Optimal. Leaf size=54 \[ \frac{\sqrt{\sqrt{b x^2+1}-1} \sqrt{\sqrt{b x^2+1}+1} \log \left (\cosh ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{b x} \]
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Rubi [A] time = 0.108626, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5895, 5674} \[ \frac{\sqrt{\sqrt{b x^2+1}-1} \sqrt{\sqrt{b x^2+1}+1} \log \left (\cosh ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 5895
Rule 5674
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+b x^2} \cosh ^{-1}\left (\sqrt{1+b x^2}\right )} \, dx &=\frac{\left (\sqrt{-1+\sqrt{1+b x^2}} \sqrt{1+\sqrt{1+b x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x} \cosh ^{-1}(x)} \, dx,x,\sqrt{1+b x^2}\right )}{b x}\\ &=\frac{\sqrt{-1+\sqrt{1+b x^2}} \sqrt{1+\sqrt{1+b x^2}} \log \left (\cosh ^{-1}\left (\sqrt{1+b x^2}\right )\right )}{b x}\\ \end{align*}
Mathematica [A] time = 0.0894676, size = 54, normalized size = 1. \[ \frac{\sqrt{\sqrt{b x^2+1}-1} \sqrt{\sqrt{b x^2+1}+1} \log \left (\cosh ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{b x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\rm arccosh} \left (\sqrt{b{x}^{2}+1}\right ) \right ) ^{-1}{\frac{1}{\sqrt{b{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} + 1} \operatorname{arcosh}\left (\sqrt{b x^{2} + 1}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12005, size = 80, normalized size = 1.48 \begin{align*} \frac{\sqrt{b x^{2}} \log \left (\log \left (\sqrt{b x^{2} + 1} + \sqrt{b x^{2}}\right )\right )}{b 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 \frac{1}{\sqrt{b x^{2} + 1} \operatorname{acosh}{\left (\sqrt{b x^{2} + 1} \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 \frac{1}{\sqrt{b x^{2} + 1} \operatorname{arcosh}\left (\sqrt{b x^{2} + 1}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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