Optimal. Leaf size=42 \[ -\frac{1}{6 x^{3/2}}-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0147932, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6098, 51, 63, 206} \[ -\frac{1}{6 x^{3/2}}-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2 \sqrt{x}}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6098
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}\left (\sqrt{x}\right )}{x^3} \, dx &=-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{4} \int \frac{1}{(1-x) x^{5/2}} \, dx\\ &=-\frac{1}{6 x^{3/2}}-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{4} \int \frac{1}{(1-x) x^{3/2}} \, dx\\ &=-\frac{1}{6 x^{3/2}}-\frac{1}{2 \sqrt{x}}-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{4} \int \frac{1}{(1-x) \sqrt{x}} \, dx\\ &=-\frac{1}{6 x^{3/2}}-\frac{1}{2 \sqrt{x}}-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{6 x^{3/2}}-\frac{1}{2 \sqrt{x}}-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0191363, size = 58, normalized size = 1.38 \[ -\frac{1}{6 x^{3/2}}-\frac{\coth ^{-1}\left (\sqrt{x}\right )}{2 x^2}-\frac{1}{2 \sqrt{x}}-\frac{1}{4} \log \left (1-\sqrt{x}\right )+\frac{1}{4} \log \left (\sqrt{x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 37, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\rm arccoth} \left (\sqrt{x}\right )}-{\frac{1}{4}\ln \left ( -1+\sqrt{x} \right ) }-{\frac{1}{6}{x}^{-{\frac{3}{2}}}}-{\frac{1}{2}{\frac{1}{\sqrt{x}}}}+{\frac{1}{4}\ln \left ( 1+\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985373, size = 49, normalized size = 1.17 \begin{align*} -\frac{3 \, x + 1}{6 \, x^{\frac{3}{2}}} - \frac{\operatorname{arcoth}\left (\sqrt{x}\right )}{2 \, x^{2}} + \frac{1}{4} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{4} \, \log \left (\sqrt{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59093, size = 107, normalized size = 2.55 \begin{align*} \frac{3 \,{\left (x^{2} - 1\right )} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\right ) - 2 \,{\left (3 \, x + 1\right )} \sqrt{x}}{12 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.8067, size = 160, normalized size = 3.81 \begin{align*} \frac{3 x^{\frac{7}{2}} \operatorname{acoth}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} - 6 x^{\frac{5}{2}}} - \frac{3 x^{\frac{5}{2}} \operatorname{acoth}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} - 6 x^{\frac{5}{2}}} - \frac{3 x^{\frac{3}{2}} \operatorname{acoth}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} - 6 x^{\frac{5}{2}}} + \frac{3 \sqrt{x} \operatorname{acoth}{\left (\sqrt{x} \right )}}{6 x^{\frac{7}{2}} - 6 x^{\frac{5}{2}}} - \frac{3 x^{3}}{6 x^{\frac{7}{2}} - 6 x^{\frac{5}{2}}} + \frac{2 x^{2}}{6 x^{\frac{7}{2}} - 6 x^{\frac{5}{2}}} + \frac{x}{6 x^{\frac{7}{2}} - 6 x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (\sqrt{x}\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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