Integrand size = 10, antiderivative size = 117 \[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=-\frac {5}{48} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}-\frac {5}{72} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}-\frac {1}{18} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{5/2}-\frac {5 \text {arccosh}\left (\sqrt {x}\right )}{48}+\frac {1}{3} x^3 \text {arccosh}\left (\sqrt {x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6017, 12, 329, 336, 54} \[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} x^3 \text {arccosh}\left (\sqrt {x}\right )-\frac {5 \text {arccosh}\left (\sqrt {x}\right )}{48}-\frac {1}{18} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{5/2}-\frac {5}{72} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}-\frac {5}{48} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x} \]
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Rule 12
Rule 54
Rule 329
Rule 336
Rule 6017
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\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 \text {arccosh}\left (\sqrt {x}\right )}{48}+\frac {1}{3} x^3 \text {arccosh}\left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{144} \left (-\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x} \left (15+10 x+8 x^2\right )+48 x^3 \text {arccosh}\left (\sqrt {x}\right )-30 \text {arctanh}\left (\sqrt {\frac {-1+\sqrt {x}}{1+\sqrt {x}}}\right )\right ) \]
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {x^{3} \operatorname {arccosh}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (8 \sqrt {x -1}\, x^{\frac {5}{2}}+10 x^{\frac {3}{2}} \sqrt {x -1}+15 \sqrt {x}\, \sqrt {x -1}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{144 \sqrt {x -1}}\) | \(75\) |
default | \(\frac {x^{3} \operatorname {arccosh}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (8 \sqrt {x -1}\, x^{\frac {5}{2}}+10 x^{\frac {3}{2}} \sqrt {x -1}+15 \sqrt {x}\, \sqrt {x -1}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{144 \sqrt {x -1}}\) | \(75\) |
parts | \(\frac {x^{3} \operatorname {arccosh}\left (\sqrt {x}\right )}{3}-\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}\, \left (8 \sqrt {x -1}\, x^{\frac {5}{2}}+10 x^{\frac {3}{2}} \sqrt {x -1}+15 \sqrt {x}\, \sqrt {x -1}+15 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{144 \sqrt {x -1}}\) | \(75\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.34 \[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=-\frac {1}{144} \, {\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt {x - 1} \sqrt {x} + \frac {1}{48} \, {\left (16 \, x^{3} - 5\right )} \log \left (\sqrt {x - 1} + \sqrt {x}\right ) \]
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\[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=\int x^{2} \operatorname {acosh}{\left (\sqrt {x} \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.48 \[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcosh}\left (\sqrt {x}\right ) - \frac {1}{18} \, \sqrt {x - 1} x^{\frac {5}{2}} - \frac {5}{72} \, \sqrt {x - 1} x^{\frac {3}{2}} - \frac {5}{48} \, \sqrt {x - 1} \sqrt {x} - \frac {5}{48} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \]
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Time = 0.49 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.51 \[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=\frac {1}{3} \, x^{3} \log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right ) - \frac {1}{144} \, {\left (2 \, {\left (4 \, x + 5\right )} x + 15\right )} \sqrt {x - 1} \sqrt {x} + \frac {5}{48} \, \log \left (-\sqrt {x - 1} + \sqrt {x}\right ) \]
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Timed out. \[ \int x^2 \text {arccosh}\left (\sqrt {x}\right ) \, dx=\int x^2\,\mathrm {acosh}\left (\sqrt {x}\right ) \,d x \]
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