Optimal. Leaf size=31 \[ \frac{\sqrt{-x-1} \sqrt{x}}{\sqrt{-x}}+x \text{csch}^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0076117, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6344, 12, 32} \[ \frac{\sqrt{-x-1} \sqrt{x}}{\sqrt{-x}}+x \text{csch}^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6344
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \text{csch}^{-1}\left (\sqrt{x}\right ) \, dx &=x \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{1}{2 \sqrt{-1-x}} \, dx}{\sqrt{-x}}\\ &=x \text{csch}^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x} \int \frac{1}{\sqrt{-1-x}} \, dx}{2 \sqrt{-x}}\\ &=\frac{\sqrt{-1-x} \sqrt{x}}{\sqrt{-x}}+x \text{csch}^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0094641, size = 24, normalized size = 0.77 \[ \sqrt{\frac{1}{x}+1} \sqrt{x}+x \text{csch}^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.113, size = 24, normalized size = 0.8 \begin{align*} x{\rm arccsch} \left (\sqrt{x}\right )+{(1+x){\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 = 1.00353, size = 24, normalized size = 0.77 \begin{align*} x \operatorname{arcsch}\left (\sqrt{x}\right ) + \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 = 2.57358, size = 89, normalized size = 2.87 \begin{align*} x \log \left (\frac{x \sqrt{\frac{x + 1}{x}} + \sqrt{x}}{x}\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 \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 \operatorname{arcsch}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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