Optimal. Leaf size=37 \[ \frac {\sqrt {b x^2} \sinh ^{-1}\left (\sqrt {b x^2-1}\right )^{n+1}}{b (n+1) x} \]
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Rubi [A] time = 0.07, antiderivative size = 37, 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, 5675} \[ \frac {\sqrt {b x^2} \sinh ^{-1}\left (\sqrt {b x^2-1}\right )^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
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Rule 5675
Rule 5894
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}\left (\sqrt {-1+b x^2}\right )^n}{\sqrt {-1+b x^2}} \, dx &=\frac {\sqrt {b x^2} \operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)^n}{\sqrt {1+x^2}} \, dx,x,\sqrt {-1+b x^2}\right )}{b x}\\ &=\frac {\sqrt {b x^2} \sinh ^{-1}\left (\sqrt {-1+b x^2}\right )^{1+n}}{b (1+n) x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 37, normalized size = 1.00 \[ \frac {\sqrt {b x^2} \sinh ^{-1}\left (\sqrt {b x^2-1}\right )^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 108, normalized size = 2.92 \[ \frac {\sqrt {b x^{2}} \cosh \left (n \log \left (\log \left (\sqrt {b x^{2} - 1} + \sqrt {b x^{2}}\right )\right )\right ) \log \left (\sqrt {b x^{2} - 1} + \sqrt {b x^{2}}\right ) + \sqrt {b x^{2}} \log \left (\sqrt {b x^{2} - 1} + \sqrt {b x^{2}}\right ) \sinh \left (n \log \left (\log \left (\sqrt {b x^{2} - 1} + \sqrt {b x^{2}}\right )\right )\right )}{{\left (b n + b\right )} x} \]
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 = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {\arcsinh \left (\sqrt {b \,x^{2}-1}\right )^{n}}{\sqrt {b \,x^{2}-1}}\, dx \]
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 \[ \int \frac {\operatorname {arsinh}\left (\sqrt {b x^{2} - 1}\right )^{n}}{\sqrt {b x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {asinh}\left (\sqrt {b\,x^2-1}\right )}^n}{\sqrt {b\,x^2-1}} \,d x \]
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 \[ \begin {cases} - \frac {2 x}{\pi } & \text {for}\: b = 0 \wedge n = -1 \\- i x \left (\frac {i \pi }{2}\right )^{n} & \text {for}\: b = 0 \\\int \frac {1}{\sqrt {b x^{2} - 1} \operatorname {asinh}{\left (\sqrt {b x^{2} - 1} \right )}}\, dx & \text {for}\: n = -1 \\\frac {\sqrt {b} \sqrt {x^{2}} \operatorname {asinh}{\left (\sqrt {b x^{2} - 1} \right )} \operatorname {asinh}^{n}{\left (\sqrt {b x^{2} - 1} \right )}}{b n x + b x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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