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 = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 5674
Rule 5895
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*}
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Mathematica [A] time = 0.09, size = 54, normalized size = 1.00 \[ \frac {\sqrt {\sqrt {b x^2+1}-1} \sqrt {\sqrt {b x^2+1}+1} \log \left (\cosh ^{-1}\left (\sqrt {b x^2+1}\right )\right )}{b x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.59, size = 33, normalized size = 0.61 \[ \frac {\sqrt {b x^{2}} \log \left (\log \left (\sqrt {b x^{2} + 1} + \sqrt {b x^{2}}\right )\right )}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\mathrm {arccosh}\left (\sqrt {b \,x^{2}+1}\right ) \sqrt {b \,x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {arcosh}\left (\sqrt {b x^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\mathrm {acosh}\left (\sqrt {b\,x^2+1}\right )\,\sqrt {b\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {acosh}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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