Optimal. Leaf size=117 \[ -\frac{1}{18} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{5}{72} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{1}{3} x^3 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{5}{48} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{5}{48} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0704343, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5903, 12, 323, 330, 52} \[ -\frac{1}{18} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{5}{72} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{1}{3} x^3 \cosh ^{-1}\left (\sqrt{x}\right )-\frac{5}{48} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{5}{48} \cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5903
Rule 12
Rule 323
Rule 330
Rule 52
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} \operatorname{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*}
Mathematica [A] time = 0.0539353, size = 79, normalized size = 0.68 \[ \frac{1}{144} \left (-\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \left (8 x^2+10 x+15\right ) \sqrt{x}+48 x^3 \cosh ^{-1}\left (\sqrt{x}\right )-30 \tanh ^{-1}\left (\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.035, size = 75, normalized size = 0.6 \begin{align*}{\frac{{x}^{3}}{3}{\rm arccosh} \left (\sqrt{x}\right )}-{\frac{1}{144}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 8\,{x}^{5/2}\sqrt{-1+x}+10\,{x}^{3/2}\sqrt{-1+x}+15\,\sqrt{x}\sqrt{-1+x}+15\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02065, size = 76, normalized size = 0.65 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17917, size = 128, normalized size = 1.09 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{acosh}{\left (\sqrt{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16154, size = 82, normalized size = 0.7 \begin{align*} \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 ({\left | \sqrt{x - 1} - \sqrt{x} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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