Optimal. Leaf size=31 \[ -\frac{\sqrt{-b x^2} \log \left (\cos ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{b x} \]
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Rubi [A] time = 0.0608216, antiderivative size = 31, 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 = {4835, 4640} \[ -\frac{\sqrt{-b x^2} \log \left (\cos ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 4835
Rule 4640
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+b x^2} \cos ^{-1}\left (\sqrt{1+b x^2}\right )} \, dx &=\frac{\sqrt{-b x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \cos ^{-1}(x)} \, dx,x,\sqrt{1+b x^2}\right )}{b x}\\ &=-\frac{\sqrt{-b x^2} \log \left (\cos ^{-1}\left (\sqrt{1+b x^2}\right )\right )}{b x}\\ \end{align*}
Mathematica [A] time = 0.0239597, size = 25, normalized size = 0.81 \[ \frac{x \log \left (\cos ^{-1}\left (\sqrt{b x^2+1}\right )\right )}{\sqrt{-b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arccos \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} \arccos \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 = 2.57099, size = 68, normalized size = 2.19 \begin{align*} -\frac{\sqrt{-b x^{2}} \log \left (\arccos \left (\sqrt{b x^{2} + 1}\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{acos}{\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} \arccos \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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