimal. Leaf size=87 \[ -\frac {2 \sqrt {x-\sqrt {x}}}{\sqrt {x}-1}+4 \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 123, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2056, 1549, 848, 98, 157, 54, 215, 93, 206} \begin {gather*} -\frac {2 \sqrt {x}}{\sqrt {x-\sqrt {x}}}+\frac {4 \sqrt {\sqrt {x}-1} \sqrt [4]{x} \sinh ^{-1}\left (\sqrt {\sqrt {x}-1}\right )}{\sqrt {x-\sqrt {x}}}-\frac {\sqrt {2} \sqrt {\sqrt {x}-1} \sqrt [4]{x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt {\sqrt {x}-1}}\right )}{\sqrt {x-\sqrt {x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 98
Rule 157
Rule 206
Rule 215
Rule 848
Rule 1549
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(-1+x) \sqrt {-\sqrt {x}+x}} \, dx &=\frac {\left (\sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \int \frac {\sqrt [4]{x}}{\sqrt {-1+\sqrt {x}} (-1+x)} \, dx}{\sqrt {-\sqrt {x}+x}}\\ &=\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^{3/2}}{\sqrt {-1+x} \left (-1+x^2\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}}\\ &=\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {x^{3/2}}{(-1+x)^{3/2} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}}\\ &=-\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-x}{\sqrt {-1+x} \sqrt {x} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}}\\ &=-\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (\sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {x} (1+x)} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}}+\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {x}} \, dx,x,\sqrt {x}\right )}{\sqrt {-\sqrt {x}+x}}\\ &=-\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}-\frac {\left (2 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt {-1+\sqrt {x}}}\right )}{\sqrt {-\sqrt {x}+x}}+\frac {\left (4 \sqrt {-1+\sqrt {x}} \sqrt [4]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {-1+\sqrt {x}}\right )}{\sqrt {-\sqrt {x}+x}}\\ &=-\frac {2 \sqrt {x}}{\sqrt {-\sqrt {x}+x}}+\frac {4 \sqrt {-1+\sqrt {x}} \sqrt [4]{x} \sinh ^{-1}\left (\sqrt {-1+\sqrt {x}}\right )}{\sqrt {-\sqrt {x}+x}}-\frac {\sqrt {2} \sqrt {-1+\sqrt {x}} \sqrt [4]{x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x}}{\sqrt {-1+\sqrt {x}}}\right )}{\sqrt {-\sqrt {x}+x}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 97, normalized size = 1.11 \begin {gather*} -\frac {\sqrt [4]{x} \left (2 \sqrt [4]{x}+4 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\sqrt {1-\sqrt {x}}\right )+\sqrt {2} \sqrt {\sqrt {x}-1} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}}{\sqrt {2} \sqrt [4]{x}}\right )\right )}{\sqrt {x-\sqrt {x}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 87, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}+4 \tanh ^{-1}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.12, size = 132, normalized size = 1.52 \begin {gather*} \frac {\sqrt {2} {\left (x - 1\right )} \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 ) + 4 \, {\left (x - 1\right )} \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) - 8 \, \sqrt {x - \sqrt {x}} {\left (\sqrt {x} + 1\right )}}{4 \, {\left (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 = 96, normalized size = 1.10 \begin {gather*} -\frac {1}{2} \, \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 ) - \frac {2}{\sqrt {x - \sqrt {x}} - \sqrt {x} + 1} - 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 = 0.11, size = 76, normalized size = 0.87
method | result | size |
derivativedivides | \(2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (1+\sqrt {x}\right )^{2}-3 \sqrt {x}-1}}\right )}{2}-\frac {2 \sqrt {\left (-1+\sqrt {x}\right )^{2}+\sqrt {x}-1}}{-1+\sqrt {x}}\) | \(76\) |
default | \(\frac {\sqrt {-\sqrt {x}+x}\, \left (2 \sqrt {2}\, \sqrt {x}\, \arctanh \left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-\sqrt {2}\, x \arctanh \left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-4 \left (-\sqrt {x}+x \right )^{\frac {3}{2}}-\sqrt {2}\, \arctanh \left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-8 \sqrt {x}\, \sqrt {-\sqrt {x}+x}-8 \sqrt {x}\, \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+4 x \sqrt {-\sqrt {x}+x}+4 x \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+4 \sqrt {-\sqrt {x}+x}+4 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )\right )}{2 \sqrt {\sqrt {x}\, \left (-1+\sqrt {x}\right )}\, \left (-1+\sqrt {x}\right )^{2}}\) | \(219\) |
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 {\sqrt {x}}{\sqrt {x - \sqrt {x}} {\left (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 = -0.01 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}\,\left (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 {\sqrt {x}}{\sqrt {- \sqrt {x} + x} \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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