Optimal. Leaf size=64 \[ \frac{1}{2} x^2 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{(-x-1)^{3/2} \sqrt{x}}{6 \sqrt{-x}}-\frac{\sqrt{-x-1} \sqrt{x}}{2 \sqrt{-x}} \]
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Rubi [A] time = 0.017491, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6346, 12, 43} \[ \frac{1}{2} x^2 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{(-x-1)^{3/2} \sqrt{x}}{6 \sqrt{-x}}-\frac{\sqrt{-x-1} \sqrt{x}}{2 \sqrt{-x}} \]
Antiderivative was successfully verified.
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Rule 6346
Rule 12
Rule 43
Rubi steps
\begin{align*} \int x \text{csch}^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{x}{2 \sqrt{-1-x}} \, dx}{2 \sqrt{-x}}\\ &=\frac{1}{2} x^2 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{x}{\sqrt{-1-x}} \, dx}{4 \sqrt{-x}}\\ &=\frac{1}{2} x^2 \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \left (-\frac{1}{\sqrt{-1-x}}-\sqrt{-1-x}\right ) \, dx}{4 \sqrt{-x}}\\ &=-\frac{\sqrt{-1-x} \sqrt{x}}{2 \sqrt{-x}}-\frac{(-1-x)^{3/2} \sqrt{x}}{6 \sqrt{-x}}+\frac{1}{2} x^2 \text{csch}^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0238208, size = 35, normalized size = 0.55 \[ \frac{1}{2} x^2 \text{csch}^{-1}\left (\sqrt{x}\right )+\frac{1}{6} \sqrt{\frac{1}{x}+1} (x-2) \sqrt{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 31, normalized size = 0.5 \begin{align*}{\frac{{x}^{2}}{2}{\rm arccsch} \left (\sqrt{x}\right )}+{\frac{ \left ( 1+x \right ) \left ( x-2 \right ) }{6}{\frac{1}{\sqrt{{\frac{1+x}{x}}}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00056, size = 46, normalized size = 0.72 \begin{align*} \frac{1}{6} \, x^{\frac{3}{2}}{\left (\frac{1}{x} + 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, x^{2} \operatorname{arcsch}\left (\sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x} \sqrt{\frac{1}{x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56279, size = 113, normalized size = 1.77 \begin{align*} \frac{1}{2} \, x^{2} \log \left (\frac{x \sqrt{\frac{x + 1}{x}} + \sqrt{x}}{x}\right ) + \frac{1}{6} \,{\left (x - 2\right )} \sqrt{x} \sqrt{\frac{x + 1}{x}} \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 \operatorname{acsch}{\left (\sqrt{x} \right )}\, dx \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 x \operatorname{arcsch}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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