Optimal. Leaf size=42 \[ \text{Unintegrable}\left (\frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2},x\right ) \]
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Rubi [A] time = 0.0417539, 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 = {} \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]
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 ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx &=\int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.744021, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.756, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b{\it Artanh} \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{-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{4 \, c x}{\sqrt{c x + 1} \sqrt{-c x + 1} b^{2} c \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) - \sqrt{c x + 1} \sqrt{-c x + 1} b^{2} c \log \left (\sqrt{c x + 1} - \sqrt{-c x + 1}\right ) + 2 \, \sqrt{c x + 1} \sqrt{-c x + 1} a b c} - \int -\frac{4}{{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) -{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \log \left (\sqrt{c x + 1} - \sqrt{-c x + 1}\right ) + 2 \,{\left (a b c^{2} x^{2} - a b\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}\,{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{1}{a^{2} c^{2} x^{2} +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} - a^{2} + 2 \,{\left (a b c^{2} x^{2} - a b\right )} \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )}, 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}{a^{2} c^{2} x^{2} - a^{2} + 2 a b c^{2} x^{2} \operatorname{atanh}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} - 2 a b \operatorname{atanh}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} + b^{2} c^{2} x^{2} \operatorname{atanh}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )} - b^{2} \operatorname{atanh}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \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 )}{\left (b \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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