Optimal. Leaf size=47 \[ \frac{\sqrt{1-\left (a+b x^n\right )^2}}{b n}+\frac{\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n} \]
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Rubi [A] time = 0.0534496, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6715, 4803, 4619, 261} \[ \frac{\sqrt{1-\left (a+b x^n\right )^2}}{b n}+\frac{\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 4803
Rule 4619
Rule 261
Rubi steps
\begin{align*} \int x^{-1+n} \sin ^{-1}\left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \sin ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \sin ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac{\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2}} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac{\sqrt{1-\left (a+b x^n\right )^2}}{b n}+\frac{\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0348918, size = 47, normalized size = 1. \[ \frac{\sqrt{1-\left (a+b x^n\right )^2}}{b n}+\frac{\left (a+b x^n\right ) \sin ^{-1}\left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{x}^{n-1}\arcsin \left ( a+b{x}^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46805, size = 53, normalized size = 1.13 \begin{align*} \frac{{\left (b x^{n} + a\right )} \arcsin \left (b x^{n} + a\right ) + \sqrt{-{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.54383, size = 132, normalized size = 2.81 \begin{align*} \frac{b x^{n} \arcsin \left (b x^{n} + a\right ) + a \arcsin \left (b x^{n} + a\right ) + \sqrt{-b^{2} x^{2 \, n} - 2 \, a b x^{n} - a^{2} + 1}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15034, size = 53, normalized size = 1.13 \begin{align*} \frac{{\left (b x^{n} + a\right )} \arcsin \left (b x^{n} + a\right ) + \sqrt{-{\left (b x^{n} + a\right )}^{2} + 1}}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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