Optimal. Leaf size=171 \[ \sqrt {-x-\sqrt {x}+1} \left (\frac {1}{4} \sqrt {\left (2 \sqrt {x}+1\right )^2}+\frac {2}{3}\right )-\frac {2}{3} \sqrt {-x-\sqrt {x}+1} x-\frac {8 \sqrt {1-\sqrt {x}}}{3}-\frac {4}{3} \sqrt {1-\sqrt {x}} \sqrt {x}-\frac {2}{3} \sqrt {-x-\sqrt {x}+1} \sqrt {x}+\frac {5}{8} i \log \left (-2 \sqrt {-x-\sqrt {x}+1}+i \sqrt {\left (2 \sqrt {x}+1\right )^2}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 102, normalized size of antiderivative = 0.60, number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {190, 43, 1341, 640, 612, 619, 216} \begin {gather*} \frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}-4 \sqrt {1-\sqrt {x}}+\frac {2}{3} \left (-x-\sqrt {x}+1\right )^{3/2}+\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}+\frac {5}{8} \sin ^{-1}\left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 216
Rule 612
Rule 619
Rule 640
Rule 1341
Rubi steps
\begin {align*} \int \left (\frac {1}{\sqrt {1-\sqrt {x}}}-\sqrt {1-\sqrt {x}-x}\right ) \, dx &=\int \frac {1}{\sqrt {1-\sqrt {x}}} \, dx-\int \sqrt {1-\sqrt {x}-x} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,\sqrt {x}\right )-2 \operatorname {Subst}\left (\int x \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+2 \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,\sqrt {x}\right )+\operatorname {Subst}\left (\int \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=-4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+\frac {5}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x-x^2}} \, dx,x,\sqrt {x}\right )\\ &=-4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {1}{8} \sqrt {5} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{5}}} \, dx,x,-1-2 \sqrt {x}\right )\\ &=-4 \sqrt {1-\sqrt {x}}+\frac {4}{3} \left (1-\sqrt {x}\right )^{3/2}+\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+\frac {5}{8} \sin ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 113, normalized size = 0.66 \begin {gather*} 2 \left (\frac {2}{3} \left (1-\sqrt {x}\right )^{3/2}-2 \sqrt {1-\sqrt {x}}\right )-2 \left (\frac {1}{2} \left (\frac {1}{4} \sqrt {-x-\sqrt {x}+1} \left (-2 \sqrt {x}-1\right )+\frac {5}{8} \sin ^{-1}\left (\frac {-2 \sqrt {x}-1}{\sqrt {5}}\right )\right )-\frac {1}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 86, normalized size = 0.50 \begin {gather*} -\frac {4}{3} \sqrt {1-\sqrt {x}} \left (2+\sqrt {x}\right )+\frac {1}{12} \left (11-2 \sqrt {x}-8 x\right ) \sqrt {1-\sqrt {x}-x}-\frac {5}{4} \tan ^{-1}\left (\frac {-1+\sqrt {1-\sqrt {x}-x}}{\sqrt {x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 100, normalized size = 0.58 \begin {gather*} -\frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 11\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {4}{3} \, {\left (\sqrt {x} + 2\right )} \sqrt {-\sqrt {x} + 1} - \frac {5}{16} \, \arctan \left (-\frac {{\left (8 \, x^{2} - {\left (16 \, x^{2} - 38 \, x + 11\right )} \sqrt {x} - 9 \, x + 3\right )} \sqrt {-x - \sqrt {x} + 1}}{4 \, {\left (4 \, x^{3} - 13 \, x^{2} + 7 \, x - 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 66, normalized size = 0.39 \begin {gather*} -\frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} - 11\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {-\sqrt {x} + 1} + \frac {5}{8} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (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 = 0.12, size = 72, normalized size = 0.42
method | result | size |
derivativedivides | \(\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}-4 \sqrt {1-\sqrt {x}}\) | \(72\) |
default | \(\frac {2 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{4}+\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}+\frac {4 \left (1-\sqrt {x}\right )^{\frac {3}{2}}}{3}-4 \sqrt {1-\sqrt {x}}\) | \(72\) |
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*} \frac {4}{3} \, {\left (-\sqrt {x} + 1\right )}^{\frac {3}{2}} - 4 \, \sqrt {-\sqrt {x} + 1} - \int \sqrt {-x - \sqrt {x} + 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 = -0.01 \begin {gather*} -\int \sqrt {1-\sqrt {x}-x}-\frac {1}{\sqrt {1-\sqrt {x}}} \,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 \left (- \frac {1}{\sqrt {1 - \sqrt {x}}}\right )\, dx - \int \sqrt {- \sqrt {x} - x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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