Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.120872, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 3.44463, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 5.385, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c^{2} x^{2} + 2 \,{\left (b^{2} c x^{4} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{4}\right )} \int \frac{c^{2} x^{2} - 2}{b^{2} c x^{5} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{5}}\,{d x} - 1}{b^{2} c x^{4} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} x^{2} + 1}}{b^{2} x^{4} \arcsin \left (c x\right )^{2} + 2 \, a b x^{4} \arcsin \left (c x\right ) + a^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{4} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} x^{2} + 1}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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