Optimal. Leaf size=51 \[ \frac {x^{5/2}}{15}+\frac {x^{3/2}}{9}+\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x}}{3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6098, 50, 63, 206} \[ \frac {x^{5/2}}{15}+\frac {x^{3/2}}{9}+\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )+\frac {\sqrt {x}}{3}-\frac {1}{3} \tanh ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 6098
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^{5/2}}{1-x} \, dx\\ &=\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {x^{3/2}}{1-x} \, dx\\ &=\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {\sqrt {x}}{1-x} \, dx\\ &=\frac {\sqrt {x}}{3}+\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{6} \int \frac {1}{(1-x) \sqrt {x}} \, dx\\ &=\frac {\sqrt {x}}{3}+\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {x}}{3}+\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15}+\frac {1}{3} x^3 \coth ^{-1}\left (\sqrt {x}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 1.16 \[ \frac {1}{90} \left (6 x^{5/2}+10 x^{3/2}+30 x^3 \coth ^{-1}\left (\sqrt {x}\right )+30 \sqrt {x}+15 \log \left (1-\sqrt {x}\right )-15 \log \left (\sqrt {x}+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 38, normalized size = 0.75 \[ \frac {1}{6} \, {\left (x^{3} - 1\right )} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{45} \, {\left (3 \, x^{2} + 5 \, x + 15\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 0.82 \[ \frac {x^{3} \mathrm {arccoth}\left (\sqrt {x}\right )}{3}+\frac {x^{\frac {5}{2}}}{15}+\frac {x^{\frac {3}{2}}}{9}+\frac {\sqrt {x}}{3}+\frac {\ln \left (-1+\sqrt {x}\right )}{6}-\frac {\ln \left (1+\sqrt {x}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 41, normalized size = 0.80 \[ \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{15} \, x^{\frac {5}{2}} + \frac {1}{9} \, x^{\frac {3}{2}} + \frac {1}{3} \, \sqrt {x} - \frac {1}{6} \, \log \left (\sqrt {x} + 1\right ) + \frac {1}{6} \, \log \left (\sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 31, normalized size = 0.61 \[ \frac {x^3\,\mathrm {acoth}\left (\sqrt {x}\right )}{3}-\frac {\mathrm {acoth}\left (\sqrt {x}\right )}{3}+\frac {\sqrt {x}}{3}+\frac {x^{3/2}}{9}+\frac {x^{5/2}}{15} \]
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 \[ \int x^{2} \operatorname {acoth}{\left (\sqrt {x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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