Optimal. Leaf size=73 \[ \frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {1}{4} \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {280, 323, 330, 52} \begin {gather*} \frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {1}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {1}{4} \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 280
Rule 323
Rule 330
Rubi steps
\begin {align*} \int \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \, dx &=\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{4} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{8} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{4} \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 87, normalized size = 1.19 \begin {gather*} \frac {\sqrt {\sqrt {x}+1} \sqrt {x} \left (2 x^{3/2}-2 x-\sqrt {x}+1\right )+2 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\frac {\sqrt {1-\sqrt {x}}}{\sqrt {2}}\right )}{4 \sqrt {\sqrt {x}-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.48, size = 404, normalized size = 5.53 \begin {gather*} \frac {-4 \sqrt {\sqrt {x}+1} \left (-1136 x^{7/2}-1096 x^{5/2}+17296 x^{3/2}-4752 x^3+7240 x^2+55360 x+28224 \sqrt {x}-18816\right )-4 \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \left (-194 x^{7/2}-6079 x^{5/2}+6992 x^{3/2}-3120 x^3-104 x^2+38632 x+74488 \sqrt {x}+32592\right )+\sqrt {3} \left (-4 \sqrt {\sqrt {x}-1} \left (656 x^{7/2}+3832 x^{5/2}-10928 x^{3/2}+3408 x^3-1192 x^2-41472 x-52416 \sqrt {x}-18816\right )-4 \left (1800 x^{7/2}-1148 x^{5/2}-23268 x^{3/2}+112 x^4+3416 x^3-6678 x^2-41440 x-10872 \sqrt {x}+10864\right )\right )}{24960 x^{3/2}+\sqrt {3} \sqrt {\sqrt {x}+1} \left (-5248 x^{3/2}-22016 x-11264 \sqrt {x}+7168\right )+\sqrt {\sqrt {x}-1} \left (9088 x^{3/2}+\sqrt {3} \sqrt {\sqrt {x}+1} \left (-896 x^{3/2}-14400 x-28672 \sqrt {x}-12416\right )+47104 x+60416 \sqrt {x}+21504\right )+1552 x^2+49408 x+13312 \sqrt {x}-12416}+\tanh ^{-1}\left (\frac {\sqrt {\sqrt {x}-1}-1}{\sqrt {3}-\sqrt {\sqrt {x}+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 52, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, {\left (2 \, x - 1\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{8} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 92, normalized size = 1.26 \begin {gather*} \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, \sqrt {x} - 10\right )} {\left (\sqrt {x} + 1\right )} + 43\right )} {\left (\sqrt {x} + 1\right )} - 39\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{3} \, {\left ({\left (2 \, \sqrt {x} - 5\right )} {\left (\sqrt {x} + 1\right )} + 9\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \frac {1}{2} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (-2 \sqrt {x -1}\, x^{\frac {3}{2}}+\ln \left (\sqrt {x}+\sqrt {x -1}\right )+\sqrt {x -1}\, \sqrt {x}\right )}{4 \sqrt {x -1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 37, normalized size = 0.51 \begin {gather*} \frac {1}{2} \, {\left (x - 1\right )}^{\frac {3}{2}} \sqrt {x} + \frac {1}{4} \, \sqrt {x - 1} \sqrt {x} - \frac {1}{4} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \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 {x}\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+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 \sqrt {x} \sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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