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 = 0.06, 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 = {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 5673
Rule 5894
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.48, size = 33, normalized size = 1.14 \[ \frac {\sqrt {b x^{2}} \log \left (\log \left (\sqrt {b x^{2} - 1} + \sqrt {b x^{2}}\right )\right )}{b x} \]
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 = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {1}{\arcsinh \left (\sqrt {b \,x^{2}-1}\right ) \sqrt {b \,x^{2}-1}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b x^{2} - 1} \operatorname {arsinh}\left (\sqrt {b x^{2} - 1}\right )}\,{d x} \]
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 \[ \frac {\ln \left (\mathrm {asinh}\left (\sqrt {b\,x^2-1}\right )\right )\,\sqrt {x^2}}{\sqrt {b}\,x} \]
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 \[ \int \frac {1}{\sqrt {b x^{2} - 1} \operatorname {asinh}{\left (\sqrt {b x^{2} - 1} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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