Optimal. Leaf size=86 \[ -\frac {1}{8} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {3}{16} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {3}{16} \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5903, 12, 323, 330, 52} \[ -\frac {1}{8} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {3}{16} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {3}{16} \cosh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 323
Rule 330
Rule 5903
Rubi steps
\begin {align*} \int x \cosh ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {x^{3/2}}{2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{4} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {3}{16} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {3}{16} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {3}{32} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {3}{16} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {3}{16} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {3}{16} \cosh ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.86 \[ \frac {1}{16} \left (8 x^2 \cosh ^{-1}\left (\sqrt {x}\right )-\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} (2 x+3) \sqrt {x}-6 \tanh ^{-1}\left (\sqrt {\frac {\sqrt {x}-1}{\sqrt {x}+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.62, size = 35, normalized size = 0.41 \[ -\frac {1}{16} \, {\left (2 \, x + 3\right )} \sqrt {x - 1} \sqrt {x} + \frac {1}{16} \, {\left (8 \, x^{2} - 3\right )} \log \left (\sqrt {x - 1} + \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 55, normalized size = 0.64 \[ \frac {1}{2} \, x^{2} \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{16} \, {\left (2 \, x + 3\right )} \sqrt {x - 1} \sqrt {x} + \frac {3}{16} \, \log \left (-\sqrt {x - 1} + \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 65, normalized size = 0.76 \[ \frac {x^{2} \mathrm {arccosh}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {-1+x}+3 \sqrt {x}\, \sqrt {-1+x}+3 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{16 \sqrt {-1+x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 46, normalized size = 0.53 \[ \frac {1}{2} \, x^{2} \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{8} \, \sqrt {x - 1} x^{\frac {3}{2}} - \frac {3}{16} \, \sqrt {x - 1} \sqrt {x} - \frac {3}{16} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]
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 \[ \int x\,\mathrm {acosh}\left (\sqrt {x}\right ) \,d x \]
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 \[ \int x \operatorname {acosh}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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