Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]
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Rubi [A] time = 0.249598, 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 = {} \[ \int \frac{1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.142004, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b{\rm arccosh} \left (cx\right )} \left ( -{c}^{2}{x}^{2}+1 \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*} \int \frac{1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\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}}{a c^{4} x^{4} - 2 \, a c^{2} x^{2} +{\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \operatorname{arcosh}\left (c x\right ) + a}, 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 (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}\, 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 (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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