Optimal. Leaf size=35 \[ \frac{\sqrt{c x-1} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt{1-c x}} \]
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Rubi [A] time = 0.220579, antiderivative size = 48, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {5713, 5674} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5674
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{-1+c x} \sqrt{1+c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.105507, size = 54, normalized size = 1.54 \[ \frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt{-(c x-1) (c x+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 55, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }{c \left ({c}^{2}{x}^{2}-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c^{2} x^{2} + 1}{\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 [B] time = 1.90049, size = 136, normalized size = 3.89 \begin{align*} -\frac{\sqrt{c^{2} x^{2} - 1} \sqrt{-c^{2} x^{2} + 1} \log \left (\frac{b \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + a}{b}\right )}{b c^{3} x^{2} - b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c^{2} x^{2} + 1}{\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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