Optimal. Leaf size=38 \[ \frac{\sqrt{-b x^2} \sin ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]
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Rubi [A] time = 0.067233, antiderivative size = 38, 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 = {4834, 4641} \[ \frac{\sqrt{-b x^2} \sin ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
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Rule 4834
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sin ^{-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{\sin ^{-1}(x)^n}{\sqrt{1-x^2}} \, dx,x,\sqrt{1+b x^2}\right )}{b x}\\ &=\frac{\sqrt{-b x^2} \sin ^{-1}\left (\sqrt{1+b x^2}\right )^{1+n}}{b (1+n) x}\\ \end{align*}
Mathematica [A] time = 0.0450492, size = 38, normalized size = 1. \[ \frac{\sqrt{-b x^2} \sin ^{-1}\left (\sqrt{b x^2+1}\right )^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \arcsin \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27725, size = 105, normalized size = 2.76 \begin{align*} \frac{\sqrt{-b x^{2}} \arcsin \left (\sqrt{b x^{2} + 1}\right )^{n} \arcsin \left (\sqrt{b x^{2} + 1}\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{\arcsin \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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