Optimal. Leaf size=67 \[ \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{\sqrt {x}}-2 \sqrt {\sqrt {x}-1} \sqrt {x} \sqrt {\sqrt {x}+1}+2 \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 67, 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 = {327, 280, 330, 52} \begin {gather*} \frac {2 \left (\sqrt {x}-1\right )^{3/2} \left (\sqrt {x}+1\right )^{3/2}}{\sqrt {x}}-2 \sqrt {\sqrt {x}-1} \sqrt {x} \sqrt {\sqrt {x}+1}+2 \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 280
Rule 327
Rule 330
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{x^{3/2}} \, dx &=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{\sqrt {x}}-2 \int \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx\\ &=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{\sqrt {x}}-2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{\sqrt {x}}-2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \left (-1+\sqrt {x}\right )^{3/2} \left (1+\sqrt {x}\right )^{3/2}}{\sqrt {x}}-2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+2 \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 74, normalized size = 1.10 \begin {gather*} \frac {2 \left (-\frac {\sqrt {\sqrt {x}+1} \left (\sqrt {x}-1\right )}{\sqrt {x}}-2 \sqrt {1-\sqrt {x}} \sin ^{-1}\left (\frac {\sqrt {1-\sqrt {x}}}{\sqrt {2}}\right )\right )}{\sqrt {\sqrt {x}-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.13, size = 184, normalized size = 2.75 \begin {gather*} \frac {\left (\sqrt {\sqrt {x}-1}-1\right ) \left (\sqrt {3}-\sqrt {\sqrt {x}+1}\right ) \left (\sqrt {\sqrt {x}-1}+\sqrt {3} \sqrt {\sqrt {x}+1}-\sqrt {x}-2\right )}{\left (\sqrt {3} \sqrt {\sqrt {x}+1} \sqrt {\sqrt {x}-1}-2 \sqrt {\sqrt {x}-1}+2 \sqrt {3} \sqrt {\sqrt {x}+1}-2 \sqrt {x}-3\right ) \sqrt {x}}-8 \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 = 55, normalized size = 0.82 \begin {gather*} -\frac {x \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) + 2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + 2 \, x}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 47, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {\sqrt {x}-1}\, \sqrt {\sqrt {x}+1}\, \left (\sqrt {x}\, \ln \left (\sqrt {x}+\sqrt {x -1}\right )-\sqrt {x -1}\right )}{\sqrt {x -1}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 27, normalized size = 0.40 \begin {gather*} -\frac {2 \, \sqrt {x - 1}}{\sqrt {x}} + 2 \, \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 [B] time = 6.25, size = 129, normalized size = 1.93 \begin {gather*} 8\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )-\frac {\frac {5\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{2\,{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}+\frac {1}{2}}{\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}}-\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{2\,\left (\sqrt {\sqrt {x}+1}-1\right )} \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 {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}{x^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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