Optimal. Leaf size=117 \[ -\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}-\frac {5}{48} \cosh ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6017, 12, 329,
336, 54} \begin {gather*} -\frac {1}{18} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{72} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}-\frac {5}{48} \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^2 \cosh ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \int \frac {x^{5/2}}{2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^{5/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{36} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=-\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{96} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=-\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )-\frac {5}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}-\frac {5}{48} \cosh ^{-1}\left (\sqrt {x}\right )+\frac {1}{3} x^3 \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 79, normalized size = 0.68 \begin {gather*} \frac {1}{144} \left (-\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \left (15+10 x+8 x^2\right )+48 x^3 \cosh ^{-1}\left (\sqrt {x}\right )-30 \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 = 0.04, size = 75, normalized size = 0.64
method | result | size |
derivativedivides | \(\frac {x^{3} \mathrm {arccosh}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (8 \sqrt {-1+x}\, x^{\frac {5}{2}}+10 x^{\frac {3}{2}} \sqrt {-1+x}+15 \sqrt {x}\, \sqrt {-1+x}+15 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{144 \sqrt {-1+x}}\) | \(75\) |
default | \(\frac {x^{3} \mathrm {arccosh}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (8 \sqrt {-1+x}\, x^{\frac {5}{2}}+10 x^{\frac {3}{2}} \sqrt {-1+x}+15 \sqrt {x}\, \sqrt {-1+x}+15 \ln \left (\sqrt {x}+\sqrt {-1+x}\right )\right )}{144 \sqrt {-1+x}}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 56, normalized size = 0.48 \begin {gather*} \frac {1}{3} \, x^{3} \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{18} \, \sqrt {x - 1} x^{\frac {5}{2}} - \frac {5}{72} \, \sqrt {x - 1} x^{\frac {3}{2}} - \frac {5}{48} \, \sqrt {x - 1} \sqrt {x} - \frac {5}{48} \, \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 = 0.36, size = 40, normalized size = 0.34 \begin {gather*} -\frac {1}{144} \, {\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt {x - 1} \sqrt {x} + \frac {1}{48} \, {\left (16 \, x^{3} - 5\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^{2} \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 = 1.59, size = 60, normalized size = 0.51 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{144} \, {\left (2 \, {\left (4 \, x + 5\right )} x + 15\right )} \sqrt {x - 1} \sqrt {x} + \frac {5}{48} \, \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^2\,\mathrm {acosh}\left (\sqrt {x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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