Optimal. Leaf size=86 \[ -\frac {3}{16} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {1}{8} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {3}{16} \cosh ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} x^2 \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.03, 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 = {6017, 12, 329,
336, 54} \begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 329
Rule 336
Rule 6017
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} \text {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 = 0.03, size = 74, normalized size = 0.86 \begin {gather*} \frac {1}{16} \left (-\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} (3+2 x)+8 x^2 \cosh ^{-1}\left (\sqrt {x}\right )-6 \tanh ^{-1}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.01, size = 65, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {x^{2} \mathrm {arccosh}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {-1+x}+3 \sqrt {x}\, \sqrt {-1+x}+3 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{16 \sqrt {-1+x}}\) | \(65\) |
default | \(\frac {x^{2} \mathrm {arccosh}\left (\sqrt {x}\right )}{2}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (2 x^{\frac {3}{2}} \sqrt {-1+x}+3 \sqrt {x}\, \sqrt {-1+x}+3 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{16 \sqrt {-1+x}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 46, normalized size = 0.53 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 35, normalized size = 0.41 \begin {gather*} -\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 ) \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 x \operatorname {acosh}{\left (\sqrt {x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.58, size = 55, normalized size = 0.64 \begin {gather*} \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 ) \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 x\,\mathrm {acosh}\left (\sqrt {x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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