Optimal. Leaf size=97 \[ \frac{2 \sqrt{1-c x} \text{Unintegrable}\left (\frac{1}{x^3 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c x^2 \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.493488, 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 = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{x^2 \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^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{(1-c x) \sqrt{1+c x} \sqrt{1-c^2 x^2}}{b c x^2 \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x^3 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 9.88206, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.145, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \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^{4} + \sqrt{c x + 1} \sqrt{c x - 1} a b c^{2} x^{3} - a b c x^{2} +{\left (b^{2} c^{3} x^{4} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x^{3} - b^{2} c x^{2}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{{\left (3 \,{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} c x + 2 \,{\left (2 \, 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^{5} x^{7} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{5} - 2 \, a b c^{3} x^{5} + a b c x^{3} + 2 \,{\left (a b c^{4} x^{6} - a b c^{2} x^{4}\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{7} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{5} - 2 \, b^{2} c^{3} x^{5} + b^{2} c x^{3} + 2 \,{\left (b^{2} c^{4} x^{6} - b^{2} c^{2} x^{4}\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^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arcosh}\left (c x\right ) + a^{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{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{2} \left (a + b \operatorname{acosh}{\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 \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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