Optimal. Leaf size=29 \[ \frac{\sqrt{b x^2} \log \left (\sinh ^{-1}\left (\sqrt{b x^2-1}\right )\right )}{b x} \]
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Rubi [A] time = 0.0615313, antiderivative size = 29, 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 = {5894, 5673} \[ \frac{\sqrt{b x^2} \log \left (\sinh ^{-1}\left (\sqrt{b x^2-1}\right )\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 5894
Rule 5673
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+b x^2} \sinh ^{-1}\left (\sqrt{-1+b x^2}\right )} \, dx &=\frac{\sqrt{b x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \sinh ^{-1}(x)} \, dx,x,\sqrt{-1+b x^2}\right )}{b x}\\ &=\frac{\sqrt{b x^2} \log \left (\sinh ^{-1}\left (\sqrt{-1+b x^2}\right )\right )}{b x}\\ \end{align*}
Mathematica [A] time = 0.0208927, size = 24, normalized size = 0.83 \[ \frac{x \log \left (\sinh ^{-1}\left (\sqrt{b x^2-1}\right )\right )}{\sqrt{b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\it Arcsinh} \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{arsinh}\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 = 2.4263, size = 80, normalized size = 2.76 \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{asinh}{\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{arsinh}\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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