Optimal. Leaf size=42 \[ \frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0687473, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4643, 4641} \[ \frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 4643
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{\sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0434918, size = 42, normalized size = 1. \[ \frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 52, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{4\,ca \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}} \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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3}}{a^{2} c x^{2} - c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{3}{\left (a x \right )}}{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \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 (a x\right )^{3}}{\sqrt{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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