Optimal. Leaf size=35 \[ -\frac{1}{2} \sqrt{x} \sqrt{x+1}+x \sinh ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} \sinh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0107196, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {5900, 12, 1958, 50, 54, 215} \[ -\frac{1}{2} \sqrt{x} \sqrt{x+1}+x \sinh ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} \sinh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5900
Rule 12
Rule 1958
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int \sinh ^{-1}\left (\sqrt{x}\right ) \, dx &=x \sinh ^{-1}\left (\sqrt{x}\right )-\int \frac{1}{2} \sqrt{\frac{x}{1+x}} \, dx\\ &=x \sinh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \sqrt{\frac{x}{1+x}} \, dx\\ &=x \sinh ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \int \frac{\sqrt{x}}{\sqrt{1+x}} \, dx\\ &=-\frac{1}{2} \sqrt{x} \sqrt{1+x}+x \sinh ^{-1}\left (\sqrt{x}\right )+\frac{1}{4} \int \frac{1}{\sqrt{x} \sqrt{1+x}} \, dx\\ &=-\frac{1}{2} \sqrt{x} \sqrt{1+x}+x \sinh ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{2} \sqrt{x} \sqrt{1+x}+\frac{1}{2} \sinh ^{-1}\left (\sqrt{x}\right )+x \sinh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0375523, size = 33, normalized size = 0.94 \[ \frac{1}{2} \left ((2 x+1) \sinh ^{-1}\left (\sqrt{x}\right )-\sqrt{\frac{x}{x+1}} (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 24, normalized size = 0.7 \begin{align*}{\frac{1}{2}{\it Arcsinh} \left ( \sqrt{x} \right ) }+x{\it Arcsinh} \left ( \sqrt{x} \right ) -{\frac{1}{2}\sqrt{x}\sqrt{1+x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75239, size = 31, normalized size = 0.89 \begin{align*} x \operatorname{arsinh}\left (\sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x + 1} \sqrt{x} + \frac{1}{2} \, \operatorname{arsinh}\left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61349, size = 92, normalized size = 2.63 \begin{align*} \frac{1}{2} \,{\left (2 \, x + 1\right )} \log \left (\sqrt{x + 1} + \sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x + 1} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.359289, size = 29, normalized size = 0.83 \begin{align*} - \frac{\sqrt{x} \sqrt{x + 1}}{2} + x \operatorname{asinh}{\left (\sqrt{x} \right )} + \frac{\operatorname{asinh}{\left (\sqrt{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.57915, size = 54, normalized size = 1.54 \begin{align*} x \log \left (\sqrt{x + 1} + \sqrt{x}\right ) - \frac{1}{2} \, \sqrt{x^{2} + x} - \frac{1}{4} \, \log \left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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