Optimal. Leaf size=29 \[ \frac {\sqrt {b x^2} \log \left (\sinh ^{-1}\left (\sqrt {-1+b x^2}\right )\right )}{b x} \]
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Rubi [A]
time = 0.04, antiderivative size = 29, 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 = {5871, 5782}
\begin {gather*} \frac {\sqrt {b x^2} \log \left (\sinh ^{-1}\left (\sqrt {b x^2-1}\right )\right )}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 5782
Rule 5871
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} \text {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*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.83 \begin {gather*} \frac {x \log \left (\sinh ^{-1}\left (\sqrt {-1+b x^2}\right )\right )}{\sqrt {b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\arcsinh \left (\sqrt {x^{2} b -1}\right ) \sqrt {x^{2} b -1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 33, normalized size = 1.14 \begin {gather*} \frac {\sqrt {b x^{2}} \log \left (\log \left (\sqrt {b x^{2} - 1} + \sqrt {b x^{2}}\right )\right )}{b x} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {b x^{2} - 1} \operatorname {asinh}{\left (\sqrt {b x^{2} - 1} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 23, normalized size = 0.79 \begin {gather*} \frac {\ln \left (\mathrm {asinh}\left (\sqrt {b\,x^2-1}\right )\right )\,\sqrt {x^2}}{\sqrt {b}\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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