Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.0285917, 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 = {} \[ \int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx &=\int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 13.9247, size = 0, normalized size = 0. \[ \int \frac{1}{(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.233, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arccos \left ( cx \right ) \right ) ^{2}} \left ( dx \right ) ^{-{\frac{3}{2}}}}\, 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{\frac{1}{2} \,{\left (b^{2} c d^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a b c d^{2} x^{2}\right )} \sqrt{d} \int \frac{{\left (c^{2} x^{2} - 3\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \sqrt{x}}{a b c^{3} d^{2} x^{5} - a b c d^{2} x^{3} +{\left (b^{2} c^{3} d^{2} x^{5} - b^{2} c d^{2} x^{3}\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )}\,{d x} + \sqrt{c x + 1} \sqrt{-c x + 1} \sqrt{d} \sqrt{x}}{b^{2} c d^{2} x^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a b c d^{2} x^{2}} \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{d x}}{b^{2} d^{2} x^{2} \arccos \left (c x\right )^{2} + 2 \, a b d^{2} x^{2} \arccos \left (c x\right ) + a^{2} d^{2} x^{2}}, 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{1}{\left (d x\right )^{\frac{3}{2}} \left (a + b \operatorname{acos}{\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{1}{\left (d x\right )^{\frac{3}{2}}{\left (b \arccos \left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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