Optimal. Leaf size=50 \[ -\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{2} \cosh ^{-1}\left (\sqrt {x}\right )+x \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6016, 12, 329,
336, 54} \begin {gather*} -\frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+x \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 329
Rule 336
Rule 6016
Rubi steps
\begin {align*} \int \cosh ^{-1}\left (\sqrt {x}\right ) \, dx &=x \cosh ^{-1}\left (\sqrt {x}\right )-\int \frac {\sqrt {x}}{2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=x \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+x \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+x \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{2} \cosh ^{-1}\left (\sqrt {x}\right )+x \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(273\) vs. \(2(50)=100\).
time = 3.21, size = 273, normalized size = 5.46 \begin {gather*} -\frac {2 \left (4 \sqrt {1+\sqrt {x}} \left (-12-24 \sqrt {x}+x+5 x^{3/2}\right )+\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \left (-84-10 \sqrt {x}+28 x+7 x^{3/2}\right )+\sqrt {3} \left (28+70 \sqrt {x}+18 x-14 x^{3/2}-4 x^2-4 \sqrt {-1+\sqrt {x}} \left (-12-8 \sqrt {x}+5 x+3 x^{3/2}\right )\right )\right )}{56-16 \sqrt {3} \sqrt {1+\sqrt {x}} \left (2+3 \sqrt {x}\right )+\sqrt {-1+\sqrt {x}} \left (96-8 \sqrt {3} \sqrt {1+\sqrt {x}} \left (7+2 \sqrt {x}\right )+80 \sqrt {x}\right )+112 \sqrt {x}+28 x}+x \cosh ^{-1}\left (\sqrt {x}\right )+2 \tanh ^{-1}\left (\frac {-1+\sqrt {-1+\sqrt {x}}}{\sqrt {3}-\sqrt {1+\sqrt {x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 49, normalized size = 0.98
method | result | size |
derivativedivides | \(x \,\mathrm {arccosh}\left (\sqrt {x}\right )-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{2 \sqrt {-1+x}}\) | \(49\) |
default | \(x \,\mathrm {arccosh}\left (\sqrt {x}\right )-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (\sqrt {x}\, \sqrt {-1+x}+\ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{2 \sqrt {-1+x}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 33, normalized size = 0.66 \begin {gather*} x \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} - \frac {1}{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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Fricas [A]
time = 0.35, size = 28, normalized size = 0.56 \begin {gather*} \frac {1}{2} \, {\left (2 \, x - 1\right )} \log \left (\sqrt {x - 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 29, normalized size = 0.58 \begin {gather*} - \frac {\sqrt {x} \sqrt {x - 1}}{2} + x \operatorname {acosh}{\left (\sqrt {x} \right )} - \frac {\operatorname {acosh}{\left (\sqrt {x} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 47, normalized size = 0.94 \begin {gather*} x \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{2} \, \sqrt {x - 1} \sqrt {x} + \frac {1}{2} \, \log \left (-\sqrt {x - 1} + \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 40, normalized size = 0.80 \begin {gather*} -2\,\sqrt {x}\,\mathrm {acosh}\left (\sqrt {x}\right )\,\left (\frac {1}{4\,\sqrt {x}}-\frac {\sqrt {x}}{2}\right )-\frac {\sqrt {x}\,\sqrt {\sqrt {x}-1}\,\sqrt {\sqrt {x}+1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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