Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]
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Rubi [A] time = 0.456095, 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 = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
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 \cosh ^{-1}(c x)\right )^2} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x}}{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [F] time = 180.008, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [A] time = 0.431, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( a+b{\rm arccosh} \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{{\left ({\left (c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{3} x^{3} - c x\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{3} x^{6} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x^{5} - a b c x^{4} +{\left (b^{2} c^{3} x^{6} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x^{5} - b^{2} c x^{4}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} - \int \frac{{\left ({\left (2 \, c^{3} x^{3} - 5 \, c x\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} + 2 \,{\left (2 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 2\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (2 \, c^{5} x^{5} - 5 \, c^{3} x^{3} + 3 \, c x\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{5} x^{9} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{7} - 2 \, a b c^{3} x^{7} + a b c x^{5} + 2 \,{\left (a b c^{4} x^{8} - a b c^{2} x^{6}\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{9} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{7} - 2 \, b^{2} c^{3} x^{7} + b^{2} c x^{5} + 2 \,{\left (b^{2} c^{4} x^{8} - b^{2} c^{2} x^{6}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}\,{d x} \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} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname{arcosh}\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 [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 = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} x^{2} + 1}}{{\left (b \operatorname{arcosh}\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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