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 = 0.0665338, antiderivative size = 37, 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 = {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 5894
Rule 5675
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*}
Mathematica [A] time = 0.0396998, size = 37, normalized size = 1. \[ \frac{\sqrt{b x^2} \sinh ^{-1}\left (\sqrt{b x^2-1}\right )^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.177, size = 0, normalized size = 0. \begin{align*} \int{ \left ({\it Arcsinh} \left ( \sqrt{b{x}^{2}-1} \right ) \right ) ^{n}{\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{\operatorname{arsinh}\left (\sqrt{b x^{2} - 1}\right )^{n}}{\sqrt{b x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.78902, size = 282, normalized size = 7.62 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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{\operatorname{arsinh}\left (\sqrt{b x^{2} - 1}\right )^{n}}{\sqrt{b x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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